ENGG 2400 Lecture Notes - Lecture 13: Characteristic Polynomial, Damping Ratio, Step Response

• School: University of Guelph
• Department: Engineering
• Course: ENGG 2400
• Professor: Julie Vale

Document Summary

Second order step and impulse: damping and oscillation, characteristic polynomial, z, n , d , %os , tmax. Divide original equation by (cid:3041)(cid:2870: (cid:1877)(cid:4666)(cid:1872)(cid:4667)= ((cid:3117)(cid:3289)(cid:3118))(cid:3118)+(cid:4672)(cid:3118)(cid:3289)(cid:4673)+(cid:2869, (cid:4666)(cid:4667)= (cid:2869)(cid:3118), (cid:3041)= (cid:1863)/ (cid:3041) is the undamped natural frequency k is the static gain z is the damping coefficient: z = 0, k = 1/k. Relating the resonance equation to the above equations. Example: (cid:883)6(cid:1877)(cid:4662)+(cid:885)(cid:1877)=(cid:886)(cid:884)(cid:1876, (cid:4666)(cid:883)6+(cid:885)(cid:4667)(cid:1877)=(cid:886)(cid:884)(cid:1876, (cid:883)6+(cid:885) is the characteristic polynomial. Finding the roots of (cid:2870)+(cid:884)(cid:1878)(cid:3041)+(cid:3041)(cid:2870) results in: = (cid:1878)(cid:3041) (cid:1862)(cid:3041) (cid:883) (cid:1878)(cid:2870, z relates to how the system is dampened. The roots of the characteristic polynomial are the poles of h(d: z=0 for the undamped case, z is between 0 and 1 for the underdamped case, z=1 for the critically damped case, z>1 for the overdamped case. %overshoot is the (cid:858)y(cid:859) value that the goes above the steady state (cid:858)k(cid:859) The time where the maximum value occurs is found using.