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ECE-205 : Dynamical Systems Homework #
Due : Tuesday March 23 at the beginning of class
Exam 1, Thursday September 23
1) For this problem, consider six second order systems described by the following differential equations:
( ) 9 ( ) 20 ( ) 20 ( ) ( ) 10 ( ) 25 ( ) 25 ( ) ( ) 4 ( ) 13 ( ) 13 ( ) ( ) 6 ( ) 8 ( ) 8 ( ) ( ) 6 ( ) 9 ( ) 9 ( ) ( ) 6 ( ) 13 ( ) 13 ( )
y t y t y t Kx t y t y t y t Kx t y t y t y t Kx t y t y t y t Kx t y t y t y t Kx t y t y t y t Kx t
a) Assume the systems are initially at rest and input is a step, x t ( ) Au t ( ), determine expressions for the system output by finding the forced and unforced responses and then solving for the unknown coefficients just as we did in class.
b) For the systems with real roots, show that your solution meets the two initial conditions ( y (0) y (0) 0 ).
For the systems with complex roots, determine and n from the governing differential equation, and show
that your solution agrees with the form
2 ( ) 1 1 si n( ) 1
y t KA e^ ^ n^ t d t
cos ^1 ( ) d n 1 ^2
Do not assume this is the form of the solution, but use it to check your answer.
Answers: 4 5 5 5 2 4 2 3 5 3
sin(3 56.3 )]
sin(2 33.
[1 1.
( ) [1 1. 03 8 7 )]
t t t t t o t t t t t o
y t KA KAe KAe y t KA KAe y t KA e
y t K
KAte t y t KA KAe KAe A KAe K y t K
te e
A
A t
2) The response of a second order system is
y t ( ) 1 1.0050 e t sin(9.95 0 0 t 1.4706 rad)
a) Take the derivative of this function to determine the time at which the maximum occurs (the time to peak)
b) Determine the maximum value of this function (the value at the time to peak)
c) Determine the percent overshoot using your answer to (b)
d) For this response determine , d , and n
e) Compute the percent overshoot using the formula
PO e ^1 ^ ^2 100%
and verify your answer to c.
3) One of the methods that can be used to identify and n for mechanical systems the log-decrement method,
which we will derive in this problem. If our system is at rest and we provide the mass with an initial displacement away from equilibrium, the response due to this displacement can be written
x t 1 ( ) Ae ^ n^ t cos( d t )
where
x t 1 ( ) = displacement of the mass as a function of time = damping ratio
n = natural frequency
d = damped frequency = n 1 ^2
After the mass is released, the mass will oscillate back and forth with period given by d^2 d
T ^ , so if we
measure the period of the oscillation ( Td ) we can estimate d .Let's assume t 0 is the time of one peak of the
cosine. Since the cosine is periodic, subsequent peaks will occur at times given by tn t 0 nTd , where n is an integer.
a) Show that 1 0 1
n T nd n
x t (^) e x t
^
b) If we define the log decrement as
1 0 1
ln (^ ) ( (^) n )
x t (^) x t ^ show that we can compute the damping ratio as
Figure 2. Initial condition response for second order system B.
4) (Prelab) Consider the following one degree of freedom system we will be utilizing this term:
a) Draw a freebody diagram of the forces on the mass.
b) Show that the equations of motion can be written:
m x t 1 1 ( ) c x t 1 ( ) ( k 1 (^) k 2 (^) ) x t 1 ( ) F t ( )
or
2 1 1
n n
x t^ x t x t KF t
c) What are the damping ratio , the natural frequency n , and the static gain K in terms of m 1 , k 1 , k 2 , and c 1?
-6 0 0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
12
Time (seconds)
Displacement (cm)
