Apresentação PowerPoint - Sistemas de Amortecimento de Oscilações em Edifícios: Análise e Escolha

Nesta atividade você deverá atuar como um membro de uma empresa que projeta sistemas de amortecimento de oscilações e será também oa responsável pela escolha de qual sistema será implementado Para isso você deverá elaborar um relatório detalhado de modo a explicitar e embasar a sua tomada de decisão Dado o cenário proposto você deverá por meio de uma apresentação em PowerPoint com pelo menos seis slides expor duas propostas para a implementação de um sistema de redução das oscilações em um edifício arranhacéu focando nas seguintes características nível de redução das oscilações sensação de conforto ao usuário confiabilidade e custo de implementação Para isso os slides devem conter itens obrigatórios tais como Primeiro slide capa com seus dados e os dados do trabalho Dado o cenário proposto você deverá por meio de uma apresentação em PowerPoint com pelo menos seis slides expor duas propostas para a implementação de um sistema de redução das oscilações em um edifício arranhacéu focando nas seguintes características nível de redução das oscilações sensação de conforto ao usuário confiabilidade e custo de implementação Para isso os slides devem conter itens obrigatórios tais como Segundo e terceiro slides descrição do primeiro sistema escolhido evidenciando suas características nível de redução das oscilações sensação de conforto ao usuário confiabilidade e custo de implementação Dado o cenário proposto você deverá por meio de uma apresentação em PowerPoint com pelo menos seis slides expor duas propostas para a implementação de um sistema de redução das oscilações em um edifício arranhacéu focando nas seguintes características nível de redução das oscilações sensação de conforto ao usuário confiabilidade e custo de implementação Para isso os slides devem conter itens obrigatórios tais como Quarto e quinto slides descrição do segundo sistema escolhido evidenciando suas características nível de redução das oscilações sensação de conforto ao usuário confiabilidade e custo de implementação VOLTAR RETORNAR MISSÃO CASE FINAL Muito bem Você chegou no último nível desta Atividade Leia atentamente o Case Final e faça o upload do seu arquivo na próxima etapa Dado o cenário proposto você deverá por meio de uma apresentação em PowerPoint com pelo menos seis slides expor duas propostas para a implementação de um sistema de redução das oscilações em um edifício arranhacéu focando nas seguintes características nível de redução das oscilações sensação de conforto ao usuário confiabilidade e custo de implementação Para isso os slides devem conter itens obrigatórios tais como Sexto slide conclusão sobre qual é o sistema de redução de oscilações mais adequado de acordo com a aplicação VOLTAR RETORNAR MISSÃO CASE FINAL Muito bem Você chegou no último nível desta Atividade Leia atentamente o Case Final e faça o upload do seu arquivo na próxima etapa Redução de oscilações em arranhacéus Nome completo do aluno 1 Nome completo do aluno 2 Prof Dr nome do professor Vibrações Mecânicas Amortecedor de massa sintonizado por pêndulo Esquemas Equações Os amortecimento por massa sintonizada AMS é um método prático e eficiente para atenuar a resposta à entrada de aceleração do solo Esta tecnologia é uma alternativa para o projeto estrutural de edifícios em regiões sísmicas O AMS é um sistema mecânico que consiste em uma massa uma mola que fornece rigidez e um amortecedor viscoso A massa é presa à estrutura através da mola e amortecedor Quando o elemento que confere rigidez ao AMS é um pêndulo o dispositivo é denominado Amortecedor de massa sintonizado por pêndulo AMSP O AMSP absorve grande parte da energia produzida por forças externas minimiza a amplitude de vibração e reduz a probabilidade de danos aos elementos estruturais O AMSP possui design muito simples e responde rapidamente ao movimento da estrutura O comprimento do pêndulo controla sua frequência natural e seu design pode integrar os amortecedores viscosos sem dificuldade Uma vantagem notável do AMSP é que o pêndulo pode oscilar em todas as direções proporcionando dissipação de energia para cargas aplicadas em diferentes direções No entanto o desempenho do AMSP na redução da vibração depende da massa comprimento e amortecimento do pêndulo entre outros Crystal Tower Nagase and Hisatoku 1990 A torre localizada em Osaka Japão tem 157 m de altura e 1860 m² em planta pesa 44000 toneladas e tem um período fundamental de aproximadamente 4s na direção nortesul e 3s na direção lesteoeste Um amortecedor de massa pendular sintonizado foi incluído na fase inicial do projeto para diminuir o movimento do edifício induzido pelo vento em cerca de 50 Seis dos nove tanques de armazenamento térmico de gelo para resfriamento e aquecimento de ar cada um pesando 90 toneladas são pendurados nas vigas superiores do telhado e usados como uma massa pendular Quatro tanques têm um comprimento de pêndulo de 4 m e deslizam na direção nortesul os outros dois tanques têm um comprimento de pêndulo de cerca de 3 m e deslizam na direção lesteoeste Amortecedores de óleo conectados aos pêndulos dissipam a energia do pêndulo O custo desse sistema de amortecedor de massa ajustado foi de cerca de US 350000 menos de 02 do custo de construção Amortecedor de massa sintonizado por pêndulo Amortecedor de massa sintonizado por translação Esquema amortecedor passivo A massa repousa sobre rolamentos que funcionam como roletes e permitem que a massa transfira lateralmente em relação ao piso Molas e amortecedores são inseridos entre a massa e os suportes verticais adjacentes que transmitem a força lateral fora de fase para o nível do piso e depois para o pórtico estrutural Os amortecedores translacionais bidirecionais são configurados com molasamortecedores em duas direções ortogonais e fornecem a capacidade de controlar o movimento estrutural em dois planos ortogonais Vários mecanismos passivos e ativos de dissipação de energia têm sido propostos e testados como meios alternativos para vibração Esquema amortecedor ativo Citicorp Center John Hancock Tower Amortecedor de massa sintonizado por translação Dois amortecedores foram adicionados à John Hancock Tower de 60 andares em Boston para reduzir a resposta ao carregamento de rajadas de vento Os amortecedores são colocados em extremidades opostas do 58º andar a 67 m de distância e se movem para neutralizar a oscilação e a torção devido à forma do edifício Cada amortecedor pesa 2700 kN e consiste em um caixa de aço com cerca de 52 m³ que se apoia em uma placa de aço de 9 m de comprimento O peso cheio de chumbo contido lateralmente por molas rígidas ancoradas nas colunas internas do edifício e controladas por cilindros servohidráulicos desliza para frente e para trás em um mancal hidrostático composto por uma fina camada de óleo forçada através de orifícios na placa de aço Sempre que a aceleração horizontal ultrapassar 0003g por dois ciclos consecutivos o sistema é ativado automaticamente Este sistema foi projetado e fabricado pela LeMessurier AssociatesSCI em associação com a MTS System Corp com um custo de cerca de 3 milhões de dólares e reduz a oscilação do edifício em 40 a 50 O Citicorp Manhattan AMS também foi projetado e fabricado pela LeMessurier AssociatesSCI em associação com a MTS System Corp Este edifício tem 279 m de altura e tem um período fundamental de cerca de 65 s com uma taxa de amortecimento inerente de 1 ao longo de cada eixo O amortecedor localizado no 63º andar da coroa da estrutura tem uma massa de 366 Mg cerca de 2 da massa modal efetiva do primeiro modo e era 250 vezes maior do que qualquer amortecedor de massa sintonizado existente no momento da instalação A LeMessurier estima que o AMS do Citicorp que custou cerca de 15 milhão de dólares economizou de 35 a 4 milhões de dólares Esta soma representa o custo de cerca de 2800 toneladas de aço estrutural que seriam necessárias para satisfazer as restrições de deflexão Conclusão Os amortecedores sintonizados por massa pendular são mais baratos possuem um design mais fácil de implementar e podem ser empregados para reduzir oscilação em múltiplas direções Possui um desempenho bastante sensível às variáveis dinâmicas do pêndulo do solo e da estrutura Os amortecedores de massa sintonizados por translação são mais robustos possuem um design um pouco mais complexo podem ser implementados em conjunto de um atuador para tornarse um sistema ativo a taxa de amortecimento pode ser mais facilmente alterada a partir da viscosidade do fluido de amortecimento em geral um óleo C H A P T E R 4 217 Tuned Mass Damper Systems 41 INTRODUCTION A tuned mass damper TMD is a device consisting of a mass a spring and a damper that is attached to a structure in order to reduce the dynamic response of the structure The frequency of the damper is tuned to a particular structural frequency so that when that frequency is excited the damper will resonate out of phase with the structural motion Energy is dissipated by the damper inertia force acting on the structure The TMD concept was first applied by Frahm in 1909 Frahm 1909 to reduce the rolling motion of ships as well as ship hull vibrations A theory for the TMD was presented later in the paper by Ormondroyd and Den Hartog 1928 followed by a detailed discussion of optimal tuning and damping parameters in Den Hartogs book on mechanical vibrations 1940 The initial theory was applicable for an undamped SDOF system subjected to a sinusoidal force excitation Extension of the theory to damped SDOF systems has been investigated by numerous researchers Significant contributions were made by Randall et al 1981 Warburton 1981 1982 Warburton and Ayorinde 1980 and Tsai and Lin 1993 This chapter starts with an introductory example of a TMD design and a brief description of some of the implementations of tuned mass dampers in building structures A rigorous theory of tuned mass dampers for SDOF systems subjected to harmonic force excitation and harmonic ground motion is discussed next Vari ous cases including an undamped TMD attached to an undamped SDOF system a damped TMD attached to an undamped SDOF system and a damped TMD attached to a damped SDOF system are considered Time history responses for a ConCh04v2fm Page 217 Thursday July 11 2002 433 PM 218 Chapter 4 Tuned Mass Damper Systems range of SDOF systems connected to optimally tuned TMD and subjected to harmonic and seismic excitations are presented The theory is then extended to MDOF systems where the TMD is used to dampen out the vibrations of a specific mode An assessment of the optimal placement locations of TMDs in building structures is included Numerous examples are provided to illustrate the level of control that can be achieved with such passive devices for both harmonic and seismic excitations 42 AN INTRODUCTORY EXAMPLE In this section the concept of the tuned mass damper is illustrated using the twomass system shown in Figure 41 Here the subscript d refers to the tuned mass damper the structure is idealized as a single degree of freedom system Introducing the following notation and defining m as the mass ratio the governing equations of motion are given by Primary mass 1 mü 2ξωi ω² u pmŭu d FIGURE 41 SDOFTMD system Section 42 An Introductory Example 219 Tuned mass 47 The purpose of adding the mass damper is to limit the motion of the structure when it is subjected to a particular excitation The design of the mass damper involves specifying the mass stiffness and damping coefficient The optimal choice of these quantities is discussed in Section 44 In this example the nearoptimal approximation for the frequency of the damper 48 is used to illustrate the design procedure The stiffnesses for this frequency combi nation are related by 49 Equation 48 corresponds to tuning the damper to the fundamental period of the structure Considering a periodic excitation 410 the response is given by 411 412 where and denote the displacement amplitude and phase shift respectively The critical loading scenario is the resonant condition The solution for this case has the following form 413 414 415 416 u d 2ξdωdu d ωd 2ud u md kd cd ωd ω kd mk p pˆ sinΩt u uˆ Ωt δ1 sin ud uˆ d Ωt δ1 δ2 sin uˆ δ Ω ω uˆ pˆ km 1 1 2ξ m 1 2ξd 2 uˆ d 1 2ξd uˆ tanδ1 2ξ m 1 2ξd tanδ2 π 2 ConCh04v2fm Page 219 Thursday July 11 2002 433 PM 220 Chapter 4 Tuned Mass Damper Systems Note that the response of the tuned mass is 90º out of phase with the response of the primary mass This difference in phase produces the energy dissipation contributed by the damper inertia force The response for no damper is given by 417 418 To compare these two cases we express Eq 413 in terms of an equivalent damping ratio 419 where 420 Equation 420 shows the relative contribution of the damper parameters to the total damping Increasing the mass ratio magnifies the damping However since the added mass also increases there is a practical limit on Decreasing the damping coefficient for the damper also increases the damping Noting Eq 414 the rela tive displacement also increases in this case and just as for the mass there is a prac tical limit on the relative motion of the damper Selecting the final design requires a compromise between these two constraints Example 41 Preliminary design of a TMD for a SDOF system Suppose and we want to add a tuned mass damper such that the equivalent damping ratio is Using Eq 420 and setting the following relation between and is obtained 1 The relative displacement constraint is given by Eq 414 2 uˆ pˆ k 1 2ξ δ1 π 2 uˆ pˆ k 1 2ξe ξe m 2 1 2ξ m 1 2ξd 2 m ξ 0 10 ξe 01 m ξd m 2 1 2ξ m 1 2ξd 2 01 uˆ d 1 2ξd uˆ ConCh04v2fm Page 220 Thursday July 11 2002 433 PM Section 42 An Introductory Example 221 Combining Eq 1 and Eq 2 and setting leads to 3 Usually is taken to be an order of magnitude greater than In this case Eq 3 can be approximated as 4 The generalized form of Eq 4 follows from Eq 420 5 Finally taking yields an estimate for 6 This magnitude is typical for The other parameters are 7 and from Eq 49 8 It is important to note that with the addition of only of the primary mass we obtain an effective damping ratio of The negative aspect is the large rela tive motion of the damper mass in this case times the displacement of the pri mary mass How to accommodate this motion in an actual structure is an important design consideration A description of some applications of tuned mass dampers to building struc tures is presented in the following section to provide additional background on this type of device prior to entering into a detailed discussion of the underlying theory ξ 0 m 2 1 uˆ d uˆ 2 01 uˆ d uˆ m 2 uˆ d uˆ 01 m 2ξe 1 uˆ d uˆ uˆ d 10uˆ m m 2 01 10 002 m ξd 1 2 uˆ uˆ d 005 kd mk 002k 2 10 10 ConCh04v2fm Page 221 Thursday July 11 2002 433 PM 222 Chapter 4 Tuned Mass Damper Systems 43 EXAMPLES OF EXISTING TUNED MASS DAMPER SYSTEMS Although the majority of applications have been for mechanical systems tuned mass dampers have been used to improve the response of building structures under wind excitation A short description of the various types of dampers and several building structures that contain tuned mass dampers follows 431 Translational Tuned Mass Dampers Figure 42 illustrates the typical configuration of a unidirectional translational tuned mass damper The mass rests on bearings that function as rollers and allow the mass to translate laterally relative to the floor Springs and dampers are inserted between the mass and the adjacent vertical support members which transmit the lateral outofphase force to the floor level and then into the structural frame Bidirectional translational dampers are configured with springsdampers in two orthogonal directions and provide the capability for controlling structural motion in two orthogonal planes Some examples of early versions of this type of damper are described next John Hancock Tower Engineering News Record Oct 1975 Two dampers were added to the 60story John Hancock Tower in Boston to reduce the response to wind gust loading The dampers are placed at opposite ends of the fiftyeighth story 67 m apart and move to counteract sway as well as twisting due to the shape of the building Each damper weighs 2700 kN and consists of a leadfilled steel box about 52 m square and 1 m deep that rides on a 9mlong steel plate The leadfilled weight laterally restrained by stiff springs anchored to the interior col umns of the building and controlled by servohydraulic cylinders slides back and forth on a hydrostatic bearing consisting of a thin layer of oil forced through holes in the steel plate Whenever the horizontal acceleration exceeds 0003g for two con secutive cycles the system is automatically activated This system was designed and manufactured by LeMessurier AssociatesSCI in association with MTS System Corp at a cost of around 3 million dollars and is expected to reduce the sway of the building by 40 to 50 FIGURE 42 Schematic diagram of a translational tuned mass damper md Direction of motion Support Floor beam ConCh04v2fm Page 222 Thursday July 11 2002 433 PM Section 43 Examples of Existing Tuned Mass Damper Systems 223 Citicorp Center Engineering News Record Aug 1975 McNamara 1977 Petersen 1980 The Citicorp Manhattan TMD was also designed and manufactured by LeMes surier AssociatesSCI in association with MTS System Corp This building is 279 m high and has a fundamental period of around 65 s with an inherent damping ratio of 1 along each axis The Citicorp TMD located on the sixtythird floor in the crown of the structure has a mass of 366 Mg about 2 of the effective modal mass of the first mode and was 250 times larger than any existing tuned mass damper at the time of installation Designed to be biaxially resonant on the building structure with a variable operating period of adjustable linear damping from 8 to 14 and a peak relative displacement of the damper is expected to reduce the building sway amplitude by about 50 This reduction corresponds to increasing the basic structural damping by 4 The concrete mass block is about 26 m high with a plan cross section of 91 m by 91 m and is supported on a series of twelve 60cmdiameter hydraulic pressurebalanced bearings During operation the bearings are supplied oil from a separate hydraulic pump which is capable of rais ing the mass block about 2 cm to its operating position in about 3 minutes The damper system is activated automatically whenever the horizontal acceleration exceeds 0003g for two consecutive cycles and will automatically shut itself down when the building acceleration does not exceed 000075g in either axis over a 30minute interval LeMessurier estimates Citicorps TMD which cost about 15 million dollars saved 35 to 4 million dollars This sum represents the cost of some 2800 tons of structural steel that would have been required to satisfy the deflection constraints Canadian National Tower Engineering News Record 1976 The 102m steel antenna mast on top of the Canadian National Tower in Toronto 553 m high including the antenna required two lead dampers to prevent the antenna from deflecting excessively when subjected to wind excitation The damper system consists of two doughnutshaped steel rings 35 cm wide 30 cm deep and 24 m and 3 m in diameter located at elevations 488 m and 503 m Each ring holds about 9 metric tons of lead and is supported by three steel beams attached to the sides of the antenna mast Four bearing universal joints that pivot in all directions connect the rings to the beams In addition four separate hydraulically activated fluid dampers mounted on the side of the mast and attached to the center of each universal joint dissipate energy As the leadweighted rings move back and forth the hydraulic damper system dissipates the input energy and reduces the towers response The damper system was designed by Nicolet Carrier Dressel and Asso ciates Ltd in collaboration with Vibron Acoustics Ltd The dampers are tuned to the second and fourth modes of vibration in order to minimize antenna bending loads the first and third modes have the same characteristics as the prestressed con crete structure supporting the antenna and did not require additional damping Chiba Port Tower Kitamura et al 1988 625 s 20 14 m ConCh04v2fm Page 223 Thursday July 11 2002 433 PM 224 Chapter 4 Tuned Mass Damper Systems Chiba Port Tower completed in 1986 was the first tower in Japan to be equipped with a TMD Chiba Port Tower is a steel structure 125 m high weighing 1950 metric tons and having a rhombusshaped plan with a side length of 15 m The first and second mode periods are 225 s and 051 s respectively for the x direction and 27 s and 057 s for the y direction Damping for the fundamental mode is estimated at 05 Damping ratios proportional to frequencies were assumed for the higher modes in the analysis The purpose of the TMD is to increase damping of the first mode for both the x and y directions Figure 43 shows the damper system Manu factured by Mitsubishi Steel Manufacturing Co Ltd the damper has mass ratios with respect to the modal mass of the first mode of about 1120 in the x direction and 180 in the y direction periods in the x and y directions of 224 s and 272 s respectively and a damper damping ratio of 15 The maximum relative displace ment of the damper with respect to the tower is about in each direction Reductions of around 30 to 40 in the displacement of the top floor and 30 in the peak bending moments are expected The early versions of TMDs employ complex mechanisms for the bearing and damping elements have relatively large masses occupy considerable space and are quite expensive Recent versions such as the scheme shown in Figure 44 have been designed to minimize these limitations This scheme employs a multiassem blage of elastomeric rubber bearings which function as shear springs and bitumen rubber compound BRC elements which provide viscoelastic damping capability The device is compact in size requires unsophisticated controls is multidirectional and is easily assembled and modified Figure 45 shows a fullscale damper being subjected to dynamic excitation by a shaking table An actual installation is con tained in Figure 46 FIGURE 43 Tuned mass damper for ChibaPort Tower Courtesy of J Connor 1 m ConCh04v2fm Page 224 Thursday July 11 2002 433 PM Section 43 Examples of Existing Tuned Mass Damper Systems 225 FIGURE 44 Tuned mass damper with spring and damper assemblage FIGURE 45 Deformed positiontuned mass damper Courtesy of J Connor FIGURE 46 Tuned mass damperHuis Ten Bosch Tower Nagasaki Courtesy of J Connor BRC Multistage rubber bearings Air brake Vibration direction of building Weight mass Limit switch ConCh04v2fm Page 225 Thursday July 11 2002 433 PM 226 Chapter 4 Tuned Mass Damper Systems The effectiveness of a tuned mass damper can be increased by attaching an aux iliary mass and an actuator to the tuned mass and driving the auxiliary mass with the actuator such that its response is out of phase with the response of the tuned mass Figure 47 illustrates this scheme The effect of driving the auxiliary mass is to produce an additional force that complements the force generated by the tuned mass and therefore increases the equivalent damping of the TMD we can obtain the same behavior by attaching the actuator directly to the tuned mass thereby eliminating the need for an auxiliary mass Since the actuator requires an external energy source this system is referred to as an active tuned mass damper The scope of this chapter is restricted to passive TMDs Active TMDs are discussed in Chapter 6 432 Pendulum Tuned Mass Damper The problems associated with the bearings can be eliminated by supporting the mass with cables which allow the system to behave as a pendulum Figure 48a shows a simple pendulum attached to a floor Movement of the floor excites the pendulum The relative motion of the pendulum produces a horizontal force that opposes the floor motion This action can be represented by an equivalent SDOF system that is attached to the floor as indicated in Figure 48b The equation of motion for the horizontal direction is 421 where T is the tension in the cable When is small the following approximations apply 422 Introducing these approximations transforms Eq 421 to FIGURE 47 An active tuned mass damper configuration Direction of motion Support Auxiliary mass Actuator Floor beam T sinθ Wd g u u d 0 θ ud L θ Lθ sin T Wd ConCh04v2fm Page 226 Thursday July 11 2002 433 PM Section 43 Examples of Existing Tuned Mass Damper Systems 227 FIGURE 48 A simple pendulum tuned mass damper m d u d W d L u d m d 423 and it follows that the equivalent shear spring stiffness is k eq W d L 424 The natural frequency of the pendulum is related to k eq by ω d 2 k eq m d g L 425 Noting Eq 425 the natural period of the pendulum is T d 2π Lg 426 The simple pendulum tuned mass damper concept has a serious limitation Since the period depends on L the required length for large T d may be greater than the typical story height For instance the length for T d 5 s is 62 meters whereas the story height is between 4 and 5 meters This problem can be eliminated by resorting to the scheme illustrated in Figure 49 The interior rigid link magnifies the support motion for the pendulum and results in the following equilibrium equation 228 Chapter 4 Tuned Mass Damper Systems FIGURE 49 Compound pendulum m d ü ü 1 ü d W d L u d 0 427 The rigid link moves in phase with the damper and has the same displacement amplitude Then taking u 1 u d in Eq 427 results in m d ü d W d 2L u d m d 2 ü 428 The equivalent stiffness is W d 2L and it follows that the effective length is equal to 2L Each additional link increases the effective length by L An example of a pendulumtype damper is described next Crystal Tower Nagase and Hisatoku 1990 The tower located in Osaka Japan is 157 m high and 28 m by 67 m in plan weighs 44000 metric tons and has a fundamental period of approximately 4 s in the northsouth direction and 3 s in the eastwest direction A tuned pendulum mass damper was included in the early phase of the design to decrease the windinduced motion of the building by about 50 Six of the nine air cooling and heating ice thermal storage tanks each weighing 90 tons are hung from the top roof girders and used as a pendulum mass Four tanks have a pendulum length of 4 m and slide in the northsouth direction the other two tanks have a pendulum length of about 3 m and slide in the eastwest direction Oil dampers connected to the pendulums dissipate the pendulum energy Figure 410 shows the layout of the ice storage tanks that were used as damper masses Views of the actual building and one of the tanks are presented in Figure 411 on page 230 The cost of this tuned mass damper system was around 350000 less than 02 of the construction cost Section 44 Tuned Mass Damper Theory for SDOF Systems 229 A modified version of the pendulum damper is shown in Figure 412 on page 231 The restoring force provided by the cables is generated by introducing curva ture in the support surface and allowing the mass to roll on this surface The vertical motion of the weight requires an energy input Assuming θ is small the equations for the case where the surface is circular are the same as for the conventional pen dulum with the cable length L replaced with the surface radius R 44 TUNED MASS DAMPER THEORY FOR SDOF SYSTEMS In what follows various cases ranging from fully undamped to fully damped condi tions are analyzed and design procedures are presented 441 Undamped Structure Undamped TMD Figure 413 shows a SDOF system having mass and stiffness subjected to both external forcing and ground motion A tuned mass damper with mass and stiff ness is attached to the primary mass The various displacement measures are the absolute ground motion the relative motion between the primary mass and the ground and the relative displacement between the damper and the primary mass With this notation the governing equations take the form FIGURE 410 Pendulum damper layoutCrystal Tower Takemaka Corporation 2 for structural control in G direction 4 for structural control in B direction Displacement of regenerative tanks used in structural control roof plan 4 m x x 672 m Support frame Ice storage tank 90 ton Ice storage tank Suspension material 90 m xxx 276 m 105 m Coil spring Coil spring Coil spring Stopper Elevation of regenerative tank TMD fillscale experiment system Guide roller Oil damper Oil damper m k md kd ug u ud ConCh04v2fm Page 229 Thursday July 11 2002 433 PM 230 Chapter 4 Tuned Mass Damper Systems 429 430 where is the absolute ground acceleration and is the force loading applied to the primary mass FIGURE 411 Ice storage tankCrystal Tower Courtesy of Takemaka Corporation md u d u kdud mdag mu ku kdud mag p ag p ConCh04v2fm Page 230 Thursday July 11 2002 433 PM Section 44 Tuned Mass Damper Theory for SDOF Systems 231 FIGURE 412 Rocker pendulum FIGURE 413 SDOF system coupled with a TMD The excitation is considered to be periodic of frequency Ω a g â g sin Ωt 431 232 Chapter 4 Tuned Mass Damper Systems 432 Expressing the response as 433 434 and substituting for these variables the equilibrium equations are transformed to 435 436 The solutions for and are given by 437 438 where 439 and the terms are dimensionless frequency ratios 440 441 Selecting the mass ratio and damper frequency ratio such that 442 reduces the solution to p pˆ sin Ωt u uˆ sin Ωt ud uˆ d sin Ωt mdΩ2 kd uˆ d mdΩ2uˆ mdaˆ g kduˆ d mΩ2 k uˆ maˆ g pˆ uˆ uˆ d uˆ pˆ k 1 ρd 2 D1 maˆ g k 1 m ρd 2 D1 uˆ d pˆ kd mρ2 D1 maˆ g kd m D1 D1 1 ρ2 1 ρd 2 mρ2 ρ ρ Ω ω Ω k m ρd Ω ωd Ω kd md 1 ρd 2 m 0 ConCh04v2fm Page 232 Thursday July 11 2002 433 PM Section 44 Tuned Mass Damper Theory for SDOF Systems 233 443 444 This choice isolates the primary mass from ground motion and reduces the response due to external force to the pseudostatic value A typical range for is to Then the optimal damper frequency is very close to the forcing frequency The exact relationship follows from Eq 442 445 We determine the corresponding damper stiffness with 446 Finally substituting for Eq 444 takes the following form 447 We specify the amount of relative displacement for the damper and determine with Eq 447 Given and the stiffness is found using Eq 446 It should be noted that this stiffness applies for a particular forcing frequency Once the mass damper properties are defined Eqs 437 and 438 can be used to determine the response for a different forcing frequency The primary mass will move under ground motion excitation in this case 442 Undamped Structure Damped TMD The next level of complexity has damping included in the mass damper as shown in Figure 414 The equations of motion for this case are 448 449 uˆ pˆ k uˆ d pˆ kd ρ2 maˆ g kd pˆ k m 001 01 ωd opt Ω 1 m kd opt ωd opt 2md Ω2mm 1 m kd uˆ d 1 m m pˆ k aˆ g Ω2 m m Ω mdu d cdu d kdud mdu md ag mu ku cdu d kdud mag p ConCh04v2fm Page 233 Thursday July 11 2002 433 PM The inclusion of the damping terms in Eqs 448 and 449 produces a phase shift between the periodic excitation and the response It is convenient to work initially with the solution expressed in terms of complex quantities We express the excitation as ag âg eiΩt 450 p p eiΩt 451 where âg and p are real quantities The response is taken as u ū eiΩt 452 ud ūd eiΩt 453 where the response amplitudes ū and ūd are considered to be complex quantities The real and imaginary parts of ag correspond to cosine and sinusoidal input Then the corresponding solution is given by either the real for cosine or imaginary for sine parts of u and ud Substituting Eqs 452 and 453 in the set of governing equations and cancelling eiΩt from both sides results in md Ω2 i cd Ω kd ūd md Ω2 ū md âg 454 i cd Ω kd ūd m Ω2 k ū m âg p 455 The solution of the governing equations is ū p k D2 f2 ρ2 i 2 ξa ρ f âg m k D2 1 m f2 ρ2 i 2 ξd ρ f 1 m 456 FIGURE 414 Undamped SDOF system coupled with a damped TMD system Section 44 Tuned Mass Damper Theory for SDOF Systems 235 457 where 458 459 and was defined earlier as the ratio of to see Eq 440 Converting the complex solutions to polar form leads to the following expressions 460 461 where the factors define the amplification of the pseudostatic responses and the s are the phase angles between the response and the excitation The various H and δ terms are as follows 462 463 464 465 466 Also 467 ud pˆ ρ2 kD2 aˆ gm kD2 D2 1 ρ2 f 2 ρ2 mρ2f 2 i2ξdρf 1 ρ2 1 m f ωd ω ρ Ω ω u pˆ kH1e iδ1 aˆ gm k H2e iδ2 ud pˆ kH3e i δ3 aˆ gm k H4e i δ3 H δ H1 f 2 ρ2 2 2ξdρf 2 D2 H2 1 m f 2 ρ2 2 2ξdρf 1 m 2 D2 H3 ρ2 D2 H4 1 D2 D2 1 ρ2 f 2 ρ2 mρ2f 2 2 2ξdρf 1 ρ2 1 m 2 δ1 α1 δ3 ConCh04v2fm Page 235 Thursday July 11 2002 433 PM 236 Chapter 4 Tuned Mass Damper Systems 468 469 470 471 For most applications the mass ratio is less than about Then the amplification factors for external loading and ground motion are essentially equal A similar conclusion applies for the phase shift In what follows the solution corre sponding to ground motion is examined and the optimal values of the damper prop erties for this loading condition are established An indepth treatment of the external forcing case is contained in Den Hartogs text Den Hartog 1940 Figure 415 shows the variation of with forcing frequency for specific val ues of damper mass and frequency ratio and various values of the damper damping ratio When there are two peaks with infinite amplitude located on each side of As is increased the peaks approach each other and then merge into a single peak located at The behavior of the amplitudes suggests that there is an optimal value of for a given damper configuration and or equivalently and Another key observation is that all the curves pass through two common points and Since these curves correspond to dif ferent values of the location of and must depend only on and Proceeding with this line of reasoning the expression for can be written as 472 where the a terms are functions of and Then for to be independent of the following condition must be satisfied 473 The corresponding values for are 474 δ2 α2 δ3 tanδ3 2ξdρf 1 ρ2 1 m 1 ρ2 f 2 ρ2 mρ2f 2 tanα1 2ξdρf f 2 ρ2 tanα2 2ξdρf 1 m 1 m f 2 ρ2 005 H1 H2 H2 m f ξd ξd 0 ρ 1 ξd ρ 1 ξd md kd m f P Q ξd P Q m f H2 H2 a1 2 ξd 2a2 2 a3 2 ξd 2a4 2 a2 a4 a1 2 a2 2 ξd 2 a3 2 a4 2 ξd 2 m ρ f H2 ξd a1 a2 a3 a4 H2 H2 P Q a2 a4 ConCh04v2fm Page 236 Thursday July 11 2002 433 PM FIGURE 415 Plot of H2 versus ρ Substituting for the a terms in Eq 473 we obtain a quadratic equation for ρ2 ρ4 1 m f2 1 05 m 1 m ρ2 f2 0 475 The two positive roots ρ1 and ρ2 are the frequency ratios corresponding to points P and Q Similarly Eq 474 expands to H2PQ 1 m 1 ρ122 1 m 476 Figure 415 shows different values for H2 at points P and Q For optimal behavior we want to minimize the maximum amplitude As a first step we require the values of H2 for ρ1 and ρ2 to be equal This produces a distribution that is symmetrical about ρ2 1 1 m as illustrated in Figure 416 Then by increasing the damping ratio ξd we can lower the peak amplitudes until the peaks coincide with points P and Q This state represents the optimal performance of the TMD system A further increase in ξa causes the peaks to merge and the amplitude to increase beyond the optimal value FIGURE 416 Plot of H2 versus ρ for fopt Requiring the amplitudes to be equal at P and Q is equivalent to the following condition on the roots 1 ρ12 1 m 1 ρ22 1 m 477 Then substituting for ρ1 and ρ2 using Eq 475 we obtain a relation between the optimal tuning frequency and the mass ratio fopt 1 05 m 1 m 478 ωdopt fopt ω 479 The corresponding roots and optimal amplification factors are Section 44 Tuned Mass Damper Theory for SDOF Systems 239 480 481 The expression for the optimal damping at the optimal tuning frequency is 482 Figures 417 through 420 show the variation of the optimal parameters with the mass ratio The response of the damper is defined by Eq 461 Specializing this equa tion for the optimal conditions leads to the plot of amplification versus mass ratio contained in Figure 421 A comparison of the damper motion with respect to the motion of the primary mass for optimal conditions is shown in Figure 422 FIGURE 417 Optimum tuning frequency ratio ρ1 2 opt 1 05m 1 m H2 opt 1 m 05m ξd opt m 3 05m 8 1 m 1 05m m 0 001 002 003 004 005 006 007 008 009 01 088 09 092 094 096 098 1 fopt m fopt ConCh04v2fm Page 239 Thursday July 11 2002 433 PM FIGURE 418 Input frequency ratios at which the response is independent of damping FIGURE 419 Optimal damping ratio for TMD FIGURE 420 Maximum dynamic amplification factor for SDOF system optimal tuning and damping FIGURE 421 Maximum dynamic amplification factor for TMD FIGURE 422 Ratio of maximum TMD amplitude to maximum system amplitude FIGURE 423 Response curves for amplitude of system with optimally tuned TMD FIGURE 424 Response curves for amplitude of optimally tuned TMD The maximum amplification for a damped SDOF system without a TMD undergoing harmonic excitation is given by Eq 132 H 1 2ξ1ξ² 483 Since ξ is small a reasonable approximation is H 1 2ξ 484 Expressing the optimal H₂ in a similar form provides a measure of the equivalent damping ratio ξₑ for the primary mass ξₑ 1 2H₂ₒₚₜ 485 Figure 425 shows the variation of ξₑ with the mass ratio A mass ratio of 002 is equivalent to about 5 damping in the primary system FIGURE 425 Equivalent damping ratio for optimally tuned TMD The design of a TMD involves the following steps Establish the allowable values of displacement of the primary mass and the TMD for the design loading This data provides the design values for H₂ₒₚₜ and H₄ₒₚₜ Determine the mass ratios required to satisfy these motion constraints from Figure 420 and Figure 421 Select the largest value of m Determine fₒₚₜ form Figure 417 Compute ωd ωd fₒₚₜω 486 Compute kd kd mdωd² mk fₒₚₜ² 487 Determine ξdₒₚₜ from Figure 419 Compute cd cd 2ξdₒₚₜ ωd md m fₒₚₜ 2ξdₒₚₜ ωm 488 Example 42 Design of a TMD for an undamped SDOF system Consider the following motion constraints H₂ₒₚₜ 7 1 H₄ H₂ₒₚₜ 6 2 Constraint Eq 1 requires m 005 For constraint Eq 2 we need to take m 002 Therefore m 005 controls the design The relevant parameters are m 005 fₒₚₜ 094 ξdₒₚₜ 0135 Then md 005m ωd 094ω kd m fₒₚₜ² k 0044k 443 Damped Structure Damped TMD All real systems contain some damping Although an absorber is likely to be added only to a lightly damped system assessing the effect of damping in the real system on the optimal tuning of the absorber is an important design consideration The main system in Figure 426 consists of the mass m spring stiffness k and viscous damping c The TMD system has mass md stiffness kd and viscous damping cd Considering the system to be subjected to both external forcing and ground excitation the equations of motion are md ud cd ud kd ud md u md ag 489 mu cu ku cd ud kd ud m ag p 490 FIGURE 426 Damped SDOF system coupled with a damped TMD system 246 Chapter 4 Tuned Mass Damper Systems Proceeding in the same way as for the undamped case the solution due to periodic excitation both p and ug is expressed in polar form 491 492 The various H and δ terms are defined as follows 493 494 495 496 497 498 499 4100 4101 4102 The and terms are defined by Eqs 470 and 471 In what follows the case of an external force applied to the primary mass is considered Since involves ξ we cannot establish analytical expressions for the optimal tuning frequency and optimal damping ratio in terms of the mass ratio In this case these parameters also depend on Numerical simulations can be applied u pˆ kH5 eiδ5 aˆgm k H6 eiδ6 ud pˆ kH7 e i δ7 aˆgm k H8 eiδ8 H5 f 2 ρ2 2 2ξdρf 2 D3 H6 1 m f 2 ρ2 2 2ξdρf 1 m 2 D3 H7 ρ2 D3 H8 1 2ξρ 2 D3 D3 f 2ρ2m 1 ρ2 f 2 ρ2 4ξξdfρ2 2 4 ξρ f 2 ρ2 ξdfρ 1 ρ2 1 m 2 δ5 α1 δ7 δ6 α2 δ7 δ8 α3 δ7 tanδ 7 2 ξρ f 2 ρ2 ξd fρ 1 ρ2 1 m f 2ρ2m 1 ρ2 f 2 ρ2 4ξξd f ρ2 tanα3 2ξρ α1 α2 D3 ξ ConCh04v2fm Page 246 Thursday July 11 2002 433 PM Section 44 Tuned Mass Damper Theory for SDOF Systems 247 to evaluate and for a range of given the values for and Start ing with specific values for and plots of versus can be generated for a range of and Each plot has a peak value of The particular combi nation of and that corresponds to the lowest peak value of is taken as the optimal state Repeating this process for different values of and produces the behavioral data needed to design the damper system Figure 427 shows the variation of the maximum value of for the optimal state The corresponding response of the damper is plotted in Figure 428 Adding damping to the primary mass has an appreciable effect for small Noting Eqs 491 and 492 the ratio of damper displacement to primary mass displace ment is given by 4103 Since is small this ratio is essentially independent of Figure 429 confirms this statement The optimal values of the frequency and damping ratios generated through simulation are plotted in Figures 430 and 431 Lastly using Eq 485 can be converted to an equivalent damping ratio for the primary system 4104 Figure 432 shows the variation of with and Tsai and Lin 1993 suggest equations for the optimal tuning parameters and determined by curve fitting schemes The equations are listed next for completeness 4105 4106 H5 H7 ρ m ξ f ξd m ξ H5 ρ f ξd H5 ρ H5 f ξd H5 m ξ H5 m uˆ d uˆ H7 H5 ρ2 f 2 ρ2 2 2ξdρf 2 ξ ξ H5 opt ξe 1 2H5 opt ξe m ξ f ξd f 1 05m 1 m 1 2ξ2 1 2375 1034 m 0426m ξ m 3730 16903 m 20496m ξ2 m ξd 3m 8 1 m 1 05m 0151ξ 0170ξ2 0163ξ 4980ξ2 m ConCh04v2fm Page 247 Thursday July 11 2002 433 PM FIGURE 427 Maximum dynamic amplification factor for damped SDOF system FIGURE 428 Maximum dynamic amplification factor for TMD FIGURE 429 Ratio of maximum TMD amplitude to maximum system amplitude FIGURE 430 Optimum tuning frequency ratio for TMD fopt FIGURE 431 Optimal damping ratio for TMD FIGURE 432 Equivalent damping ratio for optimally tuned TMD Section 45 Case Studies SDOF Systems 251 Example 43 Design of a TMD for a damped SDOF system Example 42 is reworked here allowing for damping in the primary system The same design motion constraints are considered 1 2 Using Figure 427 the required mass ratio for is The other opti mal values are and Then In this case there is a significant reduction in the damper mass required for this set of motion constraints The choice between including damping in the primary system versus incorporating a tuned mass damper depends on the relative costs and reli ability of the two alternatives and the nature of the structural problem A TMD system is generally more appropriate for upgrading an existing structure where access to the structural elements is difficult 45 CASE STUDIES SDOF SYSTEMS Figures 433 to 444 show the time history responses for two SDOF systems with periods of 049 s and 535 s respectively under harmonic at resonance conditions El Centro and Taft ground excitations All examples have a system damping ratio of 2 and an optimally tuned TMD with a mass ratio of 1 The excitation magni tudes have been scaled so that the peak amplitude of the response of the system without the TMD is unity The plots show the response of the system without the TMD the dotted line as well as the response of the system with the TMD the solid line Figures showing the time history of the relative displacement of the TMD with respect to the system are also presented Significant reduction in the response of the primary system under harmonic excitation is observed However optimally tuned mass dampers are relatively ineffective under seismic excitation and in some cases produce a negative effect ie they amplify the response slightly This poor performance is attributed to the ineffectiveness of tuned mass dampers for impulsive loadings as well as their inability to reach a resonant condi tion and therefore dissipate energy under random excitation These results are in close agreement with the data presented by Kaynia et al 1981 2 H5 opt 7 H7 H5 opt 6 ξ 002 m 003 fopt 0965 ξd opt 0105 md 003m ωd 0955ω kd mfopt 2 k 0027k ConCh04v2fm Page 251 Thursday July 11 2002 433 PM FIGURE 433 Response of SDOF to harmonic excitation FIGURE 434 Relative displacement of TMD under harmonic excitation FIGURE 435 Response of SDOF to El Centro excitation FIGURE 436 Relative displacement of TMD under El Centro excitation FIGURE 437 Response of SDOF to Taft excitation FIGURE 438 Relative displacement of TMD under Taft excitation T 535 s ξ 002 m 001 Without TMD With TMD FIGURE 439 Response of SDOF to harmonic excitation FIGURE 440 Relative displacement of TMD under harmonic excitation T 535 s ξ 002 m 001 Without TMD With TMD FIGURE 441 Response of SDOF to El Centro excitation FIGURE 442 Relative displacement of TMD under El Centro excitation T 535 s ξ 002 m 001 Without TMD With TMD FIGURE 443 Response of SDOF to Taft excitation FIGURE 444 Relative displacement of TMD under Taft excitation 46 TUNED MASS DAMPER THEORY FOR MDOF SYSTEMS The theory of a SDOF system presented earlier is extended here to deal with a MDOF system having a number of tuned mass dampers located throughout the structure Numerical simulations which illustrate the application of this theory to the set of example building structures used as the basis for comparison of the different schemes throughout the text are presented in the next section FIGURE 445 2DOF system with TMD A 2DOF system having a damper attached to mass 2 is considered first to introduce the key ideas The governing equations for the system shown in Figure 445 are The key step is to combine Eqs 4107 and 4108 and express the resulting equation in a form similar to the SDOF case defined by Eq 490 This operation reduces the problem to an equivalent SDOF system for which the theory of Section 44 is applicable The approach followed here is based on transforming the original matrix equation to scalar modal equations Introducing matrix notation Eqs 4107 and 4108 are written as where the various matrices are Section 46 Tuned Mass Damper Theory for MDOF Systems 259 4111 4112 4113 4114 We substitute for in terms of the modal vectors and coordinates 4115 The modal vectors satisfy the following orthogonality relations see Eq 2211 4116 Defining modal mass stiffness and damping terms 4117 4118 4119 expressing the elements of as 4120 and assuming damping is proportional to stiffness 4121 U u1 u2 M m1 m2 K k1 k2 k2 k2 k2 C c1 c2 c2 c2 c2 U U Φ1q1 Φ2q2 Φj TKΦi δijωj 2Φj TMΦi m j Φj TMΦj k j Φj TKΦj ωj 2m j cj Φj TCΦj Φj Φj Φj1 Φj2 C αK ConCh04v2fm Page 259 Thursday July 11 2002 433 PM 260 Chapter 4 Tuned Mass Damper Systems we obtain a set of uncoupled equations for and 4122 With this assumption the modal damping ratio is given by 4123 Equation 4122 represents two equations Each equation defines a particular SDOF system having mass stiffness and damping equal to and Since a TMD is effective for a narrow frequency range we have to decide on which modal resonant response is to be controlled with the TMD Once this decision is made the analysis can proceed using the selected modal equation and the initial equation for the TMD ie Eq 4109 Suppose the first modal response is to be controlled Taking in Eq 4122 leads to 4124 In general is obtained by superposing the modal contributions 4125 However when the external forcing frequency is close to the first mode response will dominate and it is reasonable to assume 4126 Solving for 4127 and then substituting in Eq 4124 we obtain q1 q2 m jq j cjq j k jqj Φj1 p1 m1ag j 1 2 Φj2 p2 m2ag kdud cdu d ξj cj 2ωjm j αωj 2 m k ξ j 1 m 1q 1 c1q 1 k 1q1 Φ11p1 Φ12p2 m1Φ11 m2Φ12 ag Φ12 kdud cdu d u2 u2 Φ12q1 Φ22q2 ω1 u2 Φ12q1 q1 q1 1 Φ12 u2 ConCh04v2fm Page 260 Thursday July 11 2002 433 PM Section 46 Tuned Mass Damper Theory for MDOF Systems 261 4128 where and represent the equivalent SDOF parameters for the combination of mode 1 and node 2 the node at which the TMD is attached Their definition equations are 4129 4130 4131 4132 4133 Equations 4109 and 4128 are similar in form to the SDOF equations treated in the previous section Both set of equations are compared next TMD equation 4134 Primary mass equation 4135 Taking 4136 m 1e u 2 c1e u 2 k 1e u2 kdud cdu d p 1e Γ1e m 1e ag m 1e c1e k 1e p 1e Γ1e m 1e 1 Φ12 2 m 1 k 1e 1 Φ12 2 k 1 c1e αk 1e p 1e Φ11p1 Φ12 p2 Φ12 Γ1e Φ12 m 1 m1Φ11 m2Φ22 md u d cd u d kd ud md u ag versus md u d cd u d kd ud md u 2 ag mu cu ku cd u d kd ud p mag versus m 1e u 2 c1e u 2 k 1e u2 cd u d kd ud p 1e Γ1em 1eag u2 u m 1e m c1e c k 1e k p 1e p Γ1e Γ ConCh04v2fm Page 261 Thursday July 11 2002 433 PM 262 Chapter 4 Tuned Mass Damper Systems transforms the primary mass equation for the MDOF case to 4137 which differs from the corresponding SDOF equation by the factor Γ Therefore the solution for ground excitation generated earlier has to be modified to account for the presence of Γ The generalized solution is written in the same form as the SDOF case We need only to modify the terms associated with ie H6 H8 and δ6 δ8 Their expanded form is as follows 4138 4139 4140 4141 4142 4143 where is defined by Eq 497 and is given by Eq 4101 From this point on we proceed as described in Section 44 The mass ratio is defined in terms of the equivalent SDOF mass 4144 Given and we find the tuning frequency and damper damping ratio using Figures 430 and 431 The damper parameters are determined with 4145 4146 4147 mu cu ku cd u d kd ud p Γmag ag H6 Γ m f 2 Γρ2 2 2ξdρf Γ m 2 D3 H8 1 ρ2 Γ 1 2 2ξρ 2 D3 tana2 2ξdfρ Γ m f 2 Γ m Γρ2 tana3 2ξρ 1 Γ 1 ρ2 δ6 a2 δ7 δ8 a3 δ7 D3 δ7 m md m 1e m ξ1 md m m 1e ωd foptω1 cd 2ξd opt ωdmd ConCh04v2fm Page 262 Thursday July 11 2002 433 PM Section 46 Tuned Mass Damper Theory for MDOF Systems 263 Expanding the expression for the damper mass 4148 shows that we should select the TMD location to coincide with the maximum amplitude of the mode shape that is being controlled In this case the first mode is the target mode and is the maximum amplitude for This derivation can be readily generalized to allow for tuning on the th modal frequency We write Eq 4127 as 4149 where is either or The equivalent parameters are 4150 4151 Given and we specify and find the optimal tuning with 4152 Example 44 Design of a TMD for a damped MDOF system To illustrate the foregoing procedure a 2DOF system having is con sidered Designing the system for a fundamental period of and a uniform deformation fundamental mode profile yields the following stiffnesses refer to Example 16 Requiring a fundamental mode damping ratio of and taking damping propor tional to stiffness the corresponding is md m m 1e m Φ1 TMΦ1 Φ12 2 Φ12 Φ1 i qi 1 Φi2 u2 i 1 2 m ie 1 Φi2 2 m i k ie ωi 2m ie m ie ξi m ωd foptωi m1 m2 1 T1 1s k1 12π2 11844 k2 8π2 7896 2 C αK α α 2ξ1 ω1 002 π 00064 ConCh04v2fm Page 263 Thursday July 11 2002 433 PM 264 Chapter 4 Tuned Mass Damper Systems The mass stiffness and damping matrices for these design conditions are Performing an eigenvalue analysis yields the following frequencies and mode shapes The corresponding modal mass stiffness and damping terms are The optimal parameters for a TMD located at node 2 having a mass ratio of and tuned to a specific mode are as follows Mode 1 optimum location is node 2 Mode 2 optimum location is node 1 M 1 0 0 1 K 19739 7896 7896 7896 C 126 051 051 051 ω1 628 rads ω2 1539 rads Φ1 05 10 Φ2 10 05 m 1 Φ1 TMΦ1 125 m 2 Φ2 TMΦ2 125 k 1 Φ1 TKΦ1 4935 k 2 Φ2 TKΦ2 29609 c1 Φ1 TCΦ1 032 c2 Φ2 TCΦ2 190 ξ1 c1 2ω1m 1 002 ξ2 c2 2ω2m 2 0049 001 fopt 0982 ξd opt 0062 md 00125 kd 04754 cd 00096 fopt 0972 ξd opt 0068 md 00125 kd 27974 cd 00254 ConCh04v2fm Page 264 Thursday July 11 2002 433 PM Section 46 Tuned Mass Damper Theory for MDOF Systems 265 This result is for the damper located at node 2 When located at node 1 the mass and stiffness are reduced 75 The general case of a MDOF system with a tuned mass damper connected to the nth degree of freedom is treated in a similar manner Using the notation defined previously the jth modal equation can be expressed as 4153 where denotes the modal force due to ground motion and external forcing and is the element of corresponding to the nth displacement variable To control the ith modal response we set in Eq 4153 and introduce the approximation 4154 This leads to the following equation for 4155 where 4156 4157 4158 4159 The remaining steps are the same as described previously We specify and determine the optimal tuning and damping values with Figures 430 and 431 and then compute and 4160 4161 The optimal mass damper for mode is obtained by selecting such that is the maximum element in m jq j cjq j k jqj p j Φjn kdud cdu d j 1 2 p j Φjn Φj j i qi 1 Φin un un m ie u n cie u n k ie un p ie kdud cdu d m ie 1 Φin 2 M i 1 Φin 2 Φi TMΦi k ie ωi 2m ie cie αk ie p ie 1 Φin p i m ξi md ωd md m m ie m Φin 2 Φi TMΦi ωd foptωi i n Φin Φi ConCh04v2fm Page 265 Thursday July 11 2002 433 PM 266 Chapter 4 Tuned Mass Damper Systems Example 45 Design of TMDs for a simply supported beam Consider the simply supported beam shown in Figure E45a The modal shapes and frequencies for the case where the cross sectional properties are con stant and the transverse shear deformation is negligible are 1 2 We obtain a set of N equations in terms of N modal coordinates by expressing the transverse displacement ux t as 3 and substituting for u in the principle of virtual displacements specialized for negli gible transverse shear deformation see Eq 2157 4 Substituting for 5 FIGURE E45a x x P y u EI constant L Φn x nπx L sin ωn2 EI ρm nπ L 4 n 1 2 u qi t Φj x j 1 N M δχ x d 0 L b δu dx δχ δχ x2 2 d d δu ConCh04v2fm Page 266 Thursday July 11 2002 433 PM Section 46 Tuned Mass Damper Theory for MDOF Systems 267 and taking 6 leads to the following equations 7 Lastly we substitute for M and b in terms of and q and evaluate the inte grals The expressions for M and b are 8 9 Noting the orthogonality properties of the modal shape functions 10 11 the modal equations uncouple and reduce to 12 where 13 14 15 δu δqjΦj MΦj xx dx bΦj x d j 1 2 N Φ M EIχ EI qlΦl xx l 1 N b ρmu b x t ρm Φlq l b x t l 1 N ΦjΦk x d 0 L δjk L 2 Φj xx Φk xx x d 0 L jπ L 4 δjk L 2 m jq j k jqj p j m j Lρm 2 k j EI jπ L 4L 2 p j b jπx L sin x d 0 L ConCh04v2fm Page 267 Thursday July 11 2002 433 PM When the external loading consists of a concentrated force applied at the location x x see Figure E45a the corresponding modal load for the jth mode is In this example the force is considered to be due to a mass attached to the beam as indicated in Figure E45b The equations for the tuned mass and the force are FIGURE E45b Suppose we want to control the ith modal response with a tuned mass damper attached at x x Taking j equal to i in Eqs 12 and 13 the ith modal equation has the form Assuming the response is dominated by the ith mode ux t is approximated by and Eq 19 is transformed to an equation relating u and ud Section 46 Tuned Mass Damper Theory for MDOF Systems 269 where 22 The remaining steps utilize the results generated for the SDOF undamped structure damped TMD system considered in Section 43 We use and as the mass and stiffness parameters for the primary system To illustrate the procedure consider the damper to be located at midspan and the first mode is to be controlled Taking i 1 and the corresponding parameters are 23 24 25 We specify the equivalent damping ratio and determine the required mass ratio from Figure 432 For example taking requires The other parameters corresponding to follow from Figures 429 430 and 431 26 27 28 Using these parameters the corresponding expression for the damper properties are 29 30 31 32 m ie 1 iπx L sin 2 m i m ie k ie x L 2 iπx L sin 1 m ie m 1 Lρm 2 k ie k 1 EIL 2 π L 4 ξe ξe 006 m 003 m 003 fopt ωd ω1 0965 ξd opt 0105 uˆ d uˆ 5 md 003m 1 ωd 0965ω1 kd ωd2md cd 2ξdωdmd ConCh04v2fm Page 269 Thursday July 11 2002 433 PM 270 Chapter 4 Tuned Mass Damper Systems Once and are specified the damper properties can be evaluated For example consider the beam to be a steel beam having the following properties 33 The beam parameters are 34 Applying Eqs 29 through 32 results in 35 The total mass of the girder is 20000 kg Adding 300 kg which is just 15 of the total mass produces an effective damping ratio of 006 for the first mode response The mode shape for the second mode has a null point at x L2 and therefore locating a tuned mass at this point would have no effect on the second modal response The optimal locations are and Taking and i 2 we obtain 36 37 38 39 The procedure from here on is the same as before We specify and determine the required mass ratio and then the frequency and damping parameters It is of interest to compare the damper properties corresponding to the same equivalent damping ratio Taking the damper properties for the example steel beam are m 1 ω1 L 20 m ρm 1000 kg m I 8 10 4 m4 E 2 1011 N m2 m 1 10 000 kg ω1 987 rad s md 300 kg ωd 952 r s kd 27 215 N m cd 5998 N s m x L 4 x 3L 4 x L 4 iπx L sin 1 m 2e m 2 Lρm 2 k 2e k 2 8EIL π L 4 ω22 16EI ρm π L 4 ξe ξe 006 ConCh04v2fm Page 270 Thursday July 11 2002 433 PM md 300 kg kd 435 440 Nm cd 2400 N sm The required damper stiffness is an order of magnitude greater than the corresponding value for the first mode response 47 CASE STUDIES MDOF SYSTEMS This section presents shear deformation profiles for the standard set of building examples defined in Table 24 A single TMD is placed at the top floor and tuned to either the first or second mode The structures are subjected to harmonic ground acceleration with a frequency equal to the fundamental frequency of the buildings as well as scaled versions of El Centro and Taft ground accelerations As expected significant reduction in the response is observed for the harmonic excitations see Figures 446 through 449 The damper is generally less effective for seismic excitation versus harmonic excitation see Figures 450 through 461 Results for the low period structures show more influence of the damper which is to be expected since the response is primarily due to the first mode This data indicates that a TMD is not the optimal solution for controlling the motion due to seismic excitation FIGURE 446 Maximum shear deformation for Building 1 FIGURE 447 Maximum shear deformation for Building 2 FIGURE 448 Maximum shear deformation for Building 3 FIGURE 449 Maximum shear deformation for Building 4 FIGURE 450 Maximum shear deformation for Building 1 FIGURE 451 Maximum shear deformation for Building 1 FIGURE 452 Maximum shear deformation for Building 2 Maximum shear deformation for Building 2 Maximum shear deformation for Building 3 Maximum shear deformation for Building 3 Maximum shear deformation for Building 3 Maximum shear deformation for Building 3 Maximum shear deformation for Building 4 FIGURE 459 Maximum shear deformation for Building 4 FIGURE 460 Maximum shear deformation for Building 4 FIGURE 461 Maximum shear deformation for Building 4 PROBLEMS Problem 41 Verify Eqs 413 through 417 Hint Express p u and ud in complex form p p eiΩt u u eiΩt ud ūd eiΩt and solve Eqs 46 and 47 for ū and ūd Then take ū û eiδ1 ūd ûd eiδ1 δ2 ω ωd Ω Problem 42 Refer to Eqs 414 and 420 Express ξe as a function of m ξ and ûûd Take ξ 005 and plot ξe versus m for a representative range of the magnitude of the displacement ratio ûûd Problem 43 Figure 47 illustrates an active tuned mass damper configuration The damper can be modeled with the 2DOF system shown in Figure P43 The various terms are as follows us is the total displacement of the support attached to the floor beam Fa is the selfequilibrating force provided by the actuator md kd cd are parameters for the damper mass ka and ma are parameters for the auxiliary mass a Derive the governing equation for md and ma Also determine an expression for the resultant force R that the system applies to the floor beam b Consider ma to be several orders of magnitude smaller than md eg ma 001md Also take the actuator force to be a linear function of the relative velocity of the damper mass Fa ca ud Specialize the equations for this case How would you interpret the contribution of the actuator force to the governing equation for the damper mass Problem 44 Design a pendulum damper system having a natural period of 6 seconds and requiring less than 4 meters of vertical space Problems 281 Problem 45 The pendulum shown in Figure P45 is connected to a second mass which is free to move horizontally The connection between mass 1 and mass 2 carries only shear Derive an equation for the period of the compound pendulum and the length of an equivalent simple pendulum Assume the links are rigid Problem 46 Refer to Figure 412 Establish the equations of motion for the mass con sidering to be small Verify that the equivalent stiffness is equal to Problem 47 Refer to Figure 415 and Eq 476 Derive the corresponding expression for starting with Eq 462 and using the same reasoning strategy Considering the mass ratio to be less than 003 estimate the difference in the optimal values for the various parameters Problem 48 Generate plots of versus for ranging from 0 to 02 and Compare the results with the plots shown in Figure 423 Problem 49 Consider a system composed of an undamped primary mass and a tuned mass damper The solution for periodic force excitation is given by see Eqs 452 to 471 FIGURE P45 m1 L1 m2 L2 md θ Wd R H1 P Q m H1 ρ ξd m 001 f 09876 ConCh04v2fm Page 281 Thursday July 11 2002 433 PM 282 Chapter 4 Tuned Mass Damper Systems 1 2 3 4 5 6 7 The formulation for the optimal damper properties carried out in Section 43 was based on minimizing the peak value of H1 actually H2 but H1 behaves in a sim ilar way ie on controlling the displacement of the primary mass Suppose the design objective is to control the acceleration of the primary mass Noting Eqs 1 and 3 the acceleration is given by 8 9 Substituting for k transforms Eq 9 to 10 where 11 Investigate the behavior of with and If it behaves similar to as shown in Figure 415 describe how you would establish the optimal values for the various parameters and also how you would design a tuned mass system when is specified u ueiΩt ud udeiΩt u p kH1eiδ1 ud p kH3eiδ3 H1 f 2 ρ2 2 2ξdρf 2 D2 H3 ρ2 D2 D2 1 ρ2 f 2 ρ2 mρ2f 2 2 2ξdρf 1 ρ2 1 m 2 u a aeiΩt a pΩ2 k H1ei δ1 π a p m H 1 ei δ1 π H 1 ρ2H1 H1 ρ f m ξd H2 H 1 ConCh04v2fm Page 282 Thursday July 11 2002 433 PM Problems 283 Problem 410 Design a TMD for a damped SDOF system having The design motion constraints are a b c Repeat part b considering to be equal to 005 Problem 411 This problem concerns the design of a tunedmass damper for a damped single degree of freedom system The performance criteria are a Determine the damper properties for a system having 10000 kg and for the following values of b Will the damper be effective for an excitation with frequency Discuss the basis for you conclusion Problem 412 Refer to Example 37 Suppose a tuned mass damper is installed at the top level at mass 5 ξ 002 H5 opt 10 H7 H5 opt 5 H5 opt 5 H7 H5 opt 5 ξ ξeq 01 uˆ d uˆ 5 m k 395 kNm ξ ξ 002 ξ 005 25π rads ConCh04v2fm Page 283 Thursday July 11 2002 433 PM 284 Chapter 4 Tuned Mass Damper Systems a Determine the damper properties such that the equivalent damping ratio for the fundamental mode is 016 Use the values of from Example 37 Assume stiffness proportional damping for c b Consider the tuned mass damper to be a pendulum attached to Figure P412 Determine and for the damper properties estab lished in part a c Repeat part a for the case where the mass damper is tuned for the sec ond mode rather than for the first mode and the desired equivalent modal damping ratio is 03 Use the same values of and assume stiffness proportional damping Problem 413 Consider a cantilever shear beam with the following properties a Model the beam as a 10DOF discrete shear beam having 5 m segments Determine the first three mode shapes and frequencies Normalize the mode shapes such that the peak amplitude is unity for each mode b Design tuned mass dampers to provide an effective modal damping ratio of 010 for the first and third modes Take and assume modal damping is proportional to stiffness Note You need to first establish the optimal location of the tuned mass dampers for the different modes Problem 414 Consider a simply supported steel beam having the following properties FIGURE P412 m k c m5 md L md m5 k5 c5 u5 L m k c H 50 m ρm 20 000 kgm DT 8 105 1 06x H kN ξ1 002 ConCh04v2fm Page 284 Thursday July 11 2002 433 PM Problems 285 a Design tuned mass damper systems that provide an equivalent damping of 005 for each of the first three modes b Repeat part a with the constraint that an individual damper mass can not exceed 300 kg Hint Utilize symmetry of a particular mode shape to locate a pair of dampers whose function is to control that mode Problem 415 Consider the simply supported beam shown in Figure P415 The beam has a uniform weight of 15 kNm and a concentrated weight at midspan of 100 kN The flexural rigidity is constant and equal to 200000 kNm2 a Assume the first mode can be approximated by Determine the governing equation for using the principle of virtual displacements b Design a tuned mass damper to provide an equivalent damping ratio of 005 for the first mode Assume no damping for the beam itself c Will the damper designed in part b be effective for the second mode Explain your answer Problem 416 Refer to Problem 325 part b Suggest a tuned mass damper for generating the required energy dissipation FIGURE 415 L 30 m ρm 1500 kg m I 1 10 2 m4 15 m 15 m Constant EI u1 W u u1 π Lx sin u1 ConCh04v2fm Page 285 Thursday July 11 2002 433 PM Redução de oscilações em arranhacéus Nome completo do aluno 1 Nome completo do aluno 2 Prof Dr nome do professor Vibrações Mecânicas Amortecedor de massa sintonizado por pêndulo Esquemas Equações Os amortecimento por massa sintonizada AMS é um método prático e eficiente para atenuar a resposta à entrada de aceleração do solo Esta tecnologia é uma alternativa para o projeto estrutural de edifícios em regiões sísmicas O AMS é um sistema mecânico que consiste em uma massa uma mola que fornece rigidez e um amortecedor viscoso A massa é presa à estrutura através da mola e amortecedor Quando o elemento que confere rigidez ao AMS é um pêndulo o dispositivo é denominado Amortecedor de massa sintonizado por pêndulo AMSP O AMSP absorve grande parte da energia produzida por forças externas minimiza a amplitude de vibração e reduz a probabilidade de danos aos elementos estruturais O AMSP possui design muito simples e responde rapidamente ao movimento da estrutura O comprimento do pêndulo controla sua frequência natural e seu design pode integrar os amortecedores viscosos sem dificuldade Uma vantagem notável do AMSP é que o pêndulo pode oscilar em todas as direções proporcionando dissipação de energia para cargas aplicadas em diferentes direções No entanto o desempenho do AMSP na redução da vibração depende da massa comprimento e amortecimento do pêndulo entre outros Crystal Tower Nagase and Hisatoku 1990 A torre localizada em Osaka Japão tem 157 m de altura e 1860 m² em planta pesa 44000 toneladas e tem um período fundamental de aproximadamente 4s na direção nortesul e 3s na direção lesteoeste Um amortecedor de massa pendular sintonizado foi incluído na fase inicial do projeto para diminuir o movimento do edifício induzido pelo vento em cerca de 50 Seis dos nove tanques de armazenamento térmico de gelo para resfriamento e aquecimento de ar cada um pesando 90 toneladas são pendurados nas vigas superiores do telhado e usados como uma massa pendular Quatro tanques têm um comprimento de pêndulo de 4 m e deslizam na direção nortesul os outros dois tanques têm um comprimento de pêndulo de cerca de 3 m e deslizam na direção lesteoeste Amortecedores de óleo conectados aos pêndulos dissipam a energia do pêndulo O custo desse sistema de amortecedor de massa ajustado foi de cerca de US 350000 menos de 02 do custo de construção Amortecedor de massa sintonizado por pêndulo Amortecedor de massa sintonizado por translação Esquema amortecedor passivo A massa repousa sobre rolamentos que funcionam como roletes e permitem que a massa transfira lateralmente em relação ao piso Molas e amortecedores são inseridos entre a massa e os suportes verticais adjacentes que transmitem a força lateral fora de fase para o nível do piso e depois para o pórtico estrutural Os amortecedores translacionais bidirecionais são configurados com molasamortecedores em duas direções ortogonais e fornecem a capacidade de controlar o movimento estrutural em dois planos ortogonais Vários mecanismos passivos e ativos de dissipação de energia têm sido propostos e testados como meios alternativos para vibração Esquema amortecedor ativo Citicorp Center John Hancock Tower Amortecedor de massa sintonizado por translação Dois amortecedores foram adicionados à John Hancock Tower de 60 andares em Boston para reduzir a resposta ao carregamento de rajadas de vento Os amortecedores são colocados em extremidades opostas do 58º andar a 67 m de distância e se movem para neutralizar a oscilação e a torção devido à forma do edifício Cada amortecedor pesa 2700 kN e consiste em um caixa de aço com cerca de 52 m³ que se apoia em uma placa de aço de 9 m de comprimento O peso cheio de chumbo contido lateralmente por molas rígidas ancoradas nas colunas internas do edifício e controladas por cilindros servohidráulicos desliza para frente e para trás em um mancal hidrostático composto por uma fina camada de óleo forçada através de orifícios na placa de aço Sempre que a aceleração horizontal ultrapassar 0003g por dois ciclos consecutivos o sistema é ativado automaticamente Este sistema foi projetado e fabricado pela LeMessurier AssociatesSCI em associação com a MTS System Corp com um custo de cerca de 3 milhões de dólares e reduz a oscilação do edifício em 40 a 50 O Citicorp Manhattan AMS também foi projetado e fabricado pela LeMessurier AssociatesSCI em associação com a MTS System Corp Este edifício tem 279 m de altura e tem um período fundamental de cerca de 65 s com uma taxa de amortecimento inerente de 1 ao longo de cada eixo O amortecedor localizado no 63º andar da coroa da estrutura tem uma massa de 366 Mg cerca de 2 da massa modal efetiva do primeiro modo e era 250 vezes maior do que qualquer amortecedor de massa sintonizado existente no momento da instalação A LeMessurier estima que o AMS do Citicorp que custou cerca de 15 milhão de dólares economizou de 35 a 4 milhões de dólares Esta soma representa o custo de cerca de 2800 toneladas de aço estrutural que seriam necessárias para satisfazer as restrições de deflexão Conclusão Os amortecedores sintonizados por massa pendular são mais baratos possuem um design mais fácil de implementar e podem ser empregados para reduzir oscilação em múltiplas direções Possui um desempenho bastante sensível às variáveis dinâmicas do pêndulo do solo e da estrutura Os amortecedores de massa sintonizados por translação são mais robustos possuem um design um pouco mais complexo podem ser implementados em conjunto de um atuador para tornarse um sistema ativo a taxa de amortecimento pode ser mais facilmente alterada a partir da viscosidade do fluido de amortecimento em geral um óleo Slide 1 Introdução do tema Apresentação dos modelos de AMS Amortecedores de massa sintonizados são dois Por pêndulo Por translação em roletes Slide 2 AMSP Amortecedor de massa sintonizado por pêndulo Esquemas Primeiro esquema mostra o esquema da dinâmica de um pêndulo Segundo esquema mostra o esquema de um sistema massamola equivalente Equações Equações de movimento não linear do pêndulo e simplificação para um sistema equivalente utilizando simplificações TEXTO extraído do ARTIGO sobre otimização de parâmetros para construção de AMSP Os amortecimento por massa sintonizada AMS é um método prático e eficiente para atenuar a resposta à entrada de aceleração do solo Esta tecnologia é uma alternativa para o projeto estrutural de edifícios em regiões sísmicas O AMS é um sistema mecânico que consiste em uma massa uma mola que fornece rigidez e um amortecedor viscoso A massa é presa à estrutura através da mola e amortecedor Quando o elemento que confere rigidez ao AMS é um pêndulo o dispositivo é denominado Amortecedor de massa sintonizado por pêndulo AMSP O AMSP absorve grande parte da energia produzida por forças externas minimiza a amplitude de vibração e reduz a probabilidade de danos aos elementos estruturais O AMSP possui design muito simples e responde rapidamente ao movimento da estrutura O comprimento do pêndulo controla sua frequência natural e seu design pode integrar os amortecedores viscosos sem dificuldade Uma vantagem notável do AMSP é que o pêndulo pode oscilar em todas as direções proporcionando dissipação de energia para cargas aplicadas em diferentes direções No entanto o desempenho do AMSP na redução da vibração depende da massa comprimento e amortecimento do pêndulo entre outros Slide 3 Estudo de caso da torre japonesa Crystal Tower TEXTO extraído do LIVRO anexado A torre localizada em Osaka Japão tem 157 m de altura e 1860 m² em planta pesa 44000 toneladas e tem um período fundamental de aproximadamente 4s na direção nortesul e 3s na direção lesteoeste Um amortecedor de massa pendular sintonizado foi incluído na fase inicial do projeto para diminuir o movimento do edifício induzido pelo vento em cerca de 50 Seis dos nove tanques de armazenamento térmico de gelo para resfriamento e aquecimento de ar cada um pesando 90 toneladas são pendurados nas vigas superiores do telhado e usados como uma massa pendular Quatro tanques têm um comprimento de pêndulo de 4 m e deslizam na direção norte sul os outros dois tanques têm um comprimento de pêndulo de cerca de 3 m e deslizam na direção lesteoeste Amortecedores de óleo conectados aos pêndulos dissipam a energia do pêndulo O custo desse sistema de amortecedor de massa ajustado foi de cerca de US 350000 menos de 02 do custo de construção Slide 4 AMST Amortecedor de massa sintonizado por translação Esquemas Primeiro esquema mostra o esquema de AMST sem atuador Primeiro esquema mostra o esquema de AMST com atuador TEXTO extraído do LIVRO anexado A massa repousa sobre rolamentos que funcionam como roletes e permitem que a massa transfira lateralmente em relação ao piso Molas e amortecedores são inseridos entre a massa e os suportes verticais adjacentes que transmitem a força lateral fora de fase para o nível do piso e depois para o pórtico estrutural Os amortecedores translacionais bidirecionais são configurados com molasamortecedores em duas direções ortogonais e fornecem a capacidade de controlar o movimento estrutural em dois planos ortogonais Vários mecanismos passivos e ativos de dissipação de energia têm sido propostos e testados como meios alternativos para vibração Slide 5 Estudo de caso de 2 edifícios americanos TEXTO extraído do LIVRO anexado Dois amortecedores foram adicionados à John Hancock Tower de 60 andares em Boston para reduzir a resposta ao carregamento de rajadas de vento Os amortecedores são colocados em extremidades opostas do 58º andar a 67 m de distância e se movem para neutralizar a oscilação e a torção devido à forma do edifício Cada amortecedor pesa 2700 kN e consiste em um caixa de aço com cerca de 52 m³ que se apoia em uma placa de aço de 9 m de comprimento O peso cheio de chumbo contido lateralmente por molas rígidas ancoradas nas colunas internas do edifício e controladas por cilindros servohidráulicos desliza para frente e para trás em um mancal hidrostático composto por uma fina camada de óleo forçada através de orifícios na placa de aço Sempre que a aceleração horizontal ultrapassar 0003g por dois ciclos consecutivos o sistema é ativado automaticamente Este sistema foi projetado e fabricado pela LeMessurier AssociatesSCI em associação com a MTS System Corp com um custo de cerca de 3 milhões de dólares e reduz a oscilação do edifício em 40 a 50 O Citicorp Manhattan AMS também foi projetado e fabricado pela LeMessurier AssociatesSCI em associação com a MTS System Corp Este edifício tem 279 m de altura e tem um período fundamental de cerca de 65 s com uma taxa de amortecimento inerente de 1 ao longo de cada eixo O amortecedor localizado no 63º andar da coroa da estrutura tem uma massa de 366 Mg cerca de 2 da massa modal efetiva do primeiro modo e era 250 vezes maior do que qualquer amortecedor de massa sintonizado existente no momento da instalação A LeMessurier estima que o AMS do Citicorp que custou cerca de 15 milhão de dólares economizou de 35 a 4 milhões de dólares Esta soma representa o custo de cerca de 2800 toneladas de aço estrutural que seriam necessárias para satisfazer as restrições de deflexão Slide 6 Conclusão Os amortecedores sintonizados por massa pendular são mais baratos possuem um design mais fácil de implementar e podem ser empregados para reduzir oscilação em múltiplas direções Possui um desempenho bastante sensível às variáveis dinâmicas do pêndulo do solo e da estrutura Os amortecedores de massa sintonizados por translação são mais robustos possuem um design um pouco mais complexo podem ser implementados em conjunto de um atuador para tornarse um sistema ativo a taxa de amortecimento pode ser mais facilmente alterada a partir da viscosidade do fluido de amortecimento em geral um óleo Heliyon 7 2021 e07221 Contents lists available at ScienceDirect Heliyon journal homepage wwwcellcomheliyon Research article Pendulum tuned mass damper optimization and performance assessment in structures with elastoplastic behavior Víctor J García a Edwin P Duque b José Antonio Inaudi cd Carmen O Márquez e Josselyn D Mera a Anita C Rios e a Facultad de Ingeniería Carrera de Ingeniería Civil Universidad Nacional de Chimborazo Riobamba Provincia de Chimborazo 060150 Ecuador b Departamento de Ingeniería Civil Universidad Técnica Particular de Loja San Cayetano Alto Calle París Loja 110150 Provincia de Loja Ecuador c Facultad de Ingeniería Universidad Católica de Córdoba Córdoba X5000 Argentina d Facultad de Ciencias Exactas Físicas y Naturales Universidad Nacional de Córdoba Córdoba X5000 Argentina e Facultad de Ingeniería Carrera de Ingeniería Ambiental Universidad Nacional de Chimborazo Riobamba Provincia de Chimborazo 060150 Ecuador ARTICLE INFO ABSTRACT Keywords Tuned mass damper Pendulum tuned mass damper Elastoplastic behavior Structure safety Seismic vulnerability Different types of tuned mass dampers TMD have been applied to reduce wind and seismic induces vibrations in buildings We analyze a pendulum tuned mass damper PTMD to reduce vibrations of structures that exhibit elastoplastic behavior subjected to ground motion excitation Using a simple dynamic model of the primary structure with and without the PTMD and a random process description of the ground acceleration the performance improvement of the structure is assessed using statistical linearization The Liapunov equation is used to estimate the meansquare response in the stationary condition of the random process and optimize PTMD parameters The optimum values of the PTMD frequency and damping ratio are defined as PTMD design values for a specific maximum seismic intensity design criterion The results show that 1 The values of the PTMD effectiveness criterion and the optimal design values of the frequency ratio are higher when the damping ratio of the primary structure decreases 2 The performance of the optimized PTMD is higher when the structure exhibits a linear hysteresis loop low seismic intensity 3 The optimized PTMD controls the development of structural plasticity reducing vulnerability 4 There is a strong dependence of the optimum PTMD parameters on the dynamic soil properties of the building foundation 5 The PTMD performance improves as its mass increases The optimum frequency ratio decreases and the damping ratio increases as the mass of the pendulum increases The PTMD designed and optimized with the proposed methodology reduces vibrations controls the development of plasticity and protects the primary structure particularly in low and mediumintensity earthquakes Corresponding author Email addresses vgarciaunacheduec vgarcia375gmailcom VJ García httpsdoiorg101016jheliyon2021e07221 Received 6 December 2020 Received in revised form 9 May 2021 Accepted 2 June 2021 24058440 2021 The Authors Published by Elsevier Ltd This is an open access article under the CC BYNCND license httpcreativecommonsorglicensesbyncnd40 Received 6 December 2020 Received in revised form 9 May 2021 Accepted 2 June 2021 24058440 2021 The Authors Published by Elsevier Ltd This is an open access article under the CC BYNCND license httpcreativecommonsorglicensesbyncnd40 In practice PTMDs are mainly used to reduce windinduced vibration of tall and slender structures wind turbine towers and highrise steel towers 9 10 Moreover PTMD has become a popular device in con trolling and reducing structural vibration due to wind loads In practical applications PTMD has shown to be effective in low and mediumintensity earthquake excitation Mainly where the fundamental vibration mode of the main structure controls most of the response of the building 6 An example of a building with a PTMD conceived to reduce windinduced vibration is the Taipei 101 Tower in the capital of Taiwan This building has 101 floors with a total height of 508 m At its inau guration the PTMD was the biggest ever built with a large solid steel sphere weighing 660 metric tons The PTMD attached to the Tapei 101 Tower reduces windinduced vibration levels and has also shown the capability to alleviate seismic induced vibrations 8 The difference between the structure response to strong intensity earthquakes and to wind load is the amount of energy dissipation in main structural ele ments The buildings motion during an earthquake induces plastic behavior in some structural members and the structure vibrates for a few seconds During a wind load the building oscillates during several mi nutes in the elastic range 11 PTMD performance deteriorates when the structure shows elasto plastic behavior When a structure enters the nonlinear range there is a loss of effective stiffness which generates a loss of tuning between the PTMD frequency and the structures primary frequency A large amount of energy dissipation is provided by yielding structural elements which imply a marginal contribution of the energy dissipation of the TMD to the building response Under stronger dynamic loading induced by earth quakes larger changes can occur in the effective stiffness of the structure because of inelastic effects potentially coupled with damage These changes cause an increase in the structural period and consequently a much more significant detuning effect 12 SungSik et al 13 showed that TMDs performance whose design parameters were optimized for an elastic structure considerably deteriorated when the structural re sponses hysteretic portion increased Sgobba and Marano 14 and Duque et al 15 reported that the TMD performance decreases when the structural hysteretic response increases even when the TMD design pa rameters are optimized for a structure with elastoplastic behavior While the achieved reductions are not significant they are not negligible in structures subjected to medium and high seismic intensities An alternative to the TMD limitations attached to an elastoplastic structure is the semiactive TMD STMD or the active TMD ATMD whose control method adapts the TMD to the structure with variables parameters STMD and ATMD are an alternative to passive TMD especially if demand reductions are to be achieved in structures that enter the plastic regime during their response A control system optimizes tune the TMD to get the best performance and adapt it to the dynamic structure regime In this regard Sung and Nagarajaiah 16 found that the STMD can effectively attenuate the seismic responses and outper form the optimal passive TMD Also these authors reported that the STMD remains tuned with the primary structure In contrast the optimal passive TMD becomes offtuned when dam age occurs Lourenco 17 described the design construction imple mentation and performance of a prototype adaptive PTMD The experimental studies results demonstrate the importance of optimizing the PTMD frequency and damping ratio to reduce structural vibrations Finding the PTMDs optimum design parameters is not a trivial task Gerges and Vickery 18 reported in design charts the optimum design parameters and the corresponding efficiency of the PTMD under both wind and earthquake dynamic loads considering an elastic response of the main structure Oliveira et al 4 found a general dimensionless optimal parameter for a PTMD considering structure elastic response they concluded that the dimensionless parameters could be employed to design a pendulum to control any tall building subjected to dynamic loads with different mass and damping ratios Hassani and Aminafshar 19 study the numerical optimization of PTMD attached to a tall building that shows elastic behavior and under horizontal earthquake excitation Deraemaeker and Soltani 10 extended Den Hartogs equal peak methods to the PTMD and observed an excellent agreement be tween the PTMD performance tuned by these analytical formulae and the numerical results obtained by the Oliveira et al 4 optimization process Colherinhas et al 20 assumed a structure elastic behavior Cloherinhas et al modeled a tower with a PTMD using Finite Element and ANSYS to find the relation between the mass length stiffness and damping coef ficient of the pendulum as a function of the high vibration amplitudes at the top of the tower Amrutha and Amritha 21 assessed the seismic response reduction by PTMD on regular highrise RC buildings using SAP200 V19 software They concluded that installing PTMD on the structure 1025 top story displacement reduction was observed Few authors have considered a PTMD optimized considering the elastoplastic behavior of the main structure and its foundation soil dy namic properties For the classical TMD this job has been done by Sgobba and Marano 14 and Duque et al 15 Also Jia and Jianwen 22 investigated the performance degradation of TMDs arising from ignoring soilstructure interaction effects They showed that a welltuned damper performs better than an offtuned one by up to 25 although an offtuned one may reduce the structure responses by up to 30 Simi larly Salvi et al 23 investigated an optimum TMDs effectiveness in reducing the linear structural response to strongmotion earthquakes by embedding soilstructure interaction within the dynamic and TMD optimization model Regarding the classical TMD and structure with an elastoplastic behavior Sgobba and Marano 14 studied the optimum design of TMD for structures with nonlinear behavior These authors use the de BoucWen model to describe the nonlinear behavior of the main struc ture The KanaiTajimi stochastic seismic model describes the earthquake ground acceleration Sgobba y Marano confirmed that the TMD reduces the amount of the hysteretic dissipated energy which directly measures damage in the structure So it is beneficial to protect buildings that develop a nonlinear behavior under severe dynamic loadings Woo et al 13 assessed the seismic response control of elastic and inelastic struc tures by using passive and semiactive TMDs They performed a numer ical analysis for a structure with hysteresis described by the BoucWen model The results indicated that the passive TMDs performance whose design parameters were optimized for an elastic structure considerably deteriorated when the hysteretic portion of the structural responses increased The semiactive TMD showed about 1540 more response reduction than the TMD Duque et al 15 found that if the TMD is optimized considering the seismic intensity and a structure with elasto plastic behavior the TMD reduces the structures displacements in seismic events While the achieved reductions are not significant they are not negligible in structures subject to high seismic intensities Although qualitatively we can expect similar results and performance for a PTMD These results cannot be quantitatively extrapolated to a PTMD More exploratory research on the PTMD performance is needed consid ering the pendulum optimum design parameters values the soils dy namic properties the primary structure dynamic properties and the primary structure behavior due to seismic intensity and design parameters The purpose of this study is to analyze the PTMD performance when its design parameters are optimized considering 1 the main structure dynamic properties frequency and damping 2 the structure exhibits elastoplastic behavior depending on the intensity PGA of the seismic excitation 3 the main structure is on soils with different dynamic properties soft medium and firm soil The equations of motion of the system were solved using Monte Carlo simulation and the stochastic pseudolinear equivalent system SPLES to achieve our objective The numerical optimization scheme applies opti mization methods to the SPLES parameters in an iterative scheme This scheme allows finding the optimal design parameters of the PTMD fre quency and damping ratio given the dynamic parameters of the main structure and the soil The validation of the stochastic pseudo linear equivalent system was performed by comparing the solution using Monte VJ García et al Heliyon 7 2021 e07221 2 Carlo and the solution using the stationary regime of the stochastic pseudo linear equivalent system Finally case study results are presented for seismic excitation records measured during Pedernales Ecuador 2016 earthquake 2 Methodology Seismic excitation is represented through a stationary random process of filtered white noise KanaiTajimi filter KTF and the main structures elastoplastic behavior represented by the Bouc Wen model BWM A SDOF model modeled the primary structure 21 KanaiTajimi filter The KTF is a model frequently used to represent a seismic acceleration and achieve artificial accelerograms KanaiTajimis model considers the earthquake represented by a spectrum of filtered white Gaussian noise 24 Kanai 25 and Tajimi 26 showed that a secondorder linear oscillator is suitable to filter white noise and obtain a spectrum that match frequency content of registered accelerograms Figure 1 Therefore the filter parameters are related to the soil characteristic and consequently to different frequency contents of the ground acceleration signal Figure 1 The acceleration process is characterized by its power spectral density PSD named excitation power spectral density EPSD The white noise PSD S0 used in the model scales EPSD The EPSD depends on the filter frequency ωf and filter damping coefficient ξf Filter parameters are calibrated to be representative of different soil conditions A set of frequency and damping coefficient values reported in the literature is listed in Table 1 Unless otherwise specified in our study we consider the white noise PSD obtained with the KTF considering the values of ωf and ξf reported by Sues et al 24 The parameters ωf and ξf are associated with soil dynamics characteristics However their values depend on the distance to the epicenter earthquake magnitude and soil rigidity among other factors In the context of our research the parameters ωf and ξf do not provide information on the soilstructure interaction and only serve to generate three different EPSD Eq 1 models the KTF where xf xf xf represents the relative acceleration velocity and displacement of the filter respectively and W represents the absolute bedrock acceleration which is modeled as white noise with a constant PSD S0 Table 1 Soil dynamic parameters Source Soil profile description Frequency ωf rads Damping ξf 27 Medium 20 05 28 Firm 20 065 Soft 45 01 24 Soft 109 096 Medium 165 08 Firm 169 094 values assumed for the BWM hysteretic model were A ¼ n ¼ 1β ¼ γ ¼ 05 and α ¼ 05 since with them the model captures the response of structures and structural members exposed to earthquakes 36 23 Stochastic linearization of the hysteresis BoucWen model The statistical linearization method replaces Eq 7 by the equivalent linear form given in the Eq 8 37 38 39 so that the mean square error is minimized 25 Dynamic equations of the combined system The combined system consists of the passive PTMD model attached to the structure Figure 3a The passive PTMD device consists of a pendulum with a viscous damper cd The pendulum is a solid sphere of mass md connected by a cable of length l to the structure We assume that the pendulum rotational inertia and the cable mass are negligible and the angle theta is small The pendulum oscillation frequency is omegad2 gl where g represents the acceleration of gravity The primary structure model is an SDOF model with mass ms stiffness ks and damping cs Considering the set of forces acting on the structure and the PTMD with viscous damping the force on the mass ms and the damping cs is proportional to the relative movement xs xm xg xm xs xg 19 The dynamic equilibrium equation of md Figure 3b projected on the horizontal direction leads to Eq 20 md xm cos theta md theta double dot md g sin theta cs theta dot cos theta 0 20 When rotations theta are small cos theta approx 1 and sin theta approx theta and considering the Eq 19 we can rewrite the Eq 20 into Eq 21 xs double dot theta double dot g theta cdmd theta dot xg double dot 21 The following nondimensional parameters should be defined 1 Ratio between the mass of the PTMD and the structures mass mu mdms 2 The ratio between the frequency of the PTMD and the frequency of the structures main mode of vibration f omegadomegas 3 Stiffness ki mi omegai2 4 Damping ratio xi ci2 sqrtki mi Replacing cdmd 2 xid omegad and Eq 2 in the Eq 21 xs double dot theta double dot g theta 2 xid omegad theta dot omegaf2 xf 2 xif omegaf xf dot 22 Considering the forces acting on the mass of the primary structure we have ms xm double dot md xm double dot md l theta dot cos theta ks xm xg cs xm dot xs dot ks 1 alpha z 23 Replacing Eqs 19 and 21 as well as the terms csms 2 xis omegas ksms omegas2 mu mdms in Eq 23 theta double dot 21 mu xid omega d theta dot 1 mu g l theta alpha omegas2 l xs 2 xis omegas l xs dot omegas2 1 alpha l z 24 Replacing dimensionless terms gl omegad2 and omegas2 l omegas2 omegad2 g omegas l omegas omegad2 g in Eq 24 theta double dot 21 mu xid omegad theta dot 1mu omegad2 theta alpha omegas2 omegad2 g xs dot 2 xis omegas omegad2 g xs dot 1 alpha omegas2 omegad2 g z 25 Replacing the Eq 25 and the relationship l g omegad2 in Eq 22 xs double dot 2 mu xid omegad g double dot mu g theta alpha omegas2 xs 2 xis omegas xs dot omegas2 1 alpha z omegaf2 xf 2 xif omegaf xf dot 26 Therefore the statespace formulation of the linearized model of the combined primary structure and PTMD is given in Eqs 27 28 and 29 X dot AX BW 27 X dot theta dot xs dot xf dot z dot theta double dot xs double dot xf double dot 28 X theta xs xf z dot theta xs dot xf dot 29 W 2pi S0 S0 4 9pi xif PGA2 wf 1 4 xif2 A 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 Keq 0 Ceq 0 1mu omegad2 alpha omegas2 omegad2 g 1alpha omegas2 omegad2 g 21mu xid omegad 2 xis omegas omegad2 g 0 mu g alpha omegas2 omegaf2 omegas2 1 alpha 2 mu xid g omegad 2 xis omegas 2 xif omegaf 0 0 omegaf2 0 0 0 2 xif omegaf 29 26 PTMD design parameters optimization and validation The PTMD design parameters optimization is performed for a stationary random ground acceleration process The optimization problem is handled via a numerical search algorithm The algorithm finds the parameters of the SPLES and computes expected performance with the linearized model The search for optimum PTMD design parameters is performed by solving the dynamic equation for a given mean maximum ground acceleration and finds the optimum design parameter in a steadystate condition Model validation is performed by comparing the time solution Monte Carlo simulation and the SPLES response 261 Monte Carlo simulation The Monte Carlo simulation is a method for estimating the exact response statistics of randomly excited nonlinear systems within any desired confidence level This approach is applicable for the estimation of the stationary and nonstationary response statistics The stochastic differential equation governing the systems motion is interpreted as an infinite set of deterministic differential equations For each member of this set the input is a sample function of the excitation process The output is the corresponding sample function of the response process 39 Thus to perform the Monte Carlo simulation the equations of motion are formulated in state space These equations are solved using the fourthorder RungeKutta method and discretetime whitenoise realizations for wt Where Delta t is the load discretization interval and fk is a vector of Gaussian random numbers We used 1000 sample records estimating the response statistics within acceptable engineering confidence levels Figure 3 a Representation of the combined primary structure and PTMD b Illustration of the forces acting on the PTMD mass with a viscous damper standard deviation of the displacement of the unprotected structure 15 The values of sigmaxs and sigmax0 are obtained from the covariance matrix Pxx OF 1 sigmaxs sigmax0 33 The PTMD design parameters are optimized by numerically solving Eq 33 for several frequency factor values f omegadomegas and several damping coefficient values xid Therefore we obtain a matrix OFij which contains the performance indicator OF values for the frequency and damping combinations assumed for the PTMD Figure 5 illustrates the optimization procedure for various PGA The maximum effectiveness OFmax is giving by fopt xiopt values A closed look at Figure 4b shows the optimum frequency ratio trend fopt when we keep the damping ratio constant and equal to xiopt Figure 5a Figure 5a shows a peak around fopt while Figure 5b shows a highly asymmetric peak around xiopt The symmetric peak a Figure 5a suggests that the effectiveness of the PTMD is more sensitive to the frequency ratio f values when the damping ratio is equal to xiopt Figure 5a The asymmetric peak around xiopt suggests that the PTMD effectiveness is very sensitive to the damping ratio before reaching the optimum value xiopt In contrast the PTMD effectiveness is robust to changes in the damping ratio when the damping ratio value is bigger than the xiopt Figure 5b We develop a model validation to assess equivalent linear modal accuracy for estimating the nonlinear model stochastic response Model validation involves comparing the time domain simulation Monte Carlo simulation and the SPLES response Therefore we consider a SDOF system that represents a structure described with the following parameters omegas 10 rads xis 002 A 1 beta 025 alpha 05 and n 1 The soil profile is described as a firm soil with omegaf 169 rads and xis 094 Table 1 The mass ratio between the PTMD and primary structure is considered equal to 01 mu 01 The PTMD design parameters are fopt 0853 and the xiopt 0153 We have considered two levels of seismic intensity PGA 001 g Figure 6a and PGA 06 g Figure 6b and one thousand observations samples at each time Figure 6a and b show the structural response in terms of its standard deviation of the base displacement sigmaxs Figure 6a and b suggest no significant differences between the response of the nonlinear modal computed by Monte Carlo simulation and the estimation computed using the SPLES model when the system reaches the stationary state Furthermore Figure 6c shows no differences between the optimum frequency ratio achieved by the Monte Carlo simulation and solving the equivalent Figure 4 Optimization algorithm a The workflow of the optimization process b An illustration of the objective function characteristic response surface and the optimal OF f ξd values Figure 5 a An illustration of how the PTMD effectiveness changes if we keep constants the damping ratio and change de frequency ratio b An illustration of how the PTMD effectiveness changes if we keep constants the frequency ratio and change the damping ratio Figure 6 Comparison of the standard deviation of the primary system base displacement obtained by Monte Carlo simulation and the SPLES and 1000 samples at each instant a System response to PGA ¼ 001 g b System response to PGA ¼ 06 g c Structural response σxs calculated by the Monte Carlo method and solving the SPLES while searching the PTMD optimum design parameters VJ García et al Heliyon 7 2021 e07221 7 linear system Thus we use the equivalent linear system response in the steadystate to find the PTMD optimum design parameters and the mean structural response for different seismic intensities 27 PTMD seismic performance a case study The primary structure response with an optimized PTMD and without PTMD are compared upon been subjected to the Pedernales earthquake Thus the PTMD effectiveness was verified using the Pedernales earth quakes horizontal acceleration registered by the APED station at Ped ernales Ecuador The APED station is localized at Latitude 0068 Longitude 80057 Altitude 15 masl Epicentral distance 36 km PGA EW 138049 cms2 PGA NS 81270 cms2 and PGA Z 72738 cms2 Event 0001 Date UTM 2016 4 16 Registration time 1858 local time Component EW Sampling frequency 100 Hz Units cms2 40 3 Results 31 Optimum PTMD design parameters The PTMD design parameters depend mainly on the level of incursion into the elastoplastic behavior by the primary structure Figure 7 The mean value of the standard deviation of the primary structure base displacement σxs is related to the level of elastoplastic incursion Therefore the PTMD is highly effective when its design parameters have been optimized to show the best performance in the elastic region σxs mean values close to zero Figure 7a In comparison the PTMD effec tiveness design value decreases when the PTMD design parameters are optimized considering its performance in the region with high elasto plastic behavior high σxs mean values Figure 7a The structure with the lowest damping ratio naturally removes more vibratory energy and reduces oscillation amplitude Unlike a structure that remains in the elastic range a structure that has a substantial incursion into the plastic range dissipates much energy due to the plastic behavior of its structural elements so that the energy dissipated by the PTMD is marginal Furthermore the results show that the removal of vibratory energy is enhanced with the PTMD The PTMD effectiveness decreases as the pri mary structure damping ratio increases However these results suggest that a low structure damping ratio makes the PTMD more effective when the structure exhibits elastoplastic behavior Therefore the primary structure ductility and softening development affect the PTMD perfor mance mainly when the structure exhibits a high level of incursion into the elastoplastic behavior Figure 7b shows how the design value of the PTMD frequency ratio decreased when the considered degree of incursion into the elastoplastic behavior increases However the primary structure damping ratio slightly affects the design frequency ratio An increment of the primary structure damping ratio decreases the optimum design frequency ratio The PTMD optimum design damping ratio looks less sensitive to the primary structure damping ratio Figure 7c Similarly the optimum design PTMD damping ratio increases and changes according to the seismic intensity design criterion However the optimum design PTMD damping ratio does not change with the soils dynamic properties used in the simulation The PTMD design parameters for structures with damping ratios of 002 and 005 show that they are not affected by the dynamic parameters of the three soil conditions considered reported by Sues et al 24 32 PTMDs effectiveness design criterion The PTMD exhibits high effectivity above 45 reducing primary structure base displacement when the maximum seismic design criterion is low However the PTMD effectiveness values decrease between 10 and 20 when the maximum seismic design criterion increases Figure 8b The high effectiveness design criterion values still signifi cant especially at high values of the standard deviation of the structure displacement high PGA values Figure 8b Even when the standard Figure 7 Optimum PTMD design parameters effectiveness OF a frequency ratio b and damping ratio c versus the standard deviation of the primary structure displacement seismic intensity criterion VJ García et al Heliyon 7 2021 e07221 8 Figure 8 a Primary structure hysteresis cycle without PTMD to different design seismic intensities criterion b PTMDs effectiveness in reducing the primary structure base displacement c Primary structure hysteresis cycle with PTMD to different design seismic intensities The primary structure parameters were ωs ¼ 10 rads and ξs ¼ 002 The PTMD mass ratio design value was μ ¼ 01 The soils dynamic parameters were ωf ¼ 169 rads and ξf ¼ 095 Firm soil Figure 9 Optimum PTMD design parameters effectiveness OF a frequency ratio b and damping ratio c versus the standard deviation of the primary structure displacement seismic intensity criterion For these calculations we used the dynamic soil properties proposed by Greco and Marano 27 and Marano and Greco 28 Table 1 The structure frequency is ωs ¼ 10 rads and the PTMD design mass ratio is μ ¼ 01 VJ García et al Heliyon 7 2021 e07221 9 Figure 10 Optimum PTMD design parameters frequency ratio OF a damping ratio b and effectiveness c versus the design mass ratio μ pendulum mass d The optimum PTMD effectiveness design parameter values vs the standard deviation of the structures displacement and the design mass ratio μ pendulum mass The structure frequency is ωs ¼ 10 rads The soil parameters are ωf ¼ 169 rads and ξf ¼ 095 Firm soil Figure 11 a The displacement history of the primary structure without and with the optimized PTMD b Primary structure hysteresis loops without and with the optimized PTMD c A short segment magnification of the primary structure time response VJ García et al Heliyon 7 2021 e07221 10 deviation of the structure displacement is high these relatively high design effectiveness values are of great relevance to reduce structural damage in structural elements Figure 8a shows the elastoplastic behavior of the primary structure response without the PTMD when the maximum seismic design criterion changes from low to high seismic intensity Similarly Figure 8c shows the primary structure response with the PTMD Figure 8a and c display elastoplastic behavior characteristic hysteresis loop The area enclosed by each loop is a measure of the energy dissipated due to plasticity in structural members When the seismic intensity criterion value is close to zero the hysteresis loops are slim and the energy dissipated is minimum This hysteresis loop occurs when the primary structure is loaded within its elastic range The high PTMD effectiveness in this elastic range justifies its frequent application to reduce vibration in the highly elastic primary structure However when the primary structure is loaded at high levels of inelastic behavior the dissipated energy becomes more apparent The hysteresis loops are significantly larger Moreover the loops display the charac teristic pointed shape shown in Figures 8a and c nonlinear hysteresis loops 39 However the center of symmetry of the hysteresis loops does not remain centered at the origin of the coordinated axis We expected a reduction in the hysteresis loops area of the primary structure with the optimized PTMD Figure 8c However their hyster esis loops show a shifting of the center of the loop This shifting made it harder to observe a reduction in the loops areas The presence of this area suggests that the PTMD controls the development of structural plasticity and protects the primary structures safety see Figure 8c PGA ¼ 005 We will study the optimized PTMD attached to an elastoplastic primary structure in a different section Furthermore Figure 8c suggests a larger shift of the center of the hysteresis loop of the structure with PTMD This shift is associated with a more significant base displacement This shift appears to be induced by the PTMD attached to an elastoplastic primary structure These larger base displacements could have a detrimental effect on the primary structure However the average standard deviation of the base displacement is reduced regarding the structure without the PTMD 33 Soils dynamic properties effects The effect of soils dynamic properties on PTMD design parameters values was studied using the soils dynamic properties reported by Greco and Marano 27 and Marano and Greco 28 Table 1 Figure 9 shows the PTMD design parameters values as a function of the mean values of the standard deviation of the primary structure base displacement Figure 9 confirms the results shown in Figure 7 for the firm and medium soil In contrast the PTMD design effectiveness value is around 8 when the primary structure is above the soft soil Reference 28 in Table 1 The substantial design effectiveness reduction may be due to the fre quency that characterizes the soils dynamic properties 45 rads This frequency value is lower than the primary structures frequency of 10 rads and the PTMDs optimization frequency of about 35 rads How ever the soils damping ratio is 01 the lowest of all soils considered in this study It has the highest damping capacity and removes oscillatory energy from the excitation Therefore the PTMD device may behave more like a dissipater than an absorber This result suggests an interac tion between the soils dynamic properties and the optimization param eters of the PTMD device which should be studied further The design effectiveness values Figure 9a display a behavior opposite to other soil considered in this study The design effectiveness value is lower in the region where the linear hysteresis loop is apparent However the effectiveness increases and reaches the value of 15 when the seismic intensity is between 03 and 04 g This value remains constant and equal to 15 for higher seismic intensity where the nonlinear hysteresis loop is evident These results suggest that PTMD performance turns up to be independent of the primary structure plas ticity development and softening The optimum design frequency ratio Figure 9b remains constant around 035 and seismic intensity inde pendent Therefore the PTMD optimum design frequency is 35 rads and this value is around 78 of the soil frequency 45 rads The PTMD design frequency value is around 35 of the primary structure frequency 10 rads These results suggest that the PTMD remains tuned with the soil it is independent of the primary structure of elastoplastic behavior The PTMD optimum design damping ratio Figure 9c shows similar behavior to the optimum design frequency ratio Figure 9b The opti mum design damping ratio values remain constant and equal to 006 well below the values obtained with the other soils studied We recall that the soil damping ratio is 01 These results show a strong interaction between the soils dynamic properties and the optimization parameters of the PTMD device which should be studied further Salvi et al 41 reported on an optimum tuned mass damper under seismic soilstructure interaction They concluded that the soilstructure interaction effects require a dedicated TMD tuning specifically in the case of a soft soilfoundation system However these results open the way for further studies and engineering the PTMD one tuned with the structure and the other tuned with the soft soil 34 Optimum design mass ratio The results in Figure 10 suggest that the PTMD design effectiveness improves when the mass ratio pendulums mass increases Figure 10a shows that the optimum frequency ratio decreases when the design mass ratio increases The optimum design frequency ratio decreases as the primary structure damping ratio increases However Figure 10b shows that the optimum damping ratio increase when the optimum mass ratio increases The results in Figure 10b suggest an increment of the optimum damping ratio value when the primary structure damping ratio increases mainly to the high value of the design mass ratio Figure 10c shows how the design effectiveness depends on the design mass ratio PGA ¼ 08 The result shows that as the design mass ratio μ increases the effec tiveness of the optimized PTMD increases asymptotically Thus as the mass of the pendulum increases the optimized PTMD exhibits better performance However the maximum value of the design effectiveness depends on the primary structure damping ration value High design effectiveness values are reaches when with a lower primary structure damping ratio Figure 10c The differences between successive design effectiveness values decrease as the pendulum mass increases Very little effectiveness is gained when the pendulum mass is greater than 10 of the structures mass μ ¼ 01 Similar results were reported by Hassani and Aminafshar 19 They reported on the optimization of PTMD in a tall building under horizontal earthquake excitation They concluded that the mass ratios desirable range is between 004 to 01 Figure 10d validates the improvement of the PTMD design effectiveness when increases the mass ratio in a wide range of seismic intensities One remarkable fact is that the design effectiveness to high seismic intensities also increases which is relevant to avoid primary structure collapse and save a life 35 PTMD performance case of study We chose the recorded ground motions of the Pedernales earthquake to verify the control effect of the PTMD optimized following the meth odology described in this study Therefore the model was evaluated using timedomain simulation The displacement history of the primary structure was calculated without PTMD and with the optimized PTMD VJ García et al Heliyon 7 2021 e07221 11 Figure 11 shows the results obtained for optimized PTMD parameters for a peak ground acceleration of 05 g Figure 11a and c show a reduction of the primary structure displacement when the optimized PTMD protects the primary structure The magnitude of the primary structure displacement is larger when the structure is not protected Similarly the energy dissipated by the primary structure without PTMD is bigger than the energy dissipated by the primary structure with PTMD The primary structure without PTMDs hysteresis loop encloses more area than that enclosed by loops of the primary structure with PTMD Figure 11b These results suggest the PTMD is absorbing energy from the seismic excitation to reduce primary structure base displacement We conclude that the optimized PTMD is controlling the primary structure plastic demand and softening 4 Conclusions We developed models to estimate the stochastic response of the primary structure exhibiting elastoplastic behavior without and with PTMD subjected to filtered whitenoise ground acceleration Two strategies were implemented studying the system The Monte Carlo method and the other by finding the SPLES The two solutions were compared over time and in the steadystate showing a high level of agreement The optimal PTMD parameters were found by integrating the equations of motion of the nonlinear modal using Monte Carlo simulation and solving the SPLES for estimation of stationary response The comparison of estimated performances showed good agreement Thus the strategy with the linearized model was selected for the study to reduce computational cost Thus the optimization consisted of finding the SPLES constants and the PTMD frequency and damping ratios that reduce the mean value of the standard deviation of the primary structures displacements The optimum values of the PTMD frequency and damping ratio are used as the PTMD design values for a specific seismic intensity design criterion From the series of numerical simulations developed the following conclusions were drawn The values of the effectiveness criterion and the optimal values of the frequency ratio are higher for low damping ratios of the main structure The performance of the optimized PTMD is higher when the structure exhibits a linear hysteresis loop The optimized PTMD controls the development of structural plasticity The optimized PTMD reduces the mean value of the standard deviation of the primary structure displacement There is a strong interaction between the optimum PTMD parameters and dynamic soil properties in a structure omegas 10 rads xis 002 005 on soft soil with a damping ratio of the order of 01 and a frequency of 45 rads This result suggests that the PTMD is more sensitive to soil effects and less sensitive to the primary structure elastoplastic behavior in these circumstances However the PTMD effectiveness in this type of soil is about 10 and independent of the seismic intensity suggesting that the properties of soft soil could hamper the PTMDs performance as an absorber of vibrational energy The results have confirmed that the performance of the PTMD improves as its mass increases The optimum frequency ratio decreases and the damping ratio increases as the mass of the pendulum increases However in practice very little is gained in the effectiveness of the PTMD if the design mass ratio exceeds the value of 01 The simulations performed in the case study show that the optimized PTMD reduces the deformation of the structure excited with the Pedernales Ecuador earthquake record The PTMD designed and optimized with the proposed methodology reduces vibrations controls the development of plasticity and protects the primary structures safety on soil for a range of dynamic properties It should be kept in mind that the simulations result from substantial simplification They shed light on understanding the PTMD performance under several dynamic conditions It is hoped that the results will serve to guide further studies on the subject Declarations Author contribution statement Víctor J García Conceived and designed the experiments Analyzed and interpreted the data Wrote the paper Edwin P Duque Performed the experiments Contributed reagents materials analysis tools or data José Antonio Inaudi Carmen O Márquez Analyzed and interpreted the data Wrote the paper Josselyn D Mera Performed the experiments Analyzed and interpreted the data Anita C Rios Contributed reagents materials analysis tools or data Funding statement This research did not receive any specific grant from funding agencies in the public commercial or notforprofit sectors Data availability statement Data will be made available on request Declaration of interests statement The authors declare no conflict of interest Additional information No additional information is available for this paper Acknowledgements The authors express their gratitude to the ViceRectorate of Postgraduate Studies and Research of the National University of Chimborazo Unach through the Interdisciplinary Studies research group Any opinions findings conclusions or recommendations expressed in this publication are those of the authors They do not necessarily reflect the views of the sponsors The data used in this work were obtained from the National Accelerometer Network RENAC of the Instituto Geofísico at the Escuela Politécnica Nacional IGEPN The installation and expansion of RENAC as operated by the IGEPN was made possible utilizing funds from the Escuela Politécnica Nacional part of Proyecto SENESCYT PIN08EPNGEO0001 Fortalecimiento del Instituto Geofísico Ampliación y Modernización del Servicio Nacional de Sismología y Vulcanología and of investment project termed Generación de Capacidades para la Difusión de Alertas Tempranas y para el Desarrollo de Instrumentos de Decisión ante las Amenazas Sísmicas y Volcánicas dirigidos al Sistema Nacional de Gestión de 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