Transient Dynamic Analysis

5.1. Definition of Transient Dynamic Analysis

Transient dynamic analysis (sometimes called time-history analysis) is a technique used to determine the dynamic response of a structure under the action of any general time-dependent loads. You can use this type of analysis to determine the time-varying displacements, strains, stresses, and forces in a structure as it responds to any combination of static, transient, and harmonic loads. The time scale of the loading is such that the inertia or damping effects are considered to be important. If the inertia and damping effects are not important, you might be able to use a static analysis instead (see Chapter 2, “Structural Static Analysis”).

The basic equation of motion solved by a transient dynamic analysis is (M){ &&u} + (C){ &u} + (K){u} = {F(t)}

where:

(M) = mass matrix (C) = damping matrix (K) = stiffness matrix

{ &&u} = nodal acceleration vector { &u} = nodal velocity vector {u} = nodal displacement vector {F(t)} = load vector

At any given time, t, these equations can be thought of as a set of "static" equilibrium equations that also take into account inertia forces ((M){ &&u}) and damping forces ((C){ &u}). The ANSYS program uses the Newmark time integration method or an improved method called HHT to solve these equations at discrete time points. The time increment between successive time points is called the integration time step.

5.2. Preparing for a Transient Dynamic Analysis

A transient dynamic analysis is more involved than a static analysis because it generally requires more computer resources and more of your resources, in terms of the “engineering” time involved. You can save a significant amount of these resources by doing some preliminary work to understand the physics of the problem. For example, you can:

1. Analyze a simpler model first. A model of beams, masses, and springs can provide good insight into the problem at minimal cost. This simpler model may be all you need to determine the dynamic response of the structure.

2. If you are including nonlinearities, try to understand how they affect the structure's response by doing a static analysis first. In some cases, nonlinearities need not be included in the dynamic analysis.

3. Understand the dynamics of the problem. By doing a modal analysis, which calculates the natural fre-quencies and mode shapes, you can learn how the structure responds when those modes are excited.

The natural frequencies are also useful for calculating the correct integration time step.

4. For a nonlinear problem, consider substructuring the linear portions of the model to reduce analysis costs. Substructuring is described in the ANSYS Advanced Analysis Techniques Guide.

5.3. Three Solution Methods

Three methods are available to do a transient dynamic analysis: full, mode superposition, and reduced. The ANSYS Professional program allows only the mode superposition method. Before we study the details of how to imple-ment each of these methods, we will examine the advantages and disadvantages of each.

5.3.1. Full Method

The full method uses the full system matrices to calculate the transient response (no matrix reduction). It is the most general of the three methods because it allows all types of nonlinearities to be included (plasticity, large deflections, large strain, and so on).

Note — If you do not want to include any nonlinearities, you should consider using one of the other methods because the full method is also the most expensive method of the three.

The advantages of the full method are:

• It is easy to use, because you do not have to worry about choosing master degrees of freedom or mode shapes.

• It allows all types of nonlinearities.

• It uses full matrices, so no mass matrix approximation is involved.

• All displacements and stresses are calculated in a single pass.

• It accepts all types of loads: nodal forces, imposed (nonzero) displacements (although not recommended), and element loads (pressures and temperatures) and allows tabular boundary condition specification via TABLE type array parameters.

• It allows effective use of solid-model loads.

The main disadvantage of the full method is that it is more expensive than either of the other methods.

For procedural information about using the full method, see Section 5.4: Performing a Full Transient Dynamic Analysis.

5.3.2. Mode Superposition Method

The mode superposition method sums factored mode shapes (eigenvectors) from a modal analysis to calculate the structure's response. This is the only method available in the ANSYS Professional program. Its advantages are:

• It is faster and less expensive than the reduced or the full method for many problems.

• Element loads applied in the preceding modal analysis can be applied in the transient dynamic analysis via the LVSCALE command, unless the modal analysis was done using PowerDynamics.

• It accepts modal damping (damping ratio as a function of mode number).

The disadvantages of the mode superposition method are:

• The time step must remain constant throughout the transient, so automatic time stepping is not allowed.

• The only nonlinearity allowed is simple node-to-node contact (gap condition).

• It does not accept imposed (nonzero) displacements.

For procedural information about using the mode superposition method, see Section 5.5: Performing a Mode Superposition Transient Dynamic Analysis.

5.3.3. Reduced Method

The reduced method condenses the problem size by using master degrees of freedom and reduced matrices.

After the displacements at the master DOF have been calculated, ANSYS expands the solution to the original full DOF set. (See Section 3.14: Matrix Reduction for a more detailed discussion of the reduction procedure.) The advantage of the reduced method is:

• It is faster and less expensive than the full method.

The disadvantages of the reduced method are:

• The initial solution calculates only the displacements at the master DOF. A second step, known as the expansion pass, is required for a complete displacement, stress, and force solution. (However, the expansion pass might not be needed for some applications.)

• Element loads (pressures, temperatures, and so on) cannot be applied. Accelerations, however, are allowed.

• All loads must be applied at user-defined master degrees of freedom. (This limits the use of solid-model loads.)

• The time step must remain constant throughout the transient, so automatic time stepping is not allowed.

• The only nonlinearity allowed is simple node-to-node contact (gap condition).

For procedural information about using the reduced method, see Section 5.6: Performing a Reduced Transient Dynamic Analysis.

5.4. Performing a Full Transient Dynamic Analysis

Note - Before reading this section, you are encouraged to become familiar with the concepts presented in Chapter 2, “Structural Static Analysis”.

We will first describe how to do a transient dynamic analysis using the full method. We will then list the steps that are different for the mode superposition and reduced methods.

The procedure for a full transient dynamic analysis (available in the ANSYS Multiphysics, ANSYS Mechanical, and ANSYS Structural products) consists of these steps:

1. Section 5.4.1: Build the Model

2. Section 5.4.2: Establish Initial Conditions 3. Section 5.4.3: Set Solution Controls

4. Section 5.4.4: Set Additional Solution Options 5. Section 5.4.5: Apply the Loads

6. Section 5.4.6: Save the Load Configuration for the Current Load Step 7. Section 5.4.7: Repeat Steps 3-6 for Each Load Step

8. Section 5.4.8: Save a Backup Copy of the Database 9. Section 5.4.9: Start the Transient Solution

10. Section 5.4.10: Exit the Solution Processor 11. Section 5.4.11: Review the Results

Section 5.4: Performing a Full Transient Dynamic Analysis

5.4.1. Build the Model

See Section 1.2: Building a Model in the ANSYS Basic Analysis Guide. For further details, see the ANSYS Modeling and Meshing Guide.

5.4.1.1. Points to Remember

Keep the following points in mind when building a model for a full transient dynamic analysis:

• You can use both linear and nonlinear elements.

• Both Young's modulus (EX) (or stiffness in some form) and density (DENS) (or mass in some form) must be defined. Material properties may be linear or nonlinear, isotropic or orthotropic, and constant or tem-perature-dependent.

Some comments on mesh density:

• The mesh should be fine enough to resolve the highest mode shape of interest.

• Regions where stresses or strains are of interest require a relatively finer mesh than regions where only displacements are of interest.

• If you want to include nonlinearities, the mesh should be able to capture the effects of the nonlinearities.

For example, plasticity requires a reasonable integration point density (and therefore a fine element mesh) in areas with high plastic deformation gradients.

• If you are interested in wave propagation effects (for example, a bar dropped exactly on its end), the mesh should be fine enough to resolve the wave. A general guideline is to have at least 20 elements per wavelength along the direction of the wave.

5.4.2. Establish Initial Conditions

Before you can perform a full transient dynamic analysis on a model, you need to understand how to establish initial conditions and use load steps.

A transient analysis, by definition, involves loads that are functions of time. To specify such loads, you need to divide the load-versus-time curve into suitable load steps. Each "corner" on the load-time curve may be one load step, as shown in Figure 5.1: “Examples of Load-Versus-Time Curves”.