cay Yea -Year- 7 measure of how cach vibration period ratio between 2 amplitude eab Juhtp E ແວວນເອິນ 1597 - 0% 13% Carbohydrate 350 Robes ray Added Sugars 58% Protale 00 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 kp affect period of vibration and decay ratio. Effect: Increasing ka slightly increases the period of vibration (7period). Explain: Effect: Increasing ka increases/decreases the decay ratio (circle one). Explain: Effect: Increasing kp increases/decreases the period (circle one). Explain: 2√mk (5.3) Effect: Increasing kp slightly increases/decreases the decay ratio. (read the explanation, then circle one). Explanation: Increasing kp increases the effective stiffness in the system which increases wn, which increases wa Decreasing decreases the logarithmic which decreases Tperiod. Increasing k, decreases since 2 TC (the numerator decreases and the denominator increases). decrement since logDecrement (6.2) √√1-(2 Decreasing logDecrement increases the decay ratio since decayRatio = e-logDecrement = (6.2) 1.12 PD Feedback control of a critically-damped mass-spring-damper system Hw 4.11 considered underdamped control systems for vehicle ride. Your engineering manager suggests critical-damping may be better at decreasing force transmitted to the vehicle's occupants. She wants you to reconsider the same mass-spring-damper system and use the same proportional-derivative (PD) feedback control law, namely f(y,y) mjj+by+ky = f(u.9) f(y,y) -kay - kpy • To ensure you understand the critical-damping design specifications, make a rough hand-sketch of the first 3 sec- onds of y(t) with y(3) = 0.025 m. Use the same initial values as in Hw 4.11, namely y(0) = 1.0 m and y(0) = 0. The standard mathematical form for a linear constant- coefficient homogeneous 2nd-order ODE is 0.9 08 0.7 m f(t) 0.5 0.4 0.3 0.2 0.1 0 0 0.5 13 Time (seconds) 2 2.5 3 ⚫ Determine the equation governing w, so y(t) is critically-damped with y(3) = 0.025 m. +2(w + w = 0 polution for u(t) in Hw 3.8. 0 t (sec) 0 y(t) (meters) 1.5 2 2 decay Ratio 0.0 0.5 0.5 * Tperiod 2.dR°.5 0.5 1.0 1.0 Tperiod 0.5 2-dR 0 1.5 1.5* Tperiod -0.25 2.dts -0.5 -1 2.0 2.0* Tperiod 0.125 2-dk2 -15 Note: In the 3rd column, put an exact numerical value for y(t). Optional: In the 4th-column, put an expression involving decayRatio. -2 0 0.25 05 0.75 time (seconds) 1.25 15 4.11 PD feedback control of an underdamped mass-spring-damper system Homework 4.9 sized leaf springs and shock absorbers to tune a vehicle's ride. The vehicle 200 kg, linear damping was modeled as a "mass-spring-damper" system, with mass m = 7948, and forcing function f(t). constant b 204.3 Ns, linear spring constant k 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. = m' mj + (b+ka) is + (k + kp) 31 = 0 1.75 2 foo m f(y, y) = -ka y - kp y (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). (b) Ride characteristics Original vehicle A. Simulates improved shocks B. Simulates stiff springs/shocks N sec m Vibration period (secs) Decay ratio ka kp 45 1.0 0.6 0 0 1.0 0.75 0.4 162.2 115.7 0.4 284.4 6387 Your manager asks: "How does the control system affect the ride?" Using only vibration period, decay ratio, and initial values y(0) = 1.0 m, y(0) = 0. sketch (by hand) the first 3 seconds of the vehicle's response and determine y(3) for each suspension system. Ride characteristics Original vehicle B. Stiff springs/shocks A. Improved shocks y(-3) 0.216 meters meters meters 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 0.25 0.5 0.75 1 1.25 15 1.75 2 2.25 2.5 2.75 Time (seconds) (c) Which control system is more effective at driving y(t) to 0? A/B. Why is it more effective? Copyright 1992-2024 Paul Mitiguy. All rights reserved. Decay ratio measures amplitude decay between successive oscillations. In 191