6 - DynamicResponse6 filled

6 dynamicresponse6 filled

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Unformatted text preview: ME575 Handouts Dynamic Response of LTI Systems • Linear System Response – Superposition Principle – Responses to Specific Inputs • • • • • Free and Forced Response Forced Response of 1st Order Systems Forced Response of 2nd Order Systems Effect of Zeros on Step Response Transient and Steady-State Response ME575 Session 6 – Dynamic Response School of Mechanical Engineering Purdue University Slide 1 Linear System Response & & y ( n ) + an −1 y ( n −1) + L + a2 && + a1 y + a0 y = bm u ( m ) + L + b1u + b0 u y • Superposition Principle Input u1 (t) Output Linear System y1 (t) y2 (t) u2 (t) k1 u1 (t) + k2 u2 (t) The response of a linear system to a complicated input can be obtained by studying how the system responds to simple inputs, such such as zero input, unit impulse , unit step, and sinusoidal inputs. zero step, and sinusoidal ME575 Session 6 – Dynamic Response School of Mechanical Engineering Purdue University Slide 2 1 ME575 Handouts Typical Responses • Free (Natural) Response – Response due to non-zero initial conditions (ICs) and zero input. non- • Forced Response – Response to non-zero input with zero ICs. non– Unit Impulse Response – Unit Step Response Forced response to unit impulse input. Forced response to unit step input (u (t) = 1). u(t) u(t) – Sinusoidal Response Forced response to sinusoidal inputs at different frequencies. The steady state sinusoidal response is call the Frequency Frequency Response. Response Time t Time t School of Mechanical Engineering Purdue University ME575 Session 6 – Dynamic Response Slide 3 Step Response of 1st Order System • Standard Form of Stable 1st Order System & y + ay = b u ⇒ & τy + y = Ku where τ : Time Constant Time K : Static (Steady State, DC) Gain Static – Unit Step Response ( u = 1 and zero ICs ) and y (t ) = K − K e −t τ y (t ) K ySS(t) = K Time t yT(t) = − K e -t/τ ME575 Session 6 – Dynamic Response School of Mechanical Engineering Purdue University Slide 4 2 ME575 Handouts Step Response of 1st Order System Normalized Unit Step Response (u = 1 & zer...
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