Single degree of freedom system

Forces acting on mass m

Equilibrium – d Alembert s principle :

Equation of motion for linear elasticity :

linear ordinary d.e.

Nonlinear case :

nonlinear ordinary d.e.

Analytical solutions of __linear__ ordinary di erential equations are available.

Dynamic response of linear undamped system due to harmonic loading:

- 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 t*n*, whereas*implicit*methods use the equation of motion at the new time step t*n+1*.

- discretization

- difference formula :

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

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

- Inversion of diagonal matrices
*M*and*C*is trivial.

- At timer
*t*= 0 we have initial conditions u*0*and &u-odot*0*. From equilibrium we find ü*0*. From equation (??) and (??) :

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

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

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

central differences :

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

In matrix form

*A*… time integration operator for discrete

For m-time steps and *L* = 0

Spectral decomposition of *A* :

*P* … orthonormal matrix; contains eigenvectors of *A*

*J* … Jordan form; eigenvalues λ*i* of *A* are stored on diagonal

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

Eigenvalues of A for the undamped equation of motion

For the damped equation of motion :

Damping reduces the critical time step.

For __varying time step sizes__ :

- 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

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

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

(asynchronous damping)

assumption :

update of accelerations

update formulas for velocities and displacements:

From (2):

From (1):

Remarks :

- starting procedure for first time step with

- 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 :

- critical time step for varying time increments