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Modal Analysis
SS-V: 2120 Two-body Symmetric Damped Oscillators System
Test No. VM13Determine the frequency and damping rate of a 1D damped oscillator.

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Verification Problems
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- Static Analysis
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  - SS-V: 2000 Free Vibration of a Wedge
    Test No. VM01 Determine the first fundamental frequency and vibration mode of a wedge constrained at the wide end.
  - SS-V: 2010 In-plane Vibration of a Cantilever Plate
    Test No. VM01 Find the fundamental frequency for in-plane vibration of a cantilevered plate.
  - SS-V: 2020 Cylindrical Shell
    Test No. VM03 Find the fundamental frequency of a cantilever cylindrical shell.
  - SS-V: 2030 Free Thin Square Plate
    Test No. VM04 Determine the first fourteen modal frequencies for a free thin square plate.
  - SS-V: 2040 Thick Skewed Plate
    Test No. VM05 Find the fundamental frequency of a thick clamped skewed plate.
  - SS-V: 2050 Thick Square Plate
    Test No. VM06 Determine the first twelve modes and frequencies for a simply supported thick square plate.
  - SS-V: 2060 In-plane Vibration of Pin-ended Cross
    Test No. VM07 Find the first eight modal frequencies for in-plane vibration of a cross with pin joints at the ends.
  - SS-V: 2070 Cantilever Beam
    Test No. VM08 Perform normal mode dynamic analysis of a cantilevered beam.
  - SS-V: 2080 Circular Ring - In-plane and Out-of-plane Vibration
    Test No. VM09 Perform normal mode dynamic analysis of an unconstrained circular ring.
  - SS-V: 2090 Triangular Wing - Fixed Vibration
    Test No. VM10 Perform a normal mode dynamic analysis of a fixed right triangular plate.
  - SS-V: 2100 Cantilevered Tapered Beam
    Test No. VM11 Shear behavior of a cantilevered tapered beam.
  - SS-V: 2110 Cantilever with Off-Center Remote Mass
    Test No. VM12 Find the frequencies of a cantilever with off center remote masses at the free end.
  - SS-V: 2120 Two-body Symmetric Damped Oscillators System
    Test No. VM13Determine the frequency and damping rate of a 1D damped oscillator.
  - SS-V: 2130 1D Damped Oscillator: Translation or Rotation
    Test No. VM14Determine the frequency and damping rate of a 1D damped oscillator.
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SS-V: 2120 Two-body Symmetric Damped Oscillators System

Test No. VM13Determine the frequency and damping rate of a 1D damped oscillator.

Definition

The material properties are:

Rigid body, m = 1 kg, k = 1 N/m, c₀ = 1 Ns/m, c₁ = 0.1 Ns/m

Modeling and Results

The spring and damper can be modeled with a shock absorber or a bushing. To restrict motion to one dimension, sliding conditions are applied on two sides of the cube-shaped bodies. Frequency and rate of damping are compared to the values from the analytical solution.

Table 1.
	Parameters	SimSolid	Analytical solution	% Difference
Mode 1	Frequency, Hz	0.15876	0.158956	0.12%
Mode 1	Damping rate, 1/s	0.049752	0.05	0.50%
Mode 2	Frequency, Hz	0.2201	0.219236	0.39%
Mode 2	Damping rate, 1/s	1.0476	1.05	0.23%

¹ Kausel E., “Advanced Structural Dynamics”, Cambridge University Press, 2017, Section 3: ”Multiple Degree of Freedom Systems” (general equations).