ME751 Advanced Computational Multibody Dynamics

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1 ME751 Advanced Computational Multibody Dynamics November 9, 2016 Antonio Recuero University of Wisconsin-Madison

2 Quote of the Day I intend to live forever, or die trying. -Groucho Marx 2

3 Before we get started On Monday, we learned: Define constraints with ANCF finite elements and rigid bodies ANCF cable element kinematics ANCF cable element strains ANCF cable element inertia forces This lecture Wrap up implementation of ANCF cable element in Chrono Plates and shells Introduce Chrono s bilinear ANCF shell element Kinematics of ANCF shell element Curvilinear initial configuration Inertia forces Internal forces 3

strain variations for virtual work 1 T T rx x rr x x 1 e rx e e2 e For bending curvature: f rx rxx, f r r, g r 3 g r x x xx x 1 f g e

4 ANCF Cable: Virtual Work of Elastic Forces The virtual of the elastic forces for the gradient-deficient beam element may be defined as where W EA EId x e x x L 1 2 rr T x x x rx r 1 and = r xx 3 x are the axial Green-Lagrange strain and bending curvature Calculation of strain variations for virtual work 1 T T rx x rr x x 1 e rx e e2 e For bending curvature: f rx rxx, f r r, g r 3 g r x x xx x 1 f g e g f 2 e e g e e T r r f r r r r T e e rxrxx rxrxx e e r r r r g 3 1 T 2 T 2 T rx rr x x 3rr x x rx e e e x xx x xx x xx T 3 x xx x xx 4

5 b. Generalized internal forces Axial class MyForcesAxial : public ChIntegrable1D<ChMatrixNM<double, 12, 1> > { public: ChElementCableANCF* element; ChMatrixNM<double, 4, 3>* d; // this is an external matrix, use pointer ChMatrixNM<double, 12, 1>* d_dt; // this is an external matrix, use pointer ChMatrixNM<double, 3, 12> Sd; ChMatrixNM<double, 1, 4> N; ChMatrixNM<double, 1, 4> Nd; ChMatrixNM<double, 1, 12> straind; Declarations needed ChMatrixNM<double, 1, 1> strain; ChMatrixNM<double, 1, 3> Nd_d; ChMatrixNM<double, 12, 12> temp; /// Evaluate (straind'*strain) at point x virtual void Evaluate(ChMatrixNM<double, 12, 1>& result, const double x) { element >ShapeFunctionsDerivatives(Nd, x); Evaluate shape function derivatives at // Sd=[Nd1*eye(3) Nd2*eye(3) Nd3*eye(3) Nd4*eye(3)] ChMatrix33<> Sdi; Sdi.FillDiag(Nd(0)); Sd.PasteMatrix(&Sdi, 0, 0); Sdi.FillDiag(Nd(1)); Sd.PasteMatrix(&Sdi, 0, 3); Sdi.FillDiag(Nd(2)); Sd.PasteMatrix(&Sdi, 0, 6); Sdi.FillDiag(Nd(3)); Sd.PasteMatrix(&Sdi, 0, 9); Nd_d = Nd * (*d); straind = Nd_d * Sd; r x xx e 5

6 b. Generalized internal forces Axial strain.matrmultiplyt(nd_d, Nd_d); strain(0, 0) += 1; strain(0, 0) *= 0.5; Calculation of strain Optional addition of damping // Add damping forces if selected if (element >m_use_damping) strain(0, 0) += (element >m_alpha) * (straind * (*d_dt))(0, 0); result.matrtmultiply(straind, strain); // result: straind'*strain } }; MyForcesAxial myformulaax; myformulaax.d = &d; myformulaax.d_dt = &vel_vector; myformulaax.element = this; Result dependent on space parameter ChMatrixNM<double, 12, 1> Faxial; ChQuadrature::Integrate1D<ChMatrixNM<double, 12, 1> >(Faxial, // result of integration will go there myformulaax, // formula to integrate 0, // start of x Integration and addition to generalized force vector 1, // end of x 5 // order of integration ); Faxial *= E * Area * length; 6 Fi = Faxial;

7 c. Generalized internal forces. Bending strain /// Integrate (k_e'*k_e) class MyForcesCurv : public ChIntegrable1D<ChMatrixNM<double, 12, 1> > { public: ChElementCableANCF* element; ChMatrixNM<double, 4, 3>* d; // this is an external matrix, use pointer ChMatrixNM<double, 12, 1>* d_dt; // this is an external matrix, use pointer ChMatrixNM<double, 3, 12> Sd; ChMatrixNM<double, 3, 12> Sdd; ChMatrixNM<double, 1, 4> Nd; ChMatrixNM<double, 1, 4> Ndd; ChMatrixNM<double, 1, 3> r_x; ChMatrixNM<double, 1, 3> r_xx; ChMatrixNM<double, 1, 12> g_e; ChMatrixNM<double, 1, 12> f_e; ChMatrixNM<double, 1, 12> k_e; ChMatrixNM<double, 3, 12> fe1; Declarations f rx rxx, f r r, r 3 g rx 1 f g e g f 2 e e g e e 3 x xx g x First and second shape functions needed /// Evaluate at point x virtual void Evaluate(ChMatrixNM<double, 12, 1>& result, const double x) { element >ShapeFunctionsDerivatives(Nd, x); element >ShapeFunctionsDerivatives2(Ndd, x); 7

8 c. Generalized internal forces. Bending strain // Sd=[Nd1*eye(3) Nd2*eye(3) Nd3*eye(3) Nd4*eye(3)] // Sdd=[Ndd1*eye(3) Ndd2*eye(3) Ndd3*eye(3) Ndd4*eye(3)] ChMatrix33<> Sdi; Sdi.FillDiag(Nd(0)); Sd.PasteMatrix(&Sdi, 0, 0); Sdi.FillDiag(Nd(1)); Sd.PasteMatrix(&Sdi, 0, 3); Sdi.FillDiag(Nd(2)); Sd.PasteMatrix(&Sdi, 0, 6); Sdi.FillDiag(Nd(3)); Sd.PasteMatrix(&Sdi, 0, 9); Sdi.FillDiag(Ndd(0)); Sdd.PasteMatrix(&Sdi, 0, 0); Sdi.FillDiag(Ndd(1)); Sdd.PasteMatrix(&Sdi, 0, 3); Sdi.FillDiag(Ndd(2)); Sdd.PasteMatrix(&Sdi, 0, 6); Sdi.FillDiag(Ndd(3)); Sdd.PasteMatrix(&Sdi, 0, 9); Shape function matrices: First and second derivative r_x.matrmultiply(nd, (*d)); // r_x=d'*nd'; (transposed) r_xx.matrmultiply(ndd, (*d)); // r_xx=d'*ndd'; (transposed) r x, r xx 8

9 c. Generalized internal forces. Bending strain r_xx.matrmultiply(ndd, (*d)); // r_xx=d'*ndd'; (transposed) // if (r_xx.length()==0) // {r_xx(0)=0; r_xx(1)=1; r_xx(2)=0;} ChVector<> vr_x(r_x(0), r_x(1), r_x(2)); ChVector<> vr_xx(r_xx(0), r_xx(1), r_xx(2)); ChVector<> vf1 = Vcross(vr_x, vr_xx); double f = vf1.length(); double g1 = vr_x.length(); double g = pow(g1, 3); double k = f / g; g_e = (Nd * (*d)) * Sd; g_e *= (3 * g1); // do: fe1=cross(sd,r_xxrep)+cross(r_xrep,sdd); for (int col = 0; col < 12; ++col) { ChVector<> Sd_i = Sd.ClipVector(0, col); fe1.pastevector(vcross(sd_i, vr_xx), 0, col); ChVector<> Sdd_i = Sdd.ClipVector(0, col); fe1.pastesumvector(vcross(vr_x, Sdd_i), 0, col); } ChMatrixNM<double, 3, 1> f1; f1.pastevector(vf1, 0, 0); Building the bending strain formula and its variation to obtain generalized bending force f rx rxx, f r r, g r 3 g rx 1 f g e g f 2 e e g e e x xx x 3 if (f == 0) f_e.matrtmultiply(f1, fe1); else { f_e.matrtmultiply(f1, fe1); f_e *= (1 / f); } 9

10 c. Generalized internal forces. k_e = (f_e * g g_e * f) * (1 / (pow(g, 2))); // result: k_e'*k result.copyfrommatrixt(k_e); // Add damping if selected by user: curvature rate if (element >m_use_damping) k += (element >m_alpha) * (k_e * (*d_dt))(0, 0); Bending strain }; } result *= k; MyForcesCurv myformulacurv; myformulacurv.d = &d; myformulacurv.d_dt = &vel_vector; myformulacurv.element = this; f rx rxx, f r r, g r 3 g r x x xx x 1 f g e g f 2 e e g e e 3 ChMatrixNM<double, 12, 1> Fcurv; ChQuadrature::Integrate1D<ChMatrixNM<double, 12, 1> >(Fcurv, // result of integration will myformulacurv, // formula to integrate 0, // start of x 1, // end of x 3 // order of integration ); Fcurv *= E * I * length; // note Iyy should be the same value (circular section assumption) Fi += Fcurv; Integrate formula with order 3. Account for initial configuration. Multiply by material constants. } // Substract contribution of initial configuration Fi = this >m_genforcevec0; 10

11 1. Shells and Plates 11

12 1. Shells 12

13 1. Shells 13

14 1. Shells 14

15 2. ANCF Shell Elements 15

16 2. ANCF Shell Elements. Types 16

17 2. ANCF Shell Elements. Types 17

18 3. Chrono s ANCF Shell Element 18

19 3. Chrono s ANCF Shell Element 19

20 3. Chrono s ANCF Shell Element Element i Membrane strains 20

21 3. Chrono s ANCF Shell Element More on this later! 21

22 4. Shell strains: Orthotropic and curvilinear reference 22

23 4. Shell strains: Orthotropic and curvilinear reference 23

24 4. Shell strains: Orthotropic and curvilinear reference Current deformed reference 24

25 4. Shell strains: Orthotropic and curvilinear reference 25

26 4. Shell strains: Orthotropic and curvilinear reference 26

27 4. Shell strains: Orthotropic and curvilinear reference Final expression! These strains are related to constitutive equations 27

28 4. Shell strains: Orthotropic and curvilinear reference 28

29 5. Generalized forces Full and reduced 29

30 5. Generalized forces Beta matrices need manipulations to accommodate strain vector 30

31 6. Jacobian of internal forces Integrand of elastic energy 31

32 7. Mass matrix Internal parameters 32

33 8. Generalized external forces: Point load 33

34 8. Generalized external forces: Pressure Normal definition: dependent on element s kinematics 34

35 8. Generalized external forces: Gravity load 35

36 9. Locking issues: Convergence 36

ME751 Advanced Computational Multibody Dynamics

ME751 Advanced Computational Multibody Dynamics November 23, 2016 Antonio Recuero University of Wisconsin-Madison Before we get started Last time, we learned: Plates and shells Introduce Chrono s bilinear