1
SOLUTIONS of ME 375 EXAM #1
Tuesday |February 18, 2002
Division Meckl 12:30 / Gong 2:30 (circle one)Name______
Instructions
(1)This is a closed book examination, but you are allowed one 8.5x11 crib sheet.
(2)You have one hour to work all three problems on the exam.
(3)Use the solution procedure we have discussed: what is given, what are you asked to find, what are your assumptions, what is your solution, does your solution make sense. You must show all of your work to receive any credit.
(4)You must write neatly and should use a logical format to solve the problems. You are encouraged to really “think” about the problems before you start to solve them. Please write your name in the top right-hand corner of each page.
(5)A table of Laplace transform pairs and properties of Laplace transformsis attached at the end of this exam set.
Problem No. 1 (50%)______
Problem No. 2 (30%)______
Problem No. 3 (20%)______
TOTAL(***/100%)______
February 18, 2003Name ______
PROBLEM NO. 1 (50%)
Modeling of a Mechanical System
Consider the mechanical system illustrated in Fig. 1.1. The mass of disk is Md and it is uniformly distributed. There is no friction between the cart and ground. There is also NO friction between the disk and the cart. We are interested in horizontal motion only and assume that the rotational angle of the disk is very small. The outputs that we are interested in are the spring force caused by spring K2and the damping force caused by damper B3.
a)Determine the degree of freedom of this system. Please specify each degree of freedom. (2%)
Solution. The degree of freedom is 3 and includes the horizontal translation of cart, the horizontal translation of mass center of disk and rotation of disk about its center of mass.
b)What is moment of inertia of disk around the center of mass? (2%)
Solution.Since the mass of disk is uniformly distributed, the moment of inertia around the center of mass is .
February 18, 2003Name ______
PROBLEM NO. 1 (Continued)
c)Clearly show free body diagram of each element in this system. Please use the positive directions indicated in Fig. 1.1. (16%)
Solution.
d)Find equations of motion (EOMs) for this system. (15%)
Solution.
Elements’ laws
……………………(1.d.1)
…………(1.d.2)
…………(1.d.3)
Cart M: ………………(1.d.4)
DiskMd: ………………(1.d.5)
Substituting (1.d.1) – (1.d.3) into (1.d.4), we have
……………(1.d.6)
Substituting (1.d.2) and (1.d.3) into (1.d.5), we have
……………(1.d.7)
which is equivalent to
……………(1.d.8)
February 18, 2003Name ______
PROBLEM NO. 1 (Continued)
e)Represent EOMs obtained in d) in MATRIX form. (5%)
Solution.
EOMs obtained in part c) can be written into the following matrix form
…………………(1.e.1)
f)Choosing a set of appropriate state variables, represent system model in state space equation. Make sure that output equation is also presented. (10%)
Solution.
Let state variable vector be . Using EOMs in part d) and noticing the spring force in (1.d.2) and damping force in (1.d.3), we have the following state-space representation:
…(1.b.1)
February 18, 2003Name ______
PROBLEM NO. 2 (30%)
Response of a dynamic system by using Laplace transform (LT) and inverse Laplace transform (ILT)
Suppose we have a system with the following model
……………………………………………….…(2.1)
where y is the output of the system and u is the input of the system. Initial conditions are ,, and . Let Y(s) and U(s) denote the Laplace transforms of y(t), and u(t), respectively.
a)Based on the given model, express Y(s) in terms of U(s) and initial conditions. (6%)
Solution. Taking Laplace transform of (2.1) yields
…(3.a.1)
Solving for Y(s), we have
………………………(3.a.2)
b)Suppose that , , and u(t)=0. Find output y(t).
Solution. Substituting the given initial conditions in equation (3.a.2 )and noticing zero input, we have
………………………………………(3.b.1)
We have the following partial fraction expansion
……………………………(3.b.2)
where coefficients ’s can be found by residual formula as follows
…………………(3.b.3)
Taking ILT of (3.b.2), we obtain the response as follows
…………………………………………………(3.b.4)
February 18, 2003Name ______
PROBLEM NO. 2 (Continued)
c)If all the initial conditions are zeros, please find the output response under the input , where is a unit step function. (14%)
Solution. The Laplace transform of the given input is
…………………………………………………………………………………(3.c.1)
Substituting (3.c.1) into (3.a.2) and noticing the zero initial conditions, we have
………………………………………(3.c.2)
We have the following partial fraction expansion:
…………………………(3.c.3)
where coefficients ’s can be found by residual formula as follows
…(3.c.4)
Taking ILT of (3.c.4), we obtain the response as follows
………………………………………………(3.c.5)
February 18, 2003Name ______
PROBLEM NO. 3 (20%)
Transfer function
Consider the mechanical system described in the following figure.
Fig. 3.1 Schematic of system
You have already obtained the equations of motion of this system as follows
…………………………………….(3.1)
where , and are displacements of mass , and , respectively, in absolute coordinates, and is the external force acting on . You are interested in the displacement . Assume that all initial conditions are zeros.
a)Express in terms of, where is Laplace transform of . (10%)
Solution.Keeping in mind zero initial conditions, taking Laplace transform of (3.1) yields
……………………(3.a.1)
From the second equation of (3.a.1), we have
…………………………………………………(3.a.2)
Substituting (3.a.2) into the first equation of (3.a.1), we obtain
…………(3.a.3)
From the last equation in (3.a.3), we arrive at
………………………(3.a.4)
February 18, 2003Name ______
PROBLEM NO. 3 (Continued)
b)Find transfer function from input to the output . (3%)
Solution. By definition of transfer function and from equation (3.a.3), it can be found that transfer function from input to the output is
…………………………(3.b.1)
c)Given the following parameter values: M1 = M2 = 1 kg, B1 = 1 N/m/s, and K2 = 1 N/m. Find the steady-state value of when the input force is a unit impulse. (7%)
Solution. When input is a unit impulse, by equation (3.a.3), the Laplace transform of output is
…………………………………………………(3.c.1)
By using final value theorem, the steady-state value of is
………………(3.c.2)