Product: ABAQUS/Standard
An eigenvalue analysis is performed on the system consisting of spring, mass, and rotary inertia elements. The spring element builds the stiffness for the translational degrees of freedom, while the mass is assigned to all six degrees of freedom (due to the mass and rotary inertia elements). To avoid solver singularities, a B31 element with negligible mass is included in the model. Composite damping values are given as parameters on the *MASS and *ROTARY INERTIA options.
Since the system is very simple, it is easy to check the value of composite damping per mode. This will be the sum of the element masses times their composite damping values projected into the mode and normalized with the generalized mass of this mode. The composite damping for each of the six requested modes will be equal to 0.01 for the values given in the input file for this test.
The linear behavior of a simple spring/mass system with mass proportional damping is tested (see system A in “Linear behavior of spring and dashpot elements,” Section 2.6.2 of the ABAQUS Benchmarks Manual). The MASS element (m = 0.02588) is attached to a SPRING1 element; therefore, the system is grounded. The value of the mass proportional damping parameter (
= 4.6367852) was taken such that the damping in the system (
) is the same as in Problem I in “Linear behavior of spring and dashpot elements,” Section 2.6.2 of the ABAQUS Benchmarks Manual, when a dashpot element (c = 0.12) is used to provide damping.
Force balance on the system yields a second-order linear differential equation for a single degree of freedom damped oscillator whose solution is identical to the one presented in Problem I in “Linear behavior of spring and dashpot elements,” Section 2.6.2 of the ABAQUS Benchmarks Manual.
The results for the displacements of the mass match those in “Linear behavior of spring and dashpot elements,” Section 2.6.2 of the ABAQUS Benchmarks Manual.
The behavior of a simple spring/rigid body system with rotary inertia proportional damping is tested. A rigid body (one R2D2 element), with rotary inertia at its reference node and rotary inertia proportional damping, is allowed only rotation about the z-axis. The rotation of the rigid element is constrained by the two springs acting normal to it. In the first step the rigid body is rotated by 10° in a static procedure, thus developing forces in the springs. In the next dynamic step the above single degree of freedom system is allowed to oscillate freely. An additional perturbation step is included to test *LOAD CASE
Moment balance on the system yields a second-order linear differential equation for a single degree of freedom damped oscillator. The analytical exponentially decaying sinusoidal solution is obtained for the rotation of the rigid body.
Input file for rotary inertia proportional damping, perturbation step with *LOAD CASE.