Product: ABAQUS/Standard
Rayleigh damping options are provided in ABAQUS for both direct integration (*DYNAMIC) and modal superposition (*MODAL DYNAMIC) procedures. This example is intended to verify these Rayleigh damping options by comparing the ABAQUS results with an exact solution for a simple problem.
For direct integration Rayleigh damping is defined with the *DAMPING option in the material definition for those elements in which mass and stiffness proportional damping is desired. For modal dynamics analysis Rayleigh damping is defined in the MODAL DAMPING option in the step definition. For direct integration analysis Rayleigh damping can be introduced in any stress-based element, but it is not available for elements of type *SPRING; DASHPOT elements should be used in parallel with the SPRING elements for this purpose (see “Free and forced vibrations with damping,” Section 1.4.4). Elements with nonhomogeneous material damping properties are dealt with by taking a volume average of the damping coefficients. Stiffness proportional damping in nonlinear analysis is discussed in “Material damping,” Section 20.1.1 of the ABAQUS Analysis User's Manual.
The example is the simplest dynamic system: a massless truss connecting a point mass to ground. The mass is obtained by giving the material in the truss a density so that the lumped mass of the truss gives the correct point mass at the free end of the truss. The truss is initially stretched and then let go so that it undergoes vibrations of small amplitude. This is a linear problem; consequently, the response can be predicted using either the direct integration or modal dynamic procedures. These solutions are compared with each other and to the exact solution of the equation of motion.
Figure 1.4.5–1 shows the geometry. The model consists of a single truss element, type T3D2, constrained at one node and free to move only in the x-direction at its other node. The truss's mass matrix is lumped so that the system is equivalent to a spring and a lumped mass. The cross-sectional area of the truss is 645 mm2 (1 in2), and its length is 254 mm (10 in). It is made of linear elastic material, with Young's modulus 69 GPa (107 lb/in2). The density of the truss provides a lumped mass at the unrestrained end of 2.777 × 105 kg (1585 lb-s2/in).
In each case the mass is displaced by 25.4 mm (1 in) in an initial *STATIC step. It is then released in the *DYNAMIC (or *MODAL DYNAMIC) step, and the displacement response history is saved on a file for postprocessing. The time histories are plotted; and the logarithmic decrement, 
, of the peak response is calculated graphically and compared with the theoretical value.
The equation of motion for the system is
Rayleigh damping defines the damping as 
, where 
is the mass damping factor and 
is the stiffness damping factor.
Assuming a solution of the form 
, we have
is the undamped frequency of vibration (25.118 rad/sec for the parameters of this example). Critical damping occurs when the value of c causes the discriminant of this equation to be zero, so 
, as the ratio of damping to critical damping: 
and .The equation defining 
can be rewritten
We choose the damping in this case to be less than critical, so 
1 and the system can vibrate. The initial conditions are 
1 and 
0, so the dynamic part of the motion is
is the damped frequency of the system.The amplitudes of this oscillatory equation before and after one period of vibration, 
, have the ratio
Table 1.4.5–1 shows the values of calculated from ABAQUS for the various test cases examined, together with their corresponding exact solution. A sample time history from which the logarithmic decrements are calculated is shown in Figure 1.4.5–2. All the ABAQUS runs use fixed time increments of .01 seconds. The integrator used in the modal method is exact, so the results of that analysis are exact. The integrator used in the direct integration method is not exact; however, since the period of the system is 0.25 seconds, the time increment chosen gives 25 increments per cycle, so those results are also quite accurate.
Direct integration analysis, 
1.00472, 
0.0.
Modal superposition analysis, 1.00472, 0.0.
Direct integration analysis, 0.0, 1.59248 × 10–3.
Modal superposition analysis, 0.0, 1.59248 × 10–3.
Direct integration analysis, 1.00472, 1.59248 × 10–3.
Modal superposition analysis, 1.00472, 1.59248 × 10–3.
Direct integration analysis using *BEAM GENERAL SECTION, 1.00472, 0.0.
Direct integration analysis using *BEAM GENERAL SECTION, 0.0, 1.59248 × 10–3.
Direct integration analysis using *BEAM GENERAL SECTION, 1.00472, 1.59248 × 10–3.
Direct integration analysis using *SHELL GENERAL SECTION, 1.00472, 0.0.
Direct integration analysis using *SHELL GENERAL SECTION, 0.0, 1.59248 × 10–3.
Direct integration analysis using *SHELL GENERAL SECTION, 1.00472, 1.59248 × 10–3.
Direct integration analysis using substructures, 1.00472, 0.0.
Substructure generation referenced in the analysesrayleighdamping_substr_alpha.inp and rayleighdamping_overide.inp.
Direct integration analysis using substructures, 0.0, 1.59248 × 10–3.
Substructure generation referenced in the analysis rayleighdamping_substr_beta.inp.
Direct integration analysis using substructures, 1.00472, 1.59248 × 10–3.
Substructure generation referenced in the analysis rayleighdamping_substr.inp.
Tests override of damping properties on the *SUBSTRUCTURE PROPERTY option.
Uses Rayleigh damping with user elements in direct integration dynamics (*DYNAMIC).
