+ ]fi + + + J J . +a/

This result allows us to get the response to any of the three individual inputs, or any combination of them. Note that, irrespective of which input or combination is applied, only one frequency of vibration, Wn(l - e)0·5' appears. This frequency, which can be directly observed and measured in transient lab tests, is called the damped natural frequency, wn.d·

EXAMPLE: INITIAL ENERGY STORAGE Graphs of these time responses can be obtained from the above equations, or perhaps more conveniently from simulation. Let's take an example system with M = 1 kg, Ks = 1 N/m, and B = 0.2 N/(m/sec). This makes the undamped natural frequency 1 rad/sec, the damping ratio 0.20, and the damped natural frequency 0.98 rad/sec. Let's choose our initial displacement and velocity such that each corresponds to the same initial energy storage. That is, we want

For our chosen numbers, the two initial energies will be equal if we choose the initial displacement numerically equal to the initial velocity. Let's take x0 (0) = 1.0 m and (dx0 / dt)(O) = 1.0 mfsec; each initial energy will then be 0.50 joule. For our step input force, if we choose it to be 1.0 newton, then the final energy storage for this case will also be 0.50 joule.