Solvers Background: Linear Transient Dynamic

Description

The Linear Transient Dynamic solver calculates the time history of the linear dynamic response of a structure subjected to arbitrary forcing functions (loading) and/or initial conditions. The solver produces results in the time domain and considers the effects of inertia and damping.

Solution Type

Two approaches to linear transient dynamic analysis are available:

Mode Superposition

Using the mode superposition method, the responses of individual modes of the structure are calculated separately and then combined to produce the total response of the structure. The individual modes come from a natural frequency analysis.

One of the advantages of this method is that modal damping can be used (in addition to Rayleigh damping). When experimental data is available, modal damping gives a more accurate representation of the damping in the system.

The mode superposition method is best suited to structures where the lower frequencies dominate the response (e.g., earthquakes), and where relatively few modes are needed to provide good accuracy. Modal superposition is not as useful for problems such as shock loads or impacts where higher frequency modes are excited. In these cases, many modes may be required and the cost to calculate them can offset the saving in the transient solution.
Full System

This approach does not have the limitations of mode superposition, but can be computationally more expensive as all nodal displacements are numerically integrated at every time step. This method is also referred to as direct integration.

Forcing Function

Two types of loading conditions can be applied: dynamic loads and base excitation.

Loads applied to the model in the various load cases can be factored by Factor vs Time tables to represent the time history of the loads. All load types can be included. Multiple load cases can act simultaneously; they are combined according to their load factors to form the loading condition at each time step.

Multiple freedom cases can also be included and factored in the same way as the loads in the load cases, as long as the full system option is used. This provides support for time dependent enforced displacements.
Base excitations can be expressed as acceleration, velocity or displacement time histories. The base of the model refers to all restrained nodes. The time history of the excitation is specified via a direction vector and associated Factor vs Time tables. Up to three excitations may act simultaneously, each in one of the global axis directions.

Initial Conditions

Four types of initial conditions can be specified:

A static solution that specifies the initial displacements of a structure under certain static loads. This will be a Linear Static result file.
A time step in a previously run Transient Dynamic analysis, which specifies the dynamic response of the structure at a time instance. The solution will start from any selected time step available in the file. To enable the possibility of this type of restart, the node velocity and acceleration results must be available in the previously run analysis file.
Initial velocity and acceleration of all free nodes.
Initial velocity (via the Node Velocity attribute) and zero acceleration, as specified independently at each node in one or more load cases.

Procedure

The Linear Transient Dynamic solver executes the following steps:

Initialises the nodal displacement, velocity and acceleration vectors according to the specified initial conditions.
For the mode superposition method, calculates and assembles equivalent element force vectors and external nodal force vectors. If base acceleration is included, the global mass matrix is also formed for the calculation of the pseudo load vector.
For the direct integration method, calculates and assembles element stiffness, mass and damping matrices, equivalent element force vectors and external nodal force vectors. In the stiffness calculation, material temperature dependency is considered (see Special Topics: Temperature Dependence). The element geometric stiffness matrix may also be included if initial conditions are used and the effects of stress stiffening are required. Damping can be included (see Special Topics: Damping). Constraints and links are also assembled in this process, and the constant terms in enforced displacements, multi-point links and shrink links are combined and applied according to the respective factors in the relevant freedom cases. At the end of this assembly procedure, the three global matrices in the equation of dynamic equilibrium are formed:

where

= global mass matrix,

= global damping matrix,

= global stiffness matrix,

= unknown nodal displacement, velocity and acceleration vectors, respectively, and

= applied load vector (which may be time dependent) .
Loops through the specified time steps and calculates nodal displacement, velocity and acceleration using either the Wilson-theta or Newmark-beta method.
Calculates element results such as stress and strain.

Notes

The results of a transient dynamic analysis are provided as a series of solutions for discrete points in time throughout the time period of interest based on the time stepping and saving settings. The results include the nodal displacements, velocity and acceleration, as well as element stresses, strains and other quantities at each time step.
The choice of time step is important to ensure that the complete response of the structure is captured by the solution.

In general, the smaller the time step the more accurate the solution will be. However, there are practical limits on how small the time step can be; more solution steps will be required for a smaller time step and the run time will increase accordingly.

If the time step is too large, much of the higher frequency response of the structure will be missed.