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Unformatted text preview: MEEN 260 Introduction to Engineering Experimentation Homework 10: Laplace Transform, and Frequency Response Solution
Assigned: Thursday, 9 Apr. 2009 Due: Thursday, 16 Apr. 2009, 5:00pm Learning Objectives: After completing this homework assignment, you should be able to: 1) 2) 3) 4) 5) Determine the Laplace Transform of a signal using the definition, tables, or properties of the Laplace Transform Utilize the Laplace Transform to find the Transfer Function of a dynamic system represented by a system of differential equations Utilize the Laplace Transform to solve for the transient response of a dynamic system Discuss the difference between the Laplace and Fourier Transforms and their respective uses Using the Transfer Function of a system, determine and plot the associated frequency response, and determine the steady state response of a system to a harmonic input signal Homework Problems: Problem 1) Definition of Laplace Transform Using the mathematical definition, compute the Laplace transform for the function: f (t ) = 3t + t cos(2t ) Solution: From the mathematical definition, we split the function into two pieces: 3 3· We use u-v substitution (u=3t, dv=e-st) to get: ∞3 ∞ 3 3 0 0 The second piece of the function is more complicated. Recall that: · So we find: cos 2 We use u-v substitution (u=e-st, dv=cos(2t)) and get: sin 2 2 This does not give us a useful answer, so we perform a u-v substitution to the right hand side of the equation to obtain: cos 2 By rearranging the terms, we obtain: 4 4 cos 2 cos 2 cos 2 4 Next, we take the derivative of this expression with respect to s to obtain: 4 · cos 2 4 Finally, we put the terms together to obtain: 3 4 3 · cos 2 4 cos 2 Problem 2) Transfer Functions and Transient Response 2 Given the function: a) Determine the associated the transfer function b) Determine the time constant of this system c) Find and plot (Using excel/Matlab) the response, y(t), using the Laplace Transform for an input of: i) Unit impulse input, x(t) = δ(t) ii) Unit step input, x(t) = 1 for t>0, x(t) = 0 otherwise. Solution: The transfer function is derived as follows: 2 1 2 The time constant is determined as follows: 1 1 · 2 2 0.5 1 0.5 We use Matlab to determine the step and impulse response using the following code: Num = [0 1]; Den = [1 2]; g = tf(num,den) step(g) 1 Figure 1: Step Function Response of Transfer Function u=1 impulse(u*g) Figure 2: Impulse Response of Transfer Function Problem 3) Laplace vs. Fourier Transform, Frequency Response Consider a car driving over a series of speed bumps at 2 kilometers per hour. Each bump is 0.2 meters tall and 0.3 meters wide. The car’s suspension system can be represented as a massspring-damper system with spring constant, k = 4 N/m, damping coefficient b = 3 N*s/m, and mass 10 kg. Assume that the speed bumps can be approximated as a sine wave, and that the car is represented by the differential equation below, where y(t) is the vertical position of the car, and x(t) is the vertical displacement of the road/speed bumps: a) Find the associated transfer function, and determine the damping ratio and natural frequency b) Determine the complete response of the system to the speed bumps, using the transfer function and the Laplace Transform (Note: solving for the response in this manner is like assuming that the vehicle is traveling on smooth roads for t<0, and then at t=0 the car encounters the speed bumps) c) Determine the frequency response of the system, by substituting into the transfer function d) Using Matlab/Excel, plot the frequency response (magnitude vs. frequency, and phase vs. frequency) with frequency on a log scale e) Determine the steady state response of the vehicle to the speed bumps, by a. calculating the magnitude and phase of the frequency response at the appropriate frequency b. indicate where these values can be found on the plot from part d) c. write an expression for the steady state response of the vehicle, i.e. y(t) = ? (Note: solving for the response in this manner is like assuming that the vehicle has been driving over an infinite series of speed bumps for all time) f) Using Matlab/Excel, on a same plot, show the complete system response, y(t) vs. time, obtained from part b), and the steady state response, y(t) vs. time, obtained from part e). Compare these and discuss briefly what you can conclude from the plots. Solution: We begin by determining the transfer function. Using the Laplace Transform on the system equation, we get: 0.3 0.4 0.3 0.4 From formulae in the lecture notes, we derive ωn and ς as: 0.24 b 0.625 2mω To find the response of the system, we must first determine X(s). Assuming the road to be a sinusoidal input with amplitude 0.1 and frequency 11.64 radians per second, we find: 11.64 0.1 · 0.1 sin 11.64 11.64 Since the transfer function is defined through H=Y/X, then to find Y(s), we must multiply H by X. This is given by: 0.3 0.4 11.64 · 0.1 · · 0.3 0.4 11.64 ς Rearranging, we get: To find y(t), we must take the inverse Laplace of this function. Since there are no table lookups for such a function, we must decompose it by the method of partial fractions: 0.3 0.4 · 0.3 0.4 11.64 11.64 0.3 0.4 0.3 11.64 0.4* 0.3 0.4 11.64 11.64 Multiplying both sides of the equation by 0.3 0.4 11.64 11.64 11.64 , we get: 0.3 0.4 0.3 0.4 We multiply out the right hand side of the equation and group like terms such that: 0.3 · 11.64 11.64 · 0.4 0.3 · 11.64 0.4 0.3 11.64 0.4 This system of equations may be solved in matrix form using the following Matlab code: a=0.3; b=0.4; w=11.64; w2=11.64^2; M=[1,0,1,0;0,1,a,1;w2,0,b,a;0,w2,0,b]; Mi=inv(M); N=[0;0;a*w;b*w]; Mi*N This yields: 0.026, So our equation becomes: 0.1 · 0.034, 0.026, 0.027 0.026 0.027 0.026 0.034 0.3 0.4 11.64 11.64 The first term is grouped so that we may simplify it as follows: 0.026 0.034 1.31 0.026 · 0.3 0.4 0.25 0.15 Now we apply the following Laplace transform identities: cos cos 1 We arrive at the full response of the system: 0.0026 · cos 0.614 0.00023sin 11.64
. sin 1 sin 1.89sin 0.614 0.0026 cos 11.64 This is shown graphically in Figure 1, below: Figure 3: Time response of the car to speed bumps The frequency response is given as follows: 0.3 0.3 0.4 0.4 The bode plot of the transfer function is coded by: a=0.3; b=0.4; num = [0 a b]; den = [-1 a b]; g=tf(num,den) bode(g) Figure 4: Bode Plot of Transfer Function To find the steady state response of the system, we find the magnitude and phase angle at the input frequency of 11.64: 0.16 √0.09 · 11.64 | 0.026 11.64 | 0.4 11.64 0.09 · 11.64 0.3 11.64 0.3 11.64 11.64 tan tan 1.48 0.4 11.64 0.4 So for the input of 0.1 sin 11.64 We say 0.0026sin 11.64 1.48 Comparing this with the full time response, we see the comparison: Figure 5: Comparison of Transient and Steady State Response From the Matlab code: t=0:.001:20; y=0.0026.*(exp(-0.23.*t).*(cos(0.614.*t)+1.89.*sin(0.614.*t)))0.0026.*cos(11.64.*t)-0.00023.*sin(11.64.*t); z=0.0026.*sin(11.64.*t+1.48); plot(t,y,t,z) ...
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