POGAM NAME: EVISION NO.: 0 EXAMPLE 6-005 LINK DAMPE ELEMENT UNDE HAMONIC LOADING POBLEM DESCIPTION 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. The 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 U z 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 U 1 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-sec 2 /in translational mass in the U z direction is assigned to joint 1. Also a 10 kip point load is assigned to joint 1 in the positive U z direction. A nonlinear time history analysis must be performed to obtain the desired damper element behavior. For this example both a modal time history analysis case named NLMHIST1 and a direct integration time history analysis 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 analysis cases. EXAMPLE 6-005 - 1
POGAM NAME: EVISION NO.: 0 Both the NLMHIST1 and the NLDHIST1 analysis 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. GEOMETY, POPETIES AND LOADING Linear spring link element Z k s X p(t) = p o sinw n t u(t) m c d k d Harmonic load, p(t), is applied with a frequency, w n, equal to the natural frequency of the system Damper element (dashpot in series with a spring) properties are set to provide pure damping at a level equal to 5% of critical damping (ξ = 0.05) Linear Link Properties (U 1 DOF) Linear k s = 100 k/in Damper Properties (U 1 DOF) Linear k d =0 k/in Linear c d = 0 k-sec/in Nonlinear k d = 10,000 k/in Nonlinear c d = 1 k-sec/in Joint Mass m = 1 k-sec 2 /in Loading p o = 10 k w n = 10 radians/sec Active Degrees of Freedom U z only EXAMPLE 6-005 - 2
POGAM NAME: EVISION NO.: 0 DEIVATION OF DAMPE ELEMENT POPETIES The natural frequency, ω n, of the system is calculated as: k s 100 ω n = = = 10 radians/sec m 1 The damping coefficient for the damper, c d, is calculated as: cd = 2 ζω nm = 2 * 0.05 *10 *1 = 1 kip-sec/in 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, k d, sufficiently stiff. The spring stiffness should be large enough so that the characteristic time of the spring-dashpot damper element, given by τ = c d /k d, is approximately two to three orders of magnitude smaller than 1/ω n. Care must be taken not to make k d 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/ω n. Thus τ can be expressed as: cd 1 ω τ = = n k 1,000 Solving for k d yields: d k 1,000ω c = 1,000 *10 *1 = 10,000 k/in d = n d TECHNICAL FEATUES OF TESTED Damper element links Linear link elements Nonlinear modal time history analysis Nonlinear direct integration time history analysis Joint force loads EXAMPLE 6-005 - 3
POGAM NAME: EVISION NO.: 0 ESULTS COMPAISON Independent results are hand calculated using equation 3.2.6 on page 70 in Chopra 1995. esults for Model A Output Parameter Analysis Case Independent Percent Difference U z (jt 1) displ NLMHIST1-0.10488 0% at t = 0.50 sec -0.10488 in NLDHIST1-0.10480-0.08% U z (jt 1) displ NLMHIST1-0.88875 +0.02% at t = 5.00 sec -0.88858 in NLDHIST1-0.88814-0.05% U z (jt 1) displ NLMHIST1 0.99453-0.04% at t = 11.00 sec 0.99497 in NLDHIST1 0.99459-0.04% Steady-state NLMHIST1 0.99971-0.03% deformation 1.00000 in NLDHIST1 0.99964-0.04% Model B is created to demonstrate that the steady-state results can also be obtained using a steady-state analysis case. Because the steady-state analysis case is a linear analysis case and the damper element behaves differently for linear analyses and nonlinear analyses the damper element properties are different in Models A and B. The figure to the right illustrates the damper element properties used for nonlinear and linear analyses. For nonlinear analyses the damper element acts as a spring in series with a dashpot and uses the specified nonlinear spring stiffness and damping coefficient for the damper. In contrast, for linear analyses, the damper element acts as a spring in parallel with a dashpot and uses the specified linear spring stiffness and c nonlinear k nonlinear Damper Properties for Nonlinear Analyses k linear c linear Damper Properties for Linear Analyses EXAMPLE 6-005 - 4
POGAM NAME: EVISION NO.: 0 damping coefficient for the damper. This difference in nonlinear and linear behavior occurs for most types of link elements. Model B uses only a damper link element. The linear link element is not used. The linear spring stiffness for the damper is set to 100 k/in and the damping coefficient is set to 1 k-sec/in. The steady-state analysis case is defined with no hysteretic damping. If hysteretic damping were defined in the steady-state case, it would be used instead of the damping specified for the damper element. The following table presents the results obtained for Model B. The comparison with the independent results is exact. esults for Model B Output Parameter Steady-state deformation at freq = 1.5915 sec in Analysis Case Independent Percent Difference SS1 1.00000 1.00000 0% COMPUTE FILES: Example 6-005a, Example 6-005b CONCLUSION The results show an acceptable comparison with the independent results. EXAMPLE 6-005 - 5
POGAM NAME: EVISION NO.: 0 HAND CALCULATION EXAMPLE 6-005 - 6
POGAM NAME: EVISION NO.: 0 EXAMPLE 6-005 - 7
POGAM NAME: EVISION NO.: 0 EXAMPLE 6-005 - 8