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.

The model is created in the XZ plane. Only the Uz degree of freedom is active for the analysis. The link elements are modeled as single-joint link elements at joint 1. This means that one end of the link element is connected to the ground and the other end is connected to joint 1. The link elements are oriented such that their positive local 1 axes are parallel to the positive global Z axis. This is the default orientation of single joint link elements. Only U1 degree of freedom properties are defined for the link elements. The stiffness of the linear link element is 100 k/in. For linear analyses, the damper element has zero stiffness and damping properties, and for nonlinear analyses its stiffness is 10,000 k/in and its damping coefficient, c, is 1 kip-sec/in. The damping exponent is set equal to 1, meaning that the force versus velocity characteristics of the damper are linear. The derivation of those properties for the damper element is presented later in this example.

A 1 kip-sec2/in translational mass in the Uz direction is assigned to joint 1. Also a 10 kip point load is assigned to joint 1 in the positive Uz direction. A nonlinear time history analysis must be performed to obtain the desired damper element behavior. For this example both a modal time history load case named NLMHIST1 and a direct integration time history load case named NLDHIST1 are used. A sine wave function that defines the variation of the 10 kip point load over time is created for use in these load cases.

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:

The damping coefficient for the damper, cd, is calculated as:

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

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