Time integration

Equation of Motion

Single degree of freedom system

201

Forces acting on mass m

202

Equilibrium – d Alembert s principle : 203

204

Equation of motion for linear elasticity :

205 linear ordinary d.e.

Nonlinear case :

206nonlinear ordinary d.e.

Analytical solutions of linear ordinary di erential equations are available.

207

Dynamic response of linear undamped system due to harmonic loading:

208

Direct Integration of Equation of Motion

Central Difference

209

210

Substitute equations (??) and (??) into (??) :

211

212

213

214

Stability of the Central Difference Scheme

Uncouple the system of linear equations of motion into the modal equations.

215

Φ … modal matrix with M-orthonormalized eigenvectors stored in columns N uncoupled equations of motion with generalized displacements χ :

216

central differences :

217

Substitute xn and xn into equation of motion () at time t^n* :

218

In matrix form

219

A… time integration operator for discrete

For m-time steps and L = 0

220

Spectral decomposition of A :

221

P … orthonormal matrix; contains eigenvectors of A

J … Jordan form; eigenvalues λi of A are stored on diagonal

spectral radius = ρ(A) = largest eigenvalue of A = max ((diag(J))

222

Eigenvalues of A for the undamped equation of motion

223

For the damped equation of motion :

224

Damping reduces the critical time step.

For varying time step sizes :

225

Critical time step of a rog

226

Time Integration in LS-DYNA

227

228

equation of motion at time t^n for the nonlinear case:

229

230

(asynchronous damping)

assumption :

231update of accelerations

update formulas for velocities and displacements:

From (2):

232

From (1):

233

Remarks :

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