how do I solve problem 4.9 parts (a-d)?

Why is it more effective? Decay ratio measures amplitude decay between successive oscillations. In 3 seconds, control system B has 4 oscillations (1.33 Hz), whereas A has only 3 oscillations (1 Hz). Amplitude at 3 sec can be calculated by multiplying the initial displacement by the appropriate number of decay ratios, that is (d) Discuss how increasing ka and ky affects period of vibration Tperiod and decay ratio. Effect: Increasing ka slightly increases the period Explanation: Effect: Increasing ka increases / decreases the decay ratio Explanation: Effect: Increasing kp increases / decreases the period Explanation: Effect: Increasing kp slightly increases the decay ratio Explanation:4.9 PD feedback control of an underdamped mass-spring-damper system Homework 4.7 sized leaf springs and shock absorbers to tune a vehicle's ride. The vehicle was modeled as a "mass-spring-damper" system, with mass m = 200 kg, linear damping constant b = 204.3 m' AS, linear spring constant k = 7948 m, and forcing function f(t). The ODE governing displacement y(t) from equilibrium is shown below. The system is modified using the "proportional-derivative (PD) feedback-control' for f(t) shown below. System response is tuned by proper choice of the constants ka and kp. my + by + ky = f(y, y) f (y, y) = - kay - kipy my + ( b + ka) y + ( k + kp)y = 0 (a) Your engineering manager wants to compare the original ride characteristics with two "active suspension system" alternatives and asks you to determine the values of ka and kp for alternatives A and B (Note: Analyze the two alternatives using the original values of b and k). Ride characteristics Vibration period (secs) Decay ratio K N sec m kip Original vehicle 1.0 0.6 0 0 A. Simulates improved shocks 1.0 0.4 162.2 115.7 B. Simulates stiff springs/ shocks 0.75 0.4 284.4 6387 Your manager asks: "How does the control system affect the ride?" Using only vibration period, decay 0.8 0.6 ratio, and initial values y(0) = 1.0 m, y(0) = 0, sketch 0.4 (by hand) the first 3 seconds of the vehicle's response 0.2 (b) and determine y(3) for each suspension system. Ride characteristics y (t = 3) y(t) (meters) -0.2 0.216 meters -0.4 Original vehicle -0.6 A. Improved shocks meters -0.8 B. Stiff springs/shocks meters 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 (c) Which control system is more effective at driving y(t) to 0? A/B. Time (seconds) 3 201