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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

  • For nonlinear problems only numerical solutions are possible.
  • Focus is on explicit methods, in particular Central Difference method.
  • LS-DYNA uses a modification of the central di erence time integration.
  • Central difference scheme is an explicit method.
  • For explicit schemes the equation of motion is evaluated at the old time step tn, whereas implicit methods use the equation of motion at the new time step tn+1.

Central Difference

  • discretization

209

  • difference formula :

210

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

211

  • For lumped mass and damping the matrices M are diagonal.

212

  • Inversion of diagonal matrices M and C is trivial.

213

  • At timer t = 0 we have initial conditions u0 and &u-odot0. From equilibrium we find ü0. From equation (??) and (??) :

214

  • The central di erence scheme is conditionally stable, i.e. the size of the time step is limited.

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

  • The time integration is stable, if the time step size decreases.
  • The time step is bounded by the largest natural frequency of the structure.
  • For shells: bending and membrane modes are present the frequency of the membrane mode usually limits the critical time step, since membrane stiffness is much larger than bending stiffness

Critical time step of a rog

226

Time Integration in LS-DYNA

227

  • discretization In LS-DYNA actual geometry x is used instead of displacements. Thus x replaces u.
  • difference formula :

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 :

  • starting procedure for first time step with

234

  • Standard central di erence method approximates time step limit for LS-DYNA time integration scheme.
  • No stability proofs are available for time integration of nonlinear problems. Default in LS-DYNA :

235

  • critical time step for varying time increments

236