3.4.3 Responses of measuring system to typical excitations A second-order system H(S) 2 +1 (assuming its static sensitivity K-1) h sIn sant (underdamp ed, s<I) criticall y damped,s=1) (overdamped,s>1) (3.46)
A second-order system (assuming its static sensitivity K=1) 3.4.3 Responses of measuring system to typical excitations ( ) 1 2 1 2 2 + + = n n s s H s ( ) ( ) ( ) ( ) (overdampe d, 1) 1 (criticall y damped, 1) sin 1 (underdamp ed, 1) 1 1 1 2 2 2 2 2 2 − − = = = − − = − + − − − − − − t t n t n n n t n n n n h t e e h t t e h t e t (3.46)
3.4.3 Responses of measuring system to typical excitations 0.8 15=0.65 1.0 0.2 0.4 ig. 3. 19 Impulse responses for second-order system with different dampimgs
3.4.3 Responses of measuring system to typical excitations Fig. 3.19 Impulse responses for second-order system with different dampimgs
3.4.3 Responses of measuring system to typical excitations UThe unit impulse does not exist in reality. Often in engineering, an approximation is made by use of a pulse signal with very short time duration for the impulse signal Example: a shock to a system, if the shock duration is shorter than t/10, where t is the systems time constant then the shock can be considered as a unit mpule
❑The unit impulse does not exist in reality. Often in engineering, an approximation is made by use of a pulse signal with very short time duration for the impulse signal. ▪ Example: a shock to a system, if the shock duration is shorter than τ/10, where τ is the system’s time constant, then the shock can be considered as a unit impulse. 3.4.3 Responses of measuring system to typical excitations
3.4.3 Responses of measuring system to typical excitations a/(x/) qo/u/rI 000 1.0 0000.9990.100 Approximate 0010.9900.995 0.9 01|09050913 208190826 0.7 Exac↑- 500.00674000681 1000.0000500000505 0.6 0 0.4 0.3 0.2 0. 0.20.3040.50.60.7080.9 Fig. 3.2 1 Exact and approximate impulse response
3.4.3 Responses of measuring system to typical excitations Fig. 3.21 Exact and approximate impulse response
3.4.3 Responses of measuring system to typical excitations 2. Step response The unit impulse function d()= (363) The step function is 5()=( (3.64 一 The response of first-order system to a unit step input y()=1-e (365) The related laplace transform is (3.66 s(as
2. Step response The unit impulse function The step function is The response of first-order system to a unit step input The related Laplace transform is 3.4.3 Responses of measuring system to typical excitations ( ) ( ) dt d t t = (3.63) ( ) ( ) − = t ' t t dt (3.64) ( ) t y t e − = 1− (3.65) ( ) ( 1) 1 + = s s Y s (3.66)