Prova 2 Vibrações Mecânicas

BEAM DESIGN FORMULAS WITH SHEAR AND MOMENT DIAGRAMS DESIGN AID No 6 American Forest Paper Association American Wood Council AMERICAN WOOD COUNCIL The American Wood Council AWC is part of the wood products group of the American Forest Paper Association AFPA AFPA is the national trade association of the forest paper and wood products industry representing member companies engaged in growing harvesting and processing wood and wood fiber manufacturing pulp paper and paperboard products from both virgin and recycled fiber and producing engineered and traditional wood products For more information see wwwafandpaorg While every effort has been made to insure the accuracy of the information presented and special effort has been made to assure that the information reflects the stateof theart neither the American Forest Paper Association nor its members assume any responsibility for any particular design prepared from this publication Those using this document assume all liability from its use Copyright 2007 American Forest Paper Association Inc American Wood Council 1111 19th St NW Suite 800 Washington DC 20036 2024634713 awcinfoafandpaorg wwwawcorg BEAM F BEAM F BEAM F BEAM F BEAM FORMULAS WITH ORMULAS WITH ORMULAS WITH ORMULAS WITH ORMULAS WITH SHEAR AND MOMENT SHEAR AND MOMENT SHEAR AND MOMENT SHEAR AND MOMENT SHEAR AND MOMENT DIA DIA DIA DIA DIAGRAMS GRAMS GRAMS GRAMS GRAMS AMERICAN FOREST PAPER ASSOCIATION Figures 1 through 32 provide a series of shear and moment diagrams with accompanying formulas for design of beams under various static loading conditions Shear and moment diagrams and formulas are excerpted from the Western Woods Use Book 4th edition and are provided herein as a courtesy of Western Wood Products Association Intr Intr Intr Intr Introduction oduction oduction oduction oduction Notations Relative to Shear and Moment Diagrams E modulus of elasticity psi I moment of inertia in4 L span length of the bending member ft R span length of the bending member in M maximum bending moment inlbs P total concentrated load lbs R reaction load at bearing point lbs V shear force lbs W total uniform load lbs w load per unit length lbsin Δ deflection or deformation in x horizontal distance from reaction to point on beam in List of Figur List of Figur List of Figur List of Figur List of Figures es es es es Figure 1 Simple Beam Uniformly Distributed Load 4 Figure 2 Simple Beam Uniform Load Partially Distributed 4 Figure 3 Simple Beam Uniform Load Partially Distributed at One End 5 Figure 4 Simple Beam Uniform Load Partially Distributed at Each End 5 Figure 5 Simple Beam Load Increasing Uniformly to One End 6 Figure 6 Simple Beam Load Increasing Uniformly to Center 6 Figure 7 Simple Beam Concentrated Load at Center 7 Figure 8 Simple Beam Concentrated Load at Any Point 7 Figure 9 Simple Beam Two Equal Concentrated Loads Symmetrically Placed 8 Figure 10 Simple Beam Two Equal Concentrated Loads Unsymmetrically Placed 8 Figure 11 Simple Beam Two Unequal Concentrated Loads Unsymmetrically Placed 9 Figure 12 Cantilever Beam Uniformly Distributed Load 9 Figure 13 Cantilever Beam Concentrated Load at Free End 10 Figure 14 Cantilever Beam Concentrated Load at Any Point 10 Figure 15 Beam Fixed at One End Supported at Other Uniformly Distributed Load 11 Figure 16 Beam Fixed at One End Supported at Other Concentrated Load at Center 11 Figure 17 Beam Fixed at One End Supported at Other Concentrated Load at Any Point 12 Figure 18 Beam Overhanging One Support Uniformly Distributed Load 12 Figure 19 Beam Overhanging One Support Uniformly Distributed Load on Overhang 13 Figure 20 Beam Overhanging One Support Concentrated Load at End of Overhang 13 Figure 21 Beam Overhanging One Support Concentrated Load at Any Point Between Supports 14 Figure 22 Beam Overhanging Both Supports Unequal Overhangs Uniformly Distributed Load 14 Figure 23 Beam Fixed at Both Ends Uniformly Distributed Load 15 Figure 24 Beam Fixed at Both Ends Concentrated Load at Center 15 Figure 25 Beam Fixed at Both Ends Concentrated Load at Any Point 16 Figure 26 Continuous Beam Two Equal Spans Uniform Load on One Span 16 Figure 27 Continuous Beam Two Equal Spans Concentrated Load at Center of One Span 17 Figure 28 Continuous Beam Two Equal Spans Concentrated Load at Any Point 17 Figure 29 Continuous Beam Two Equal Spans Uniformly Distributed Load 18 Figure 30 Continuous Beam Two Equal Spans Two Equal Concentrated Loads Symmetrically Placed 18 Figure 31 Continuous Beam Two Unequal Spans Uniformly Distributed Load 19 Figure 32 Continuous Beam Two Unequal Spans Concentrated Load on Each Span Symmetrically Placed 19 Figure 1 Simple Beam Uniformly Distributed Load R V wl 2 Vx w ℓ2 x Mmax at center wl2 8 Mx wx 2 ℓ x Δmax at center 5wl4 384 EI Δx wx 24 EI ℓ3 2ℓx2 x3 AMERICAN WOOD COUNCIL Figure 2 Simple Beam Uniform Load Partially Distributed R1 V1 max when a c wb 2ℓ 2c b R2 V2 max when a c wb 2ℓ 2a b Vx when x a and a b R1 wx a Mmax at x a R1w R1 a R1 2w Mx when x a R1x Mx when x a and a b R1x w2 x a2 Mx when x a b R2ℓ x Figure 3 Simple Beam Uniform Load Partially Distributed at One End R1 V1 wa 2ℓ 2ℓ a R2 V2 wa2 2ℓ Vx when x a R1 wx Mmax at x R1w R12 2w Mx when x a R1x wx2 2 Mx when x a R2ℓ x Δx when x a wx 24 EI a2 2ℓ a2 2ax2 2ℓ a ℓx3 Δx when x a wa2 ℓ x 24 EI 4xℓ 2x2 a2 AMERICAN FOREST PAPER ASSOCIATION Figure 4 Simple Beam Uniform Load Partially Distributed at Each End R1 V1 w1a 2ℓ a w2c2 2ℓ R2 V2 w2c 2ℓ c w1a2 2ℓ Vx when x a R1 w1x Vx when x a and a b R1 w1a Vx when x a b R2 w2ℓ x Mmax at x R1w1 when R1 w1a R12 2w1 Mmax at x ℓ R2w2 when R2 w2c R22 2w2 Mx when x a R1x w1x2 2 Mx when x a and a b R1x w1a 22x a Mx when x a b R2ℓ x w2ℓ x2 2 Figure 5 Simple Beam Load Increasing Uniformly to One End R1 V1 W3 R2 V2 2W3 Vx W3 Wx2 l2 Mmax at x lsqrt3 5774l 2Wl9sqrt3 1283Wl Mx Wx3l2 l2 x2 Δmax at x l sqrt1 815 5193l 01304 Wl3 EI Δx Wx180EI l2 3x4 10l2x2 7l4 Figure 6 Simple Beam Load Increasing Uniformly to Center R V W2 Vx when x l2 W2l2 l2 4x2 Mmax at center Wl6 Mx when x l2 Wx 12 2x23l2 Δmax at center Wl360EI Δx Wx480EI l2 5l2 4x22 Figure 7 Simple Beam Concentrated Load at Center R V P2 Mmax at point of load Pl4 Mx when x l2 Px2 Δmax at point of load Pl348EI Δx when x l2 Px48EI 3l2 4x2 Figure 8 Simple Beam Concentrated Load at Any Point R1 V1 max when a b Pbl R2 V2 max when a b Pal Mmax at point of load Pabl Mx when x b Pbxl Δmax at x sqrtaa 2b3 when a b Paba 2b sqrt3aa 2b27EI l Δa at point of load Pa2b23EI l Δx when x a Pbx6EI l l2 b2 x2 Δx when x a Pal x6EI l 2lx x2 a2 Figure 11 Simple Beam Two Unequal Concentrated Loads Unsymmetrically Placed R1 V1 P1l a P2 b l R2 V2 P1 a P2 l b l Vx when x a and l b R1 P1 M1 max when R1 P1 R1 a M2 max when R2 P2 R2 b Mx when x a R1 x Mx when x a and l b R1 x P1x a Figure 12 Cantilever Beam Uniformly Distributed Load R V wl Vx wx Mmax at fixed end wl2 2 Mx wx2 2 Δmax at free end wl4 8EI Δx w 24EI x4 4 l3 x 3 l4 Figure 9 Simple Beam Two Equal Concentrated Loads Symmetrically Placed R V P Mmax between loads Pa Mx when x a Px Δmax at center Pa24EI 3l2 4a2 Δx when x a Px6EI 3la 3a2 x2 Δx when x a and l a Pa6EI 3lx 3x2 a2 Figure 13 Cantilever Beam Concentrated Load at Free End R V P Mmax at fixed end P l Mx P x Δmax at free end P l3 3EI Δx P 6EI 2 l3 3 l2 x x3 Figure 14 Cantilever Beam Concentrated Load at Any Point R V P Mmax at fixed end P b Mx when x a P x a Δmax at free end Pb2 6EI 3 l b Δa at point of load Pb3 3EI Δx when x a Pb2 6EI 3 l 3 x b Δx when x a P l x2 6EI 3 b l x Figure 10 Simple Beam Two Equal Concentrated Loads Unsymmetrically Placed R1 V1 max when a b Pl l a b R2 V2 max when a b Pl l b a Vx when x a and l b Pl b a M1 max when a b R1 a M2 max when a b R2 b Mx when x a R1 x Mx when x a and l b R1 x Px a Figure 15 Beam Fixed at One End Supported at Other Uniformly Distributed Load R1 V1 3wl 8 R2 V2 5wl 8 Vx R1 wx Mmax wl2 8 M1 at x 38 l 9128 wl2 Mx R1 x wx2 2 Δmax at x l16 1 33 4215 l wl4 185EI Δx wx 48EI l3 3l x2 2x3 Figure 19 Beam Overhanging One Support Uniformly Distributed Load on Overhang R1 V1 wa22ℓ R2 V1 V2 wa2ℓ 2ℓ a V2 wa Vx1 for overhang wa x1 Mmax at R2 wa22 Mx between supports wa2 x2ℓ Mx1 for overhang w2 a x12 Δmax between supports at x ℓ 3 wa2 ℓ2 183EI 03208 wa2 ℓ2 EI Δmax for overhang at x1 a wa3 24EI 4ℓ 3a Δx between supports wa2 x 12EI ℓ ℓ2 x2 Δx1 for overhang wx1 24EI 4a2 ℓ ℓ3 6a2 x1 4ax12 x13 Figure 20 Beam Overhanging One Support Concentrated Load at End of Overhang R1 V1 Pa ℓ R2 V1 V2 P ℓ ℓ a V2 P Mmax at R2 Pa Mx between supports Pax ℓ Mx1 for overhang Pa x1 Δmax between supports at x ℓ 3 Paℓ2 93EI 06415 Paℓ2 EI Δmax for overhang at x1 a Pa2 3EI ℓ a Δx between supports Pax 6EI ℓ ℓ2 x2 Δx1 for overhang Px1 6EI 2aℓ 3ax1 x12 AMERICAN FOREST PAPER ASSOCIATION Figure 23 Beam Fixed at Both Ends Uniformly Distributed Load RVwl2 Vxwl2x Mmax at ends wl212 Ml at center wl224 Mx w12 6lx l2 6x2 Δmax at center wl4384EI Δx wx224EI lx2 Figure 24 Beam Fixed at Both Ends Concentrated Load at Center RVP2 Mmax at center and ends P l 8 Mx when x l2 P8 4x l Δmax at center P l3 192EI Δx when x l 2 Px248EI 3l 4x Figure 25 Beam Fixed at Both Ends Concentrated Load at Any Point R1 V1 max when a b Pb2l3 3ab R2 V2 max when a b Pa2l3 a 3b M1 max when a b Pab2l2 M2 max when a b Pa2bl2 Ma at point of load 2Pa2b2l3 Mx when x a R1x Pab2l2 Δmax when a b at x 2al3ab 2Pa3b23EI3ab2 Δa at point of load Pa3b33EI l3 Δx when x a Pb2 x2 6EI l3 3al 3ax bx Figure 26 Continuous Beam Two Equal Spans Uniform Load on One Span R1 V1 716 wl R2 V2 V3 58 wl R3 V3 116 wl V2 916 wl Mmax at x 716 l 49512 wl2 Ml at support R2 116 wl2 Mx when x l wx16 7l 8x Figure 29 Continuous Beam Two Equal Spans Uniformly Distributed Load R₁ V₁ R₃ V₃ 3wℓ 8 R₂ 10wℓ 8 V₂ Vmax 5wℓ 8 M₁ wℓ² 8 M₂ at 3ℓ 8 9wℓ² 128 Δmax at 04215 ℓ approx from R₁ and R₃ wℓ⁴ 185EI Figure 27 Continuous Beam Two Equal Spans Concentrated Load at Center of One Span R1 V1 1332 P R2 V2 V3 1116 P R3 V3 332 P V2 1932 P Mmax at point of load 1364 Pl Ml at support R2 332 Pl Figure 30 Continuous Beam Two Equal Spans Two Equal Concentrated Loads Symmetrically Placed R₁ V₁ R₃ V₃ 5P 16 R₂ 2V₂ 11P 8 V₂ P R₁ 11P 16 Vmax V₂ M₁ 3Pℓ 16 M₂ 5Pℓ 32 Mₓ when x a R₁x A F P A American F American F American F American F American Forest P orest P orest P orest P orest Paper Association aper Association aper Association aper Association aper Association American W American W American W American W American Wood Council ood Council ood Council ood Council ood Council 1111 19th S 1111 19th S 1111 19th S 1111 19th S 1111 19th Stree tree tree tree treet NW t NW t NW t NW t NW Suit Suit Suit Suit Suite 800 e 800 e 800 e 800 e 800 W W W W Washingt ashingt ashingt ashingt ashington DC 20036 on DC 20036 on DC 20036 on DC 20036 on DC 20036 Phone 2024634 Phone 2024634 Phone 2024634 Phone 2024634 Phone 2024634777771111133333 FFFFFax 2024632 ax 2024632 ax 2024632 ax 2024632 ax 202463279 79 79 79 7911111 aaaaawwwwwcinf cinf cinf cinf cinfoafandpaorg oafandpaorg oafandpaorg oafandpaorg oafandpaorg www www www www wwwa aa aawwwwwcorg corg corg corg corg 1107 Figure 28 Continuous Beam Two Equal Spans Concentrated Load at Any Point R1 V1 Pb4l3 4l2 ala R2 V2 V3 Pa2l3 2l2 bla R3 V3 Pab4l3 la V2 Pa4l3 4l2 bla Mmax at point of load Pab4l3 4l2 ala Ml at support R2 Pab4l2 la 2a Prova de Vibracoes Mecˆanicas EMA006 turma N Parte Pratica 19 Dezembro de 2024 Nome No Na questao a seguir responda a cada um dos itens usando um editor de texto Nao envie o documento de resposta manuscrito Coloque apenas a resposta solicitada usando preferencialmente graficos e ou tabelas nao e necessaria a deducao ou desenvolvimento da ex pressao apenas o resultado SOLICITADO No documento de respostas indique claramente cada item da questao Nas respostas use sempre 3 algarismos significativos em notacao ci entıfica ou exponencial base 10 exemplo escreva 305 104 em lugar de 000030523987 O documento de respostas indicando cada item da questao deve estar no formato pdf A data de entrega e dia 15 de Janeiro ate as 20 horas na area do Moodle aberta para a prova pratica Envie o documento de resposta na area apropriada do Moodle Nao serao aceitos ou considerados outras formas de envio 1a A Figura abaixo mostra a estrutura de uma asa cujo modelo simplificado e uma viga engastada livre A rigidez de flexao e EI34 106 Nm2 Montada na longarina ha o motor da aeronave e 4 outros pontos representam a distribuicao de massa da asa M1200 kg M220 kg M320 kg M415 kg e M510 kg espacadas como indicado sendo o comprimento total L10 metros Deter mine a as equacoes de movimento expressao simbolica b as equacoes matriciais equivalentes expressao simbolica c a matriz de massa expressao simbolica d a matriz de flexibilidade L1 L2 L3 L4 L5 L5 expressao simbolica e a matriz de rigidezdeve satisfazer os criterios para ser simetrica e positiva definida f as frequˆencias naturais nao amortecidas g os modos correspondentes as frequˆencias naturais nao amortecidas matriz modal h a matriz modal normalizada pela massa i a matriz de amortecimento desacoplada Cnmatriz diagonal se para a estrutura os amortecimentos modais foram experimentalmente ξ1 008 ξ2 001 ξ3 0007 ξ4 0002 e ξ5 0002 i as amplitudes de resposta X1 X2 X3 X4 e X5 x1t X1 cosωet x2t X2 cosωet x3t X3 cosωet x4t X4 cosωet x5t X5 cosωet para uma forca de excitacao f1t F1 cosωet f2t f3t f4t f5t 0 sendo a frequˆencia de excitacao ωe 200 rads e F1 1000 N6 pontos Figura 1 Desenho esquematico da Asa 2a Use a Figura 1 e os dados do problema anterior para a estrutura engastadalivre amorteci mento modal medida experimental diagonal da matrix de amortecimento desacoplada 2ξkωk para ξ1 008 ξ2 001 ξ3 0007 ξ4 0002 e ξ5 0002 Considere o carregamento f1t Fo Ut f2t f3t f4t f5t 0 sendo Ut a funcao degrau unitario e Fo1000 N Determine a o vetor de forcas em coordenadas principais Q XFo1 ponto b trace os graficos individuais de resposta no tempo de 0 a 10 segundos para coordenadas prin cipais q1t q2t q3t q4t e q5t indique o metodo numerico empregado 4 pontos c as respostas temporais xit Xqit nas coordenadas x1 x2 x3 x4 e x5 Tracar um unico grafico com o comportamento no tempo das variaveis xii 1 2 5 O grafico de deslocamen tos deve permitir a visualizacao de todas as curvas use cores diferentes faca uma legenda 5 pontos Figure 31 Continuous Beam Two Unequal Spans Uniformly Distributed Load R₁ M₁ ℓ₁ wℓ₁ 2 R₂ wℓ₁ wℓ₂ R₁ R₃ R₃ V₄ M₁ ℓ₂ wℓ₂ 2 V₁ R₁ V₂ wℓ₁ R₁ V₃ wℓ₂ R₃ V₄ R₃ M₁ wℓ₂³ wℓ₁³ 8ℓ₁ ℓ₂ Mₓ₁ when x₁ R₁ w R₁x₁ wx₁² 2 Mₓ₂ when x₂ R₃ w R₃x₂ wx₂² 2 Figure 32 Continuous Beam Two Unequal Spans Concentrated Load on Each Span Symmetrically Placed R₁ M₁ ℓ₁ P₁ 2 R₂ P₁ P₂ R₁ R₃ R₃ M₁ ℓ₂ P₂ 2 V₁ R₁ V₂ P₁ R₁ V₃ P₂ R₃ V₄ R₃ M₁ 3 16 P₁ℓ₁² P₂ℓ₂² ℓ₁ ℓ₂ Mₘ₁ R₁a Mₘ₂ R₃b Parte Prática da 2ª Prova Na sequência os procedimentos para o caso da função de excitação ser uma função arbitrária Apenas um método não o mais simples existem outros M K e Matriz modal normalizada pela massa dos últimos exemplos numéricos enviados aos alunos matrizes a serem introduzidas na unha Cada linha sepa rada por seguida de virgula Os coeficientes dentro de uma linha devem ser separados por vírgula i1 Mmatrix 60 0 0 0 0 0 20 0 0 0 0 0 20 0 0 0 0 0 100 0 0 0 0 0 20 60 0 0 0 0 0 20 0 0 0 0 0 20 0 0 0 0 0 100 0 0 0 0 0 20 o1 Para executar um comando selecione shiftenter acionando enter apenas muda de linha na célula de comandos i2 k1e7 100000000 o2 i3 Kmatrix 4kk000k2kk00 0k3kk0 00k3kk000 k4k 40107 100000000 0 0 0 100000000 20107 100000000 0 0 0 100000000 30107 100000000 0 0 0 100000000 30107 100000000 0 0 0 100000000 40107 o3 Do scilab a Matriz Y ou matriz modal normalizada pela massa Na unha 1 i4 Ymatrix 00164628 00955126 00836241 00166549 00016179 00446632 0128517 01318098 01180637 0013256 00537029 00478075 01022436 01836128 00257492 00934066 00273954 00151232 00097803 00141541 0026157 00087942 00070628 00322006 0219435 00164628 00955126 00836241 00166549 00016179 00446632 0128517 01318098 01180637 0013256 00537029 00478075 01022436 01836128 00257492 00934066 00273954 00151232 00097803 00141541 0026157 00087942 00070628 00322006 0219435 o4 O procedimento de mudar de coordenadas premultiplicar por Y e pósmultiplicar por Y ao passar as matrizes de Massa amortecimento e rigidez para o sistema de coordenadas principais SEMPRE tem por resultado matrizes diagonais com termos diagonais conhecidos É uma propriedade decorrente da ortogonalidade das matrizes modais e o resultado é SEMPRE o mesmo os valores podem mudar com os coeficientes mas os parâmetros ωn ξn etc sempre serão encon trados no mesmo grau de liberdade n Para resolver o problema é necessário passar o vetor de forças para o sistema de coordenadas principais o vetor de forças é o único termo que realmente muda de um problema para outro i5 ftFodeltadiract ft Fo deltadiract o5 supondo a excitação pela função impulso unitário no 3o GDL Para executar dois comandos seguidos na mesma célula separe os comandos por ponto e virgula o enter muda de linha Para executar a célula de comandos acione simultaneamente shiftenter i7 Fcmatrix00100 FFoFcdeltadiract 0 0 1 0 0 o6 0 0 Fo deltadiract 0 0 o7 2 A força no sistema de coordenadas principais será i8 FcptransposeYF 00537029Fodeltadiract 00478075Fodeltadiract 01022436Fodeltadiract 01836128Fodeltadiract 00257492Fodeltadiract o8 Y é a matriz modal normalizada pela massa no maxima a transposta é transposeY e não Y A equação a ser resolvida por exemplo para o segundo grau de liberdade será i9 eq2diffqtt22ξ2ω2diffqttω2 2qt00478075Fodeltadiract qtω222ddt qtξ2ω2d2dt2 qt 00478075Fodeltadiract o9 esta é uma equação diferencial ordinária com coeficientes constantes A solução geral é conhecida i10 qteξωntFosinωnsqrt1ξ2tmωnsqrt1ξ2 qt Foetξ ωn sin t1 ξ2ωnm1 ξ2ωn o10 No sistema de coordenadas principais a massa é unitária e para o segundo grau de liberdade a rigidez é ω2 i11 q2te ξ2ω2t00478075Fosinω2sqrt1ξ2 2tω2sqrt1ξ2 2 q2t 00478075Foet ξ2 ω2 sin t1 ξ2 2ω21 ξ2 2ω2 o11 Para simplificar a solução será usado o vetor de forças assim definido i12 GtransposeYFc 00537029 00478075 01022436 01836128 00257492 o12 3 Apenas para evitar erros vamos definir as funções qnt qGnξnωnt i17 ξ1008 ξ2001 ξ30007 ξ40002 ξ50001 008 o13 001 o14 0007 o15 0002 o16 0001 o17 i18 ksiξ1ξ2ξ3ξ4ξ5 008 001 0007 0002 0001 o18 Supondo já conhecidas as frequencias naturais para este problema ver exemplo numérico i19 w 21520794 25790265 31048965 36870871 37756728 21520794 25790265 31048965 36870871 37756728 o19 i20 qξωFteξωtFsinωsqrt1ξ2tωsqrt1ξ2 qξωFt γ₀eξωt F sin ω1 ξ2 tω1 ξ2 o20 Nesse problema vamos considerar Fo1 para valores diferentes multiplique o vetor G por este valor GFo nos casos a seguir 4 i21 q1tqksi1w1G11t q1t 0002503419649882092e172166352t sin 2145181691872129t o21 as aspas garantem que a função tenha os valores corretos dos parâmetros ksi w e G i22 q2tqksi2w2G21t q2t 0001853796017772328e025790265t sin 2578897545451056t o22 i23 q3tqksi3w3G31t q3t 0003293060012153749e0217342755t sin 310482042910387t o23 i24 q4tqksi4w4G41t q4t 0004979897741680734e0073741742t sin 3687079725818426t o24 i25 q5tqksi5w5G51t q5t 6819768088646255104e0037756728t sin 3775670912163128t o25 5 i26 wxplot2dq1tt03y00050005 tempo de 0 a 3 coordenada y 0005 t26 o26 A solução temporal será então o vetor xtYqntYQ Q é o vetor ou matriz coluna com as soluções temporais i27 Qmatrixq1tq2tq3tq4tq5t 0002503419649882092e172166352t sin 2145181691872129t 0001853796017772328e025790265t sin 2578897545451056t 0003293060012153749e0217342755t sin 310482042910387t 0004979897741680734e0073741742t sin 3687079725818426t 6819768088646255104e0037756728t sin 3775670912163128t o27 Xis é o vetor com a solução temporal em cada coordenada ou GDL que é influ enciado pelo comportamento em todos os outros GDL 6 i28 XisYQ 1103370279062077106e0037756728t sin 3775670912163128t 8293969889791846105e0073741742t sin 3687079725818426t 2753791797623463104e0217342755t sin 310482042910387t 1770608775270813104e025790265t sin 2578897545451056t 412132970120789105e172166352t sin 2145181691872129t 9040284578309477106e0037756728t sin 3775670912163128t 5879451530044716104e0073741742t sin 3687079725818426t 4340575815899832104e0217342755t sin 310482042910387t 2382443028160463104e025790265t sin 2578897545451056t 1118107325066138104e172166352t sin 2145181691872129t 1756035724681701105e0037756728t sin 3775670912163128t 9143729680636762104e0073741742t sin 3687079725818426t 3366943106586431104e0217342755t sin 310482042910387t 8862535311965061105e025790265t sin 2578897545451056t 1344408951156529104e172166352t sin 2145181691872129t 9652767950350796106e0037756728t sin 3775670912163128t 4870489388296008105e0073741742t sin 3687079725818426t 4980160517580358105e0217342755t sin 310482042910387t 5078548342528005105e025790265t sin 2578897545451056t 2338359178686766104e172166352t sin 2145181691872129t 1496495810532091104e0037756728t sin 3775670912163128t 1603556952207646104e0073741742t sin 3687079725818426t 232582242538395105e0217342755t sin 310482042910387t 1630265293949341105e025790265t sin 2578897545451056t 6548194778196589105e172166352t sin 2145181691872129t o28 i33 x1tXis11 primeira linha de Xis ou solução para a primeira coorde nada x2tXis21 segunda linha de Xis ou solução para a segunda coorde nada x3tXis31 solução para a terceira coordenada x4tXis41 para a quarta coordenada x5tXis51 quinta coordenada generalisada x1t 1103370279062077106e0037756728t sin 3775670912163128t8293969889791846105e0073741742t sin 3687079725818426t2753791797623463104e0217342755t sin 310482042910387t1770608775270813104e025790265t sin 2578897545451056t412132970120789105e172166352t sin 2145181691872129t o29 x2t 9040284578309477106e0037756728t sin 3775670912163128t5879451530044716104e0073741742t sin 3687079725818426t4340575815899832104e0217342755t sin 310482042910387t2382443028160463104e025790265t sin 2578897545451056t1118107325066138104e172166352t sin 2145181691872129t o30 x3t 1756035724681701105e0037756728t sin 3775670912163128t9143729680636762104e0073741742t sin 3687079725818426t3366943106586431104e0217342755t sin 310482042910387t8862535311965061105e025790265t sin 2578897545451056t1344408951156529104e172166352t sin 2145181691872129t o31 x4t 9652767950350796106e0037756728t sin 3775670912163128t4870489388296008105e0073741742t sin 3687079725818426t4980160517580358105e0217342755t sin 310482042910387t5078548342528005105e025790265t sin 2578897545451056t2338359178686766104e172166352t sin 2145181691872129t o32 x5t 1496495810532091104e0037756728t sin 3775670912163128t1603556952207646104e0073741742t sin 3687079725818426t232582242538395105e0217342755t sin 310482042910387t1630265293949341105e025790265t sin 2578897545451056t6548194778196589105e172166352t sin 2145181691872129t o33 Plotando os resultados para t de 0 a 50 e coordenada y de 002 a 002 a função ajusta a escala para valores menores e trunca para os valores superiores aos prescritos 7 i34 wxplot2dx1tx2tt050z002002nticks100 t34 8 i35 wxplot2dx3tx4t t050 z002002 nticks100 t35 9 i36 wxplot2dx5t t040 y0001500015 nticks100 t36 10 i37 wxplot2dx1tx2tx3tx4tx5t t040 y0001500015 nticks100 t37 11