School of Mechanical Engineering Purdue University. ME375 Dynamic Response - 1

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1 Dynamic Response of Linear Systems Linear System Response Superposition Principle Responses to Specific Inputs Dynamic Response of f1 1st to Order Systems Characteristic Equation - Free Response Stable 1st Order System Response Dynamic Response of 2nd Order Systems Characteristic Equation - Free Response Stable 2nd Order System Response Transient and Steady-StateState Response ME375 Dynamic Response - 1

2 Linear System Response ( n) ( n1) ( m) y a y a ya y a y b u bu b u n1 Superposition Principle i m 1 0 Input u 1 (t) u 2 (t) k 1 u 1 (t)+k k 2 u 2 (t) Linear System Output y 1 (t) y 2 (t) The response of a linear system to a complicated input can be obtained by studying how the system responds to simple inputs, such as zero input, unit impulse, unit step, and sinusoidal inputs. ME375 Dynamic Response - 2

3 Typical Responses Free (Natural) Response Response due to non-zero initial conditions (ICs) and zero input. Forced Response Response to non-zero input with zero ICs. Unit Impulse Response Response to unit impulse input. u(t) Unit Step Response Response to unit step input (u (t) = 1). u(t) Sinusoidal Response Response to sinusoidal inputs at different frequencies. The steady state sinusoidal response is call the Frequency Response. Time t Time t ME375 Dynamic Response - 3

4 Dynamic Response of 1st Order Systems Characteristic Equation: y ay bu s a 0 Free Response [ y H (t)]: (u u = 0) y () t Ae y H (t) H at a > 0 a = 0 a < 0 e.g. y ( t ) Ae H 4 t y H (t) 0 e.g. y t Ae t ( ) H y H (t) e.g. y ( t ) Ae H ( 4 ) t Time (t) Time (t) Time (t) Q: What determines whether the free response will converge to zero? Q: How does the coefficient, a, affect the converging rate? ME375 Dynamic Response - 4

5 Response of Stable 1st Order System Stable 1st Order System y ay bu y y Ku where : : Time Constant K : Static (Steady State, DC) Gain Unit Step Response ( u = 1 and zero ICs ) yt () y () t y () t H A e t P K IC : y(0) A K A yt () y(t) K y H (t) = K e -t/ y P (t) = K Time t ME375 Dynamic Response - 5

6 Response of Stable 1st Order System Normalized Unit Step Response (u= 1 & zero ICs) 1 y y Ku 0.9 yt () K( 1 e t 0.8 ) Normalized (such that as t, 1): y n y ( t ) yn ( t ) K t ( 1 e ) Normalized Response Time [ t ] Time t ( 1 e t/ ) ME375 Dynamic Response - 6

7 Response of Stable 1st Order System Effect of Time Constant yy y Ku 1 yt K e t () ( 1 ) Normalized: y() t y () t ( e t n 1 ) K Slope at t = 0: d () dt y t n d ( dt y n( 0) Q: What is your conclusion? Norma alized Respon nse Time [sec] ME375 Dynamic Response - 7

8 Example Vehicle Acceleration b m v F m 1 v v F v max b b 160 Standing-Start Acceleration; Dodge Viper SRT Standing-Start Acceleration; Lincoln Aviator SUV Speed (MPH) Speed (MPH) Time (sec) Time (sec) ME375 Dynamic Response - 8

9 Response of Stable 1st Order System Normalized Unit Step Response yy y Ku yt K e t () ( 1 ) Normalized (such that as t, 1): 0.6 y n yt () y ( ) ( t n t 1 e ) K Initial Slope 1 y ( 0 n ) K y( 0) Norm malized Respon nse Time [ t ] ME375 Dynamic Response - 9

10 Response of Stable 1st Order System Q: How would you calculate the response of a 1st order system to a unit pulse (not unit impulse)? Q: How would you calculate the unit impulse response of a 1st order system? u(t) t 1 (Hint: superposition principle?!) Time t Q: How would you calculate the sinusoidal response of a 1st order system? ME375 Dynamic Response - 10

11 Dynamic Response of 2nd Order Systems y a 1y a0y b1u b0u Characteristic Equation: 2 s a sa Free Response [ y H (t)]: (u u = 0) Determined by the roots of the characteristic equation: Real and Distinct [ s 1 & s 2 ]: H () s t y t Ae 1 A2e 1 s2t Real and Identical [ s 1 =s 2 ]: H () s t y t A1e A2te Complex [ s 1,2 = j 1 s1t t t y ( t) e ( A cos( t) A sin( t) ) Ae cos( t ) H 1 2 ME375 Dynamic Response - 11

12 Dynamic Response of 2nd Order Systems Free Response (Two distinct real roots) s1t s2t 1 2 s t yh () t Ae A e y t Ae A e H () H () y t Ae 1 A2e 1 2 s 0 & s 0 s 0 & s 0 s 0 & s s t s t 1 2 s t Img. Img. Img. Real Real Real y H (t) y H (t) y H (t) Time (t) Time (t) Time (t) ME375 Dynamic Response - 12

13 Dynamic Response of 2nd Order Systems Free Response (Two identical real roots ) s1t s1t 1 2 y H () t Ae A te yh () t Ae 1 A2te y H () t A 1 e A 2 te s s 0 s s 0 s s s t 1 2 s t s t 1 2 s t Img. Img. Img. Real Real Real y H (t) y H (t) y H (t) Time (t) Time (t) Time (t) ME375 Dynamic Response - 13

14 Dynamic Response of 2nd Order Systems Free Response (Two complex roots) t t y t Ae cos( t t Ae cos( t t Ae H () ) y H () ) y H () cos( t ) s j & 0 s j & 0 s j & 0 1,2 1,2 1,2 t Img. Img. Img. Real Real Real y H (t) y H (t) y H (t) Time (t) Time (t) Time (t) ME375 Dynamic Response - 14

15 Example Automotive Suspension m y g k b r my by ky br kr 0.02 Response to Initial Conditions for free response: my by ky 0 b k y y y 0 m m y28y 400y 0 Amplitude Time (sec) ME375 Dynamic Response - 15

16 Dynamic Response of 2nd Order Systems Q: What part of the characteristic roots determines whether the free response is bounded, converging to zero or blowing up? Q: For a second order system, what conditions will guarantee the system to be stable? (Hint: Check the characteristic roots ) Q: If the free response of the system converges to zero, what determines the convergence rate? ME375 Dynamic Response - 16

17 Response of Stable 2nd Order System Stable 2nd Order System y a y a y bu y y 2 y K n n n u where n > 0 : Natural Frequency [rad/s] > 0 : Damping Ratio K : Static (Steady State, DC) Gain Characteristic roots s 2 2 n s n 2 0 s ( 1) n n 2 Img. n 1: 1: 1: Real n ME375 Dynamic Response - 17

18 Response of Stable 2nd Order System Unit Step Response of Under-damped damped 2nd Order Systems ( u = 1 and zero ICs ) Characteristic equation: n n n y y y K u s 2 2 n s n 2 0 s n jn (1 ) d 2 y (t) = ME375 Dynamic Response - 18

19 Response of Stable 2nd Order System Unit Step Response of 2nd Order Systems 16K 1.6K y MAX 1.4K 12K 1.2K OS Unit Step Response K 0.8K 0.6K T d 0.4K 0.2K 0 t P Time [sec] t S ME375 Dynamic Response - 19

20 Response of Stable 2nd Order System Peak Time (t P ) Time when output y(t) ) reaches its maximum value y MAX. yt () n 1 t n K e cos( dt) sin( dt) d d y () t dt Find t such that y ( t ) 0 P t P P Percent Overshoot (%OS) At peak time t P the maximum output F y y( t ) K 1e MAX The overshoot (OS)) is: OS y y MAX P SS The percent overshoot is: %OS F HG y SS HG I K J OS J100% y( 0) 12 I KJ ME375 Dynamic Response - 20

21 Response of Stable 2nd Order System Settling Time (t S ) Time required for the response to be within a specific percent of the final (steady-state) state) value. Some typical specifications for settling time are: 5%, 2% and 1%. Look at the envelope of the response: % 1% 2% 5% t S Q: What parameters in a 2nd order system affect the peak time? Q: What parameters in a 2nd order system affect the % OS? Q: What parameters in a 2nd order system affect the settling time? Q: Can you obtain the formula for a 3% settling time? ME375 Dynamic Response - 21

22 In Class Exercise Mass-Spring-Damper System K x Q: What is the static (steady-state) state) )g gain of the system? B I/O Model: M f(t) Q: How would the physical parameters (M, B, K) affect the response of the system? M x Bx K x f ( t ) (This is equivalent to asking you for the ( This is equivalent to asking you for the relationship between the physical parameters and the damping ratio, natural frequency and the static gain.) ME375 Dynamic Response - 22

23 Transient and Steady State Response Ex: Let s find the total response of a stable first order system: y 5 y 10u to a ramp input: u(t) ) = 5t with IC: y(0) = 2 Total Response Y() s U() s y(0), where U() s L 5t Transfer Function Gs () Y( s) PFE: A1 A2 A3 Y() s yt () ME375 Dynamic Response - 23

24 Transient and Steady State Responses In general, the total response of a stable LTI system ( n) ( n1) ( m) ( m1) ay a y ayaybu b u bubu n n1 1 0 m m1 1 0 m m1 bms bm 1s bs 1 b0 Ns () bm( sz1)( sz2) ( szm) Gs () n n1 ans an 1s as 1 a0 D() s an( s p1)( s p2) ( s pn) to an input u(t) ( ) can be decomposed into two parts: where Transient Response (y T (t)) yt () y() () T t y SS t Transient Response Steady State Response Contains the free response y Free (t) of the system plus a portion of the forced response. Will decay to zero at a rate that is determined by the characteristic roots (poles) of the system. Steady State Response (y SS (t)) will take the same form as the forcing input. Specifically, for a sinusoidal input, the steady state response will be a sinusoidal signal with the same frequency as the input but with different magnitude and phase. ME375 Dynamic Response - 24

25 Transient and Steady State Response Ex: Let s find the total response of a stable second order system: y 4y 3y 6u to a step input: u(t) ) = 5 with IC: y( 0) 0 and y( 0) 2 Total Response PFE: ME375 Dynamic Response - 25

26 Steady State Response Final Value Theorem (FVT) Given a signal sltf(s) s F(s), )ifthe poles of sf(s) all lie in the LHP (stable region), then f(t) ) converges to a constant value f(). f() ) can be obtained without knowing f(t) ) by using the FVT: f ( ) lim f () t lim sf ( s) t s0 Ex: : A model of a linear system is determined to be: y 4 y 1 2 y 4 u 3 u (1) if a constant input u = 5 is applied at t = 0, determine whether the output y(t) ) will converge to a constant value? (2) If the output converges, what will be its steady state value? ME375 Dynamic Response - 26

27 Steady State Response Given a stable LTI system a y a y a ya ybu b u bubu () n ( n 1) ( m) ( m1) n n m m The corresponding transfer function is m m1 bms bm1s b1sb0 bm( s z1)( s z2) ( s zm) G( s) n n1 a s a s a s a a ( s p )( s p ) ( s p ) n n n 1 2 n Steady State Value of the Free Response Recall the free response of the system is: Y F( () s () s 1 as a s asa Free n n n n1 1 0 Apply FVT: ME375 Dynamic Response - 27

28 Steady State Response Steady State Value of the Unit Impulse Response Y () s G () s U () s Apply FVT: Steady State Value of the Unit Step Response Apply FVT: Ys () Gs () Us () 1st Order Systems: Gs () b0 asa 1 0 2nd Order Systems: Gs () bsb as as a G(0) G(0) ME375 Dynamic Response - 28