I want to prove/disprove that the energy loss in an unforced spring mass damper system with ksi < 0.707 is a monotonically increasing function of d at the end of a given time T (i.e., as damping increases, the lost energy, which is positive, increases).
Mathematically, the system is m \ddot{x} + d \dot{x} + k x = 0. Thus, the corresponding lost energy := E_0 - 1/2 k x^2 + 1/2 m \dot{x}^2 or integral of
d \dot{x}^2 from t=0 to t=T. I need to show derivative of the lost energy with respect to d is positive.
1) In complex approach of frequency, a definite of loss energy can be expressed as: (1/2)*real(\omega)/imag(\omega), where \omega is the frequency of the equation of motion.
2)In qualify analysis, by plotting phase plane and using corresponding lost energy equation, you can show that it is positive or not. [Ref: Nonlinear Oscillation, Nayfeh]
If you really want to be clean, you get the analytic response of your simple equation, and inject it in your energy balance between two instants. The analysis of the resulting formula should highlight the role of each of your parameters.
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