### Overview ABAQUS offers several methods for performing dynamic analysis of problems in which inertia effects are important. Direct integration of the system must be used when nonlinear dynamic response is being studied. Implicit direct integration is provided in ABAQUS/Standard; explicit direct integration is provided in ABAQUS/Explicit. Modal methods are usually chosen for linear analyses because in direct-integration dynamics the global equations of motion of the system must be integrated through time, which makes direct-integration methods significantly more expensive than modal methods. Subspace-based methods are provided in ABAQUS/Standard and offer cost-effective approaches to the analysis of systems that are mildly nonlinear.

In ABAQUS/Standard dynamic studies of linear problems are generally performed by using the eigenmodes of the system as a basis for calculating the response. In such cases the necessary modes and frequencies are calculated first in a frequency extraction step. The mode-based procedures are generally simple to use; and the dynamic response analysis itself is usually not expensive computationally, although the eigenmode extraction can become computationally intensive if many modes are required for a large model. The eigenvalues can be extracted in a prestressed system with the “stress stiffening” effect included (the initial stress matrix is included if the base state step definition included nonlinear geometric effects), which may be necessary in the dynamic study of preloaded systems. It is not possible to prescribe nonzero displacements and rotations directly in mode-based procedures. The method for prescribing motion in mode-based procedures is explained in Transient modal dynamic analysis, Section 6.3.7.

The density must be defined for all materials used in any dynamic analysis.

### Implicit versus explicit dynamics The direct-integration dynamic procedure provided in ABAQUS/Standard uses the implicit Hilber-Hughes-Taylor operator for integration of the equations of motion, while ABAQUS/Explicit uses the central-difference operator. In an implicit dynamic analysis the integration operator matrix must be inverted and a set of nonlinear equilibrium equations must be solved at each time increment. Because displacements and velocities in an explicit dynamic analysis are calculated in terms of quantities that are known at the beginning of an increment, the global mass and stiffness matrices need not be formed and inverted, which means that each increment is relatively inexpensive compared to the increments in an implicit integration scheme. The size of the time increment in an explicit dynamic analysis is limited, however, because the central-difference operator is only conditionally stable; whereas the Hilber-Hughes-Taylor operator is unconditionally stable and, thus, there is no such limit on the size of the time increment that can be used for most analyses in ABAQUS/Standard (accuracy governs the time increment in ABAQUS/Standard).

The stability limit for the central-difference method (the largest time increment that can be taken without the method generating large, rapidly growing errors) is closely related to the time required for a stress wave to cross the smallest element dimension in the model; thus, the time increment in an explicit dynamic analysis can be very short if the mesh contains small elements or if the stress wave speed in the material is very high. The method is, therefore, computationally attractive for problems in which the total dynamic response time that must be modeled is only a few orders of magnitude longer than this stability limit; for example, wave propagation studies or some “event and response” applications. Many of the advantages of the explicit procedure also apply to slower (quasi-static) processes for cases in which it is appropriate to use mass scaling to reduce the wave speed (see Mass scaling, Section 11.7.1).

ABAQUS/Explicit offers fewer element types than ABAQUS/Standard. For example, only first-order, displacement method elements (4-node quadrilaterals, 8-node bricks, etc.) and modified second-order elements are used, and each degree of freedom in the model must have mass or rotary inertia associated with it. However, the method provided in ABAQUS/Explicit has some important advantages:

1. The analysis cost rises only linearly with problem size, whereas the cost of solving the nonlinear equations associated with implicit integration rises more rapidly than linearly with problem size. Therefore, ABAQUS/Explicit is attractive for very large problems.

2. The explicit integration method is more efficient than the implicit integration method for solving extremely discontinuous events or processes.

3. It is possible to solve complicated, very general, three-dimensional contact problems with deformable bodies in ABAQUS/Explicit.

4. Problems involving stress wave propagation can be far more efficient computationally in ABAQUS/Explicit than in ABAQUS/Standard.

In choosing an approach to a nonlinear dynamic problem you must consider the length of time for which the response is sought compared to the stability limit of the explicit method; the size of the problem; and the restriction of the explicit method to first-order, pure displacement method or modified second-order elements. In some cases the choice is obvious, but in many problems of practical interest the choice depends on details of the specific case. Experience is then the only useful guide.

### Direct-solution dynamic analysis procedures Implicit dynamic analysis: Implicit direct-integration dynamic analysis (Implicit dynamic analysis using direct integration, Section 6.3.2) is used to study (strongly) nonlinear transient dynamic response in ABAQUS/Standard.

Subspace-based explicit dynamic analysis: The subspace projection method in ABAQUS/Standard uses direct, explicit integration of the dynamic equations of equilibrium written in terms of a vector space spanned by a number of eigenvectors (Implicit dynamic analysis using direct integration, Section 6.3.2). The eigenmodes of the system extracted in a frequency extraction step are used as the global basis vectors. This method can be very effective for systems with mild nonlinearities that do not substantially change the mode shapes. It cannot be used in contact analyses.

Explicit dynamic analysis: Explicit direct-integration dynamic analysis (Explicit dynamic analysis, Section 6.3.3) is available in ABAQUS/Explicit.

Direct-solution steady-state harmonic response analysis: The steady-state harmonic response of a system can be calculated in ABAQUS/Standard directly in terms of the physical degrees of freedom of the model (Direct-solution steady-state dynamic analysis, Section 6.3.4). The solution is given as in-phase (real) and out-of-phase (imaginary) components of the solution variables (displacement, stress, etc.) as functions of frequency. The main advantage of this method is that frequency-dependent effects (such as frequency-dependent damping) can be modeled. The direct method is the most accurate but also the most expensive steady-state harmonic response procedure. The direct method can also be used if nonsymmetric terms in the stiffness are important or if model parameters depend on frequency.

Subspace-based steady-state harmonic response analysis: In this type of ABAQUS/Standard analysis the steady-state dynamic equations are written in terms of a vector space spanned by a number of eigenvectors (Subspace-based steady-state dynamic analysis, Section 6.3.9). The eigenmodes of the system extracted in a frequency extraction step are used as the global basis vectors. The method is attractive because it allows frequency-dependent effects to be modeled and is much cheaper than the direct analysis method (Direct-solution steady-state dynamic analysis, Section 6.3.4). Subspace-based steady-state harmonic response analysis can be used if the stiffness is nonsymmetric.

### Modal superposition procedures The following procedures are available only in ABAQUS/Standard:

Frequency extraction: The natural frequencies of a system can be extracted using eigenvalue analysis (Natural frequency extraction, Section 6.3.5). Frequency extraction is required prior to any of the mode-based procedures.

Modal dynamic time history analysis: The modal dynamic procedure (Transient modal dynamic analysis, Section 6.3.7) provides dynamic time history response for linear problems using modal superposition. A frequency extraction must be performed in a step prior to the modal dynamic step.

Mode-based steady-state harmonic response analysis: A steady-state dynamic analysis based on the natural modes of the system can be used to calculate a system's linearized response to harmonic excitation (Mode-based steady-state dynamic analysis, Section 6.3.8). This mode-based method is less expensive than the direct method. The solution is given as in-phase (real) and out-of-phase (imaginary) components of the solution variables (displacement, stress, etc.) as functions of frequency. When the response is based on modal superposition, a frequency extraction must be performed first.

Response spectrum analysis: A linear response spectrum analysis (Response spectrum analysis, Section 6.3.10) is often used to obtain an approximate upper bound of the peak significant response of a system to a user-supplied input spectrum (such as earthquake data) as a function of frequency. The method has a very low computational cost and provides useful information about the spectral behavior of a system. A frequency extraction must be performed in a step prior to the response spectrum step.

Random response analysis: The linearized response of a model to random excitation can be calculated based on the natural modes of the system (Random response analysis, Section 6.3.11). This procedure is used when the structure is excited continuously and the loading can be expressed statistically in terms of a “Power Spectral Density” (PSD) function. The response is calculated in terms of statistical quantities such as the mean value and the standard deviation of nodal and element variables. A frequency extraction must be performed in a step prior to the random response step.