Question

. Question 1 (40 marks) This question asks you to demonstrate your understanding of the following learning objectives LO 1.6 Express the Laplace Transform of common mathematical functions and linear ordinary differential equations using both first principles and mathematical tables. LO 1.7 Construct transfer functions for linear dynamic systems from (i) differential equations and (ii) reduction of block diagrams. LO1.8 Determine the time response of a Linear SISO system to an arbitrary input and having arbitrary initial conditions. LO 1.9 Develop system models that can be used to predict dynamic behaviour of mechanical, electrical and electromechanical systems LO 2.2 Analyse the transient response of SISO linear dynamic systems using the system's poles and zeros. LO 2.5 Construct root loci from the characteristic equation of a system using the Evans Root Locus rules and the MATLAB control systems toolbox. LO 2.6 Employ root loci to analyse the dynamic behaviour under parameter variation of SISO systems. LO2.7 Calculate the system gain to realise a specific closed loop pole from a root locus. L02.8 Construct Nyquist diagrams for SISO systems using (i) using asymptotic behaviours and (ii) the MATLAB control systems tool. Figure 1 shows a mass-spring-damper system in which y(t) denotes the displacement of the mass M from its rest position. A force f(t) is applied to the mass M as shown. Take M = 1 kg and D = D2 = 1 Ns/m. . . a) Derive the differential equation relating f(t) to y(t). b) Derive the transfer function () relating Y(s) to F(s). c) Sketch the locus of the poles of G(s) for 0 < K <0. d) Find the value of K for which the response is critically damped e) Find the value of K for which the response is marginally stable. f) Take K = 1 N/m. Suppose the f(t) = cos(t). Find the steady-state response Yss(t). g) Take K = 1 N/m. Sketch the Nyquist diagram of G(s). In answering these questions take care to show the steps in your working. Marks will be awarded for demonstrating your mastery of the learning objectives listed. LLLLLLLL 1 M y(t) LLID

Answer 1

J YS) FO 8425th This is closed loop transfer function Open loop transfer function s(s+2) We sketch root locus ask varier from 0 to a Im(s) * -2 Reis d) Value of k for critically damped system: Poles are real and equel.,. $21 comparing s²+2stk, s²4ę waston

29 Wne 2 2x 1 x Wna 2 » Whal rad/sec Kz wn² z q2 = 1 (e) Value of k for marginally stable response. &zo s=+ jun since the system has dampers, no such value of exists which will make it marginally stable. This is also evident from root locus. f) Ket Ys Fis- s²2sti fit) e cost → FS) s S?+1 S y sz Frs) 64+25+1 (8+1 (84+25+1) Yss(t) = lim sys to

lim 5-0 (5+1) (s?+25+1) g) Nyquist diagram A Im (GH) . Wza ReGH wzo Ny quist diagram is drawn for loop TF: open GS z 1 2 s$+2)

Soln- Di SH I yet) KED2 It +Kyt) = fit @ Differential eqoi:- Md?y +B+Didy dt? Given, M=1 kg, Di-Deal Nsim, 2 dy dt thy(t) = f(t) (6) Transfer function Writing in Laplace domain → dry + + 2 at s? YUSS+ 25 Ys) + K YO) z FS) → Yo (5+25+) = F(S)