1.4.4 Free and forced vibrations with damping

Product: ABAQUS/Standard  

This example is intended to provide basic verification of the frequency-dependent spring and dashpot elements available in ABAQUS.

There are several different mechanisms that can cause damping in a system. In linear viscous damping the damping force is directly proportional to the velocity. In many cases such simple expressions for the damping forces are not available directly. However, it is possible to obtain an equivalent viscous damping coefficient by equating the loss of kinetic and strain energy to the energy dissipation. Hysteretic and viscoelastic damping are two important damping mechanisms that are more complex than linear viscous damping. In the frequency domain these mechanisms can be simulated by using dashpots with viscous damping coefficients that depend on the forcing frequency. Frequency-dependent springs will also be needed for modeling viscoelastic damping.

To illustrate how to model viscous, hysteretic, and viscoelastic damping mechanisms, springs and dashpots with constant and frequency-dependent properties will be used in frequency domain dynamic analyses of one- and two-degree-of-freedom discrete mass-spring-dashpot systems. In addition, viscous damping is modeled in the time domain by using a constant dashpot coefficient.

ABAQUS also allows for spring and dashpot properties that depend on temperature and user-defined field variables. This dependence provides an easy means to vary material properties of springs and dashpots during time-domain analysis. In doing perturbation analysis (such as frequency-domain steady-state dynamic analysis) with ABAQUS, temperature and field variable variations are not permitted within an analysis step. However, since the base state temperature and field variable values for each perturbation analysis step can be changed, it is possible to perform a multiple-step perturbation analysis that uses different temperature- and field-variable-dependent material properties that correspond to the base state temperature and field variable values. This dependence feature will be illustrated in analyses 2 and 3 described below. These two analyses employ both the direct-solution and the subspace-based steady-state dynamic procedure in ABAQUS.

The one- and two-degree-of-freedom mass-spring-dashpot systems are shown in Figure 1.4.4–1. The following dynamic analyses are performed: (1) free vibration of the one-degree-of-freedom system after it is given an initial displacement and then released; (2) steady-state response to applied harmonic loading of the one-degree-of-freedom model with viscous damping; (3) steady-state response to applied harmonic loading of the one-degree-of-freedom model with hysteretic damping; and (4) steady-state response to applied harmonic loading of the two-degree-of-freedom model with viscoelastic damping. In all cases the forcing function is applied to the point mass closest to the anchor point, and numerical results are compared to the exact solutions for the system.

Problem description

Results and discussion

Input files

Reference

Table

Table 1.4.4–1 Energy balance at 0.7 seconds.

 Solution with HAFTOL = 4.448 N (1 lb)Solution with HAFTOL = 44.48 N (10 lb)
Kinetic energyN-m0.04720.0033
lb-in0.4180.029
Strain energyN-m0.04900.1943
lb-in0.4341.720
Dissipated energyN-m1.58171.3445
lb-in14.00011.900
Total energyN-m1.67801.5421
lb-in14.85213.649
Energy loss through numerical dampingN-m0.01670.1526
lb-in0.1481.351


Figures

Figure 1.4.4–1 One- and two-dof spring-mass-dashpot systems.

Figure 1.4.4–2 Displacement-time response for one-dof spring-mass-dashpot example.

Figure 1.4.4–3 Peak amplitude response for viscous damping.

Figure 1.4.4–4 Phase angle response for viscous damping.

Figure 1.4.4–5 Peak amplitude response for hysteretic damping.

Figure 1.4.4–6 Phase angle response for hysteretic damping.

Figure 1.4.4–7 Peak amplitude response for viscoelastic damping.

Figure 1.4.4–8 Phase angle response for viscoelastic damping.