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MIDDLE EAST TECHNICAL UNIVERSITY
MECHANICAL ENGINEERING DEPARTMENT
ME 304 CONTROL SYSTEMS
SPRING 2015
HOMEWORK 3
Due Date: 30.03.2015 at 17:00
Prepared by
M.Uğur DİLBEROĞLU (B-183)
You should submit your homework on its due time. No extension will be given afterward.
Problem 1:
A mechanical system shown in Figure 1 consists of three masses and other mechanical
elements as well as a pulley.
Figure 1: The mechanical system
All the elements of the system are assumed to be ideal (i.e., pure and linear). The masses of
the three blocks are denoted as 𝑚𝑚1 , 𝑚𝑚2 , and 𝑚𝑚3 . The spring coefficients are denoted as 𝑘𝑘1
and 𝑘𝑘2 . The spring forces vanish when their terminal displacements are equal. The first two
mass blocks slide on viscous lubricants at the contact surfaces. The viscous friction forces are
characterized by the coefficients 𝑏𝑏1 and 𝑏𝑏2 . Also, the first two mass blocks have a viscous
damper in between with a coefficient 𝑏𝑏. The pulley is ideal without any inertia and
dissipation effect. The horizontal force 𝐹𝐹(𝑡𝑡) applied on the first mass block is one of the
inputs of the system. The other input is the gravitational acceleration 𝑔𝑔.
Hint 1: The gravitational acceleration as a time function is 𝑔𝑔(𝑡𝑡) = 𝑔𝑔 = constant. Therefore,
its Laplace Transform is 𝐺𝐺(𝑠𝑠) = 𝑔𝑔/𝑠𝑠.
Hint 2: Define the positivity directions to be rightward for the horizontal part and downward
for the vertical part.
Figure 1 shows the system in its initial position, where the displacements 𝑥𝑥1 (𝑡𝑡) , 𝑥𝑥2 (𝑡𝑡), and
𝑦𝑦(𝑡𝑡) are yet zero.
a) Draw the necessary free body diagrams.
b) Write down all the elemental equations together with the connectivity equations (if
any required).
Problem 2:
Figure 2: The mechanical system with frictionless surface
Assume that the friction coefficients of the sliding surface are negligible. That is,
𝑏𝑏1 ≈ 𝑏𝑏2 ≈ 0
In other words, all the equations derived in the solution to Problem 1 are still valid in a
simplified form with 𝑏𝑏1 = 𝑏𝑏2 = 0.
a) For the simplified system with the frictionless surface, identify the distinct unknown
variables in the system. List the corresponding simplified versions of the elemental
equations found previously. Check whether the number of the distinct unknown variables is
equal to the number of the equations.
b) In order to express the input-output relationship for the selected output 𝑌𝑌(𝑠𝑠), determine
the transfer functions 𝐺𝐺𝑦𝑦𝑦𝑦 (𝑠𝑠) and 𝐺𝐺𝑦𝑦𝑦𝑦 (𝑠𝑠) between 𝑌𝑌(𝑠𝑠) and the inputs 𝐹𝐹(𝑠𝑠) and 𝐺𝐺(𝑠𝑠) =
𝑔𝑔/𝑠𝑠.
c) Fill in the blocks of the detailed operational block diagram of the system given below.
Indicate the variables on the relevant branches.
𝐹𝐹(𝑠𝑠)
Input 1
Mass 1
Connector
Element
(Spring 1
& damper)
Mass 2
Connector
Element
(Spring 2)
𝑔𝑔/𝑠𝑠
Input 2
Mass 3
MECHANICAL SYSTEM
𝑌𝑌(𝑠𝑠)
Output