Verification of Rayleigh damping options with direct integration and modal superposition

The equation of motion for the system is

where m is the mass, c the damping, k the stiffness, and u the displacement.

Rayleigh damping defines the damping as $c=\alpha m+\beta k$, where $\alpha$ is the mass damping factor and $\beta$ is the stiffness damping factor.

Assuming a solution of the form $u\left(t\right)=u_0\exp \lambda t$, we have

where ${\omega }_u=\sqrt{k/m}$ 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

We define the damping ratio, $\xi$, as the ratio of damping to critical damping:

The relationships in this equation are often used as a basis for choosing $\alpha$ and $\beta$.

The equation defining $\lambda$ can be rewritten

We choose the damping in this case to be less than critical, so $\xi <$1 and the system can vibrate. The initial conditions are $u\left(0\right)=$ 1 and $\dot{u}\left(0\right)=$ 0, so the dynamic part of the motion is

where ${\omega }_d={\omega }_u\sqrt{1-{\xi }^2}$ is the damped frequency of the system.

The amplitudes of this oscillatory equation before and after one period of vibration, $T=2\pi /{\omega }_d$, have the ratio

so the logarithmic decrement over n cycles of response is

Table 1 shows the values of $\delta$ 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 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.