Problem Description
In this single degree of freedom example a spring-mass-damper system is subjected to a harmonic load. The frequency of the harmonic load is chosen to be equal to the frequency of the spring-mass-damper system. The damper is assumed to provide 5% of critical damping. The displacements of the springmass-damper system at various arbitrary times and the steady-state deformation of the system are compared with results that are hand calculated using formulas presented in Chopra 1995. This model consists of a single joint, labeled joint 1, and two link elements. One of the link elements is a linear spring element and the other is a damper element.
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Both the NLMHIST1 and the NLDHIST1 load cases use an output step time size of 0.01 second and 2,550 total output steps, yielding results for 25.5 seconds, which is just over 40 cycles of loading. The sine wave function is defined for 41 cycles of loading.
Derivation of Damper Element Properties
The natural frequency, ωn, of the system is calculated as:
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If pure damping behavior is desired from the damper element, as is the case in this example, the effect of the spring can be made negligible by making its stiffness, kd, sufficiently stiff. The spring stiffness should be large enough so that the characteristic time of the spring-dashpot damper element, given by τ = cd/kd, is approximately two to three orders of magnitude smaller than 1/ωn. Care must be taken not to make kd excessively large because numerical sensitivity may result. For this example the spring stiffness is initially based on τ being three orders of magnitude smaller than 1/ωn. Thus τ can be expressed as:
Solving for kd yields:
Tested Features
- Tested Features
- Damper element links
- Linear link elements
- Nonlinear modal time history analysis
- Nonlinear direct integration time history analysis
- Joint force loads