MEEN 364 Exam 2, April 2, 2015

Problem1

Obtain the transfer function 𝐶(𝑠)/𝑅(𝑠) and 𝐶(𝑠)/𝐷(𝑠)of the system shown below.

MEEN 364 Exam 2, April 2, 2015

Problem2

Find the inverse Laplace transform (𝑔(𝑡)) of the given function 𝐺(𝑠). Using final value theorem, estimate

the final value of the function in time domain.

(Hint: the denominator has a common factor of 𝑠)

𝐺(𝑠) =5𝑠2 + 4𝑠 + 8

𝑠3 + 4𝑠

MEEN 364 Exam 2, April 2, 2015

Problem3

Consider the offset beam (mass = m, Length = L) shown below, where 𝐹𝑐 = 𝑐𝐿�̇�𝑐𝑜𝑠𝜃 and 𝐹𝑘 = 2𝑘𝐿𝑠𝑖𝑛𝜃.

The E.O.M. can be (and is) shown as

1

3𝑚𝐿2�̈� +

1

2𝑐𝐿2(𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃)�̇� + 2𝑘𝐿2 𝑠𝑖𝑛2𝜃 −

1

2𝑚𝑔𝐿𝑠𝑖𝑛𝜃 = 0

For this beam,

a) Find the equilibrium point 𝜃0, in terms of 𝑘, 𝐿, 𝑚 or any other relevant beam parameters.

b) Linearize E.O.M. about 𝜃0. Keep your answer in terms of 𝜃0 (i.e. do not substitute your answer

from part a)

MEEN 364 Exam 2, April 2, 2015

Problem4

For the fluid system given below,

a) Obtain the governing differential equations. Then put the equations into state space

representation. Use [ℎ1 ℎ2 ℎ3]𝑇as you state vector, use 𝑞𝑖and 𝑞𝑑as the inputs, and use

[𝑞1 𝑃3 𝑞𝑜]𝑇as the output (keep the order in matrices).

b) Now consider a special case. Only consider tank 3 (the last one) and suppose that 𝑅3valve is

working with turbulence. We know that in case of turbulence, the relation between output flow

and head of the tank would be 𝑄 = 𝐾√𝐻 where Q is the output flow, H is the head, and K is a

given constant (here equal 0.01). Assume at 𝑡 < 0, tank 3 is in steady state condition, and all the

inflows are equal to 𝑄𝑖 = 0.015𝑚3

𝑠. At 𝑡 = 0, inflows are closed and there is no inflow for 𝑡 ≥ 0.

Find the time required to empty the tank to half of the steady state head. The capacitance C of

the tank is 2 𝑚3.

ℎ2

MEEN 364 Exam 2, April 2, 2015

Problem5

Using a PD controller for a given plant, we want to set the proportional gain 𝐾𝑝and derivative gain 𝐾𝑑to

reach to the following desired specifications:

a) The maximum overshoot to the unit step response should be 0.2.

b) The peak time should be 1 sec.

Compute the gains that satisfy the aforementioned properties, then calculate the rise time and the settling

time (1% criterion). Assume 𝐽 = 1𝐾𝑔

𝑚2 , 𝐵 = 1 𝑁.𝑚.𝑠

𝑟𝑎𝑑