VM-LSDYNA-WB-003
VM-LSDYNA-WB-003
Response of Spring-Mass-Damper System
Overview
| Reference: |
C. M. Close & D. R. Frederick. (1994) Modeling and Analysis of Dynamic Systems (2nd ed.).New York, NY: John Wiley and Sons, 314-315. G. F. Franklin, J. D. Powell, & A. Emami-Naeini. (1994). Feedback Control of Dynamic Systems (3rd ed.). Reading, MA: Addison-Wesley Publishing, 126-127. |
| Analysis Type(s): | Explicit Dynamics with Workbench LS-DYNA |
| Element Type(s): |
Solid, Spring Connection |
| Input Files: | Link to Input Files Download Page |
Test Case
The one-degree-of-freedom system consists of a spring K and mass M with viscous damping C. There are two loading cases:
Case 1: f(t) = A = constant (step input)
Case 2: f(t) = At (ramp input)
For this underdamped system, the displacement of M for Case 1 overshoots the steady-state static displacement. The overshoot and the peak time, tp are compared to theory outlined in Close and Frederick (1994). Based on the discussion in Franklin, Powell, and Emami-Naeini (1994), the mass velocity in response to the ramp input, in theory, is equal to the mass displacement due to the step input.
| Material Properties | Geometric Properties | Loading |
|---|---|---|
|
Mass M = 1.0 kg Spring K = 16π2 N/m Damper C = 0.21545376 |
Spring Length = 1 m |
Case 1: A step force input, f (t) = 16π2 on the mass M in the +x direction. Case 2: A ramp force input, f(t) = (16π2)t, on the mass M in the +x direction. |
Analysis Assumptions and Modeling Notes
The magnitude of the step force input for Case 1 was chosen to equal the spring stiffness constant to produce a steady-state static deflection of unity. The ramp input for Case 2 was defined such that the input for Case 1 is the time derivative of the input for Case 2. The value of the stiffness constant was chosen so that the system undamped natural frequency equals 2 Hz. The damping constant was chosen to produce a damping ratio that results in a theoretical 50% overshoot of the steady-state deflection for the step input.
As outlined in Franklin, Powell, and Emami-Naeini (1994), for a single DOF system subjected to a step input, the relationship between overshoot, Mp, and damping ratio, ζ, is given by:
For the system shown in the problem sketch above:
The expression for peak time, tp, which is the time to reach xmax is given by:
where ωn is the system undamped natural frequency in units of radians per second.