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 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) The time constant is determined as follows: 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