Avibrating system has the following constants: w=40.6 lb, k=50lb/in.,and c=0.40 lb/in. per sec. determine a) the damping factor, b) the natural frequency of damped oscillation, c) derive the frequency response function (frf) and plot it as a bode plot (matlab or excel) d) find the half power bandwidth, predict damping factor via the half power bandwidth.

An external consultant recommends that a plant installs a bank of capacitors for power factor correction. this will reduce the peak electrical demand charges by an average of 93 kw every month. the plant current pays $13 per kw in peak demand charges. the capacitor bank will include 223 kw of fixed capacitors, and 183 of variable capacitors. the fixed capacitors cost $59 per kw, and the variable capacitors will cost $65 per kw. the consultant charges 21% of the equipment costs to install the capacitors. because this project will reduce the demand for the electric utility, they are prepared to provide a one-time rebate of $42 per kw of reduced demand. what is the simple payback period for this project (in years)?