4) Consider the following mass spring damper system: m2 u(t) Assuming that y, > y1, equation of motion (EOM) for this system can be obtained by applying the Newton's second law: = mỹ (i. e. RHS is positive) Then, equations are written as: m,y, = k2(y2 – Yı) – kıyı – cỷı (1) m2yz = u(t) - k2(y2 - Yı) (2) where u(t) is the input function to the system. Take m, = 1 kg, m2 = 2 kg, c = 10 Ns/m and k = 10 N/m, k2 = 20 N/m. Then, answer the following questions. ¥½(s) a) Find the transfer functions G, (s) = 1 and G2(s) = U(s) U(s) b) Use Laplace transform method and obtain the displacements in Laplace space (i.e. s domain) Y,(s) and Y2(s) utilizing initial conditions y, (0) = 0, y2(0) = 1, ỷ, (0) = 1 and y2(0) = 0. Use Cramer's rule for the solutions of Y, (s) and Y2(s). c) Using Inverse Laplace transformation, obtain displacements in time space (i.e. time domain) Y1(t) and y2(t) if, c.l. u(t) = 8(t) (Impulse function) (Ramp function starts from t=2) c.2. u(t) = {"2, 2st<∞ 0, 0st< 2 u(t) t-2 O 2 d) Plot y, (t) and y2(t) in the interval t e [0,5]s together in the same plot that you found in part c.1. e) Plot y, (t) and y½(t) in the interval t e [0,5]s together in the same plot that you found in part c.2. Hint: For the partial fractions of lengthy functions, you may use Matlab's residue () function b(s) which is useful to obtain fractions as- a(s) s-Pi s-P2 S-P3 S-Pn
4) Consider the following mass spring damper system: m2 u(t) Assuming that y, > y1, equation of motion (EOM) for this system can be obtained by applying the Newton's second law: = mỹ (i. e. RHS is positive) Then, equations are written as: m,y, = k2(y2 – Yı) – kıyı – cỷı (1) m2yz = u(t) - k2(y2 - Yı) (2) where u(t) is the input function to the system. Take m, = 1 kg, m2 = 2 kg, c = 10 Ns/m and k = 10 N/m, k2 = 20 N/m. Then, answer the following questions. ¥½(s) a) Find the transfer functions G, (s) = 1 and G2(s) = U(s) U(s) b) Use Laplace transform method and obtain the displacements in Laplace space (i.e. s domain) Y,(s) and Y2(s) utilizing initial conditions y, (0) = 0, y2(0) = 1, ỷ, (0) = 1 and y2(0) = 0. Use Cramer's rule for the solutions of Y, (s) and Y2(s). c) Using Inverse Laplace transformation, obtain displacements in time space (i.e. time domain) Y1(t) and y2(t) if, c.l. u(t) = 8(t) (Impulse function) (Ramp function starts from t=2) c.2. u(t) = {"2, 2st<∞ 0, 0st< 2 u(t) t-2 O 2 d) Plot y, (t) and y2(t) in the interval t e [0,5]s together in the same plot that you found in part c.1. e) Plot y, (t) and y½(t) in the interval t e [0,5]s together in the same plot that you found in part c.2. Hint: For the partial fractions of lengthy functions, you may use Matlab's residue () function b(s) which is useful to obtain fractions as- a(s) s-Pi s-P2 S-P3 S-Pn
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Question
![4) Consider the following mass spring damper system:
m2
u(t)
Assuming that y, > y1, equation of motion (EOM) for this system can be obtained by applying
the Newton's second law:
= mỹ (i. e. RHS is positive)
Then, equations are written as:
m,y, = k2(y2 – Yı) – kıyı – cỷı
(1)
m2yz = u(t)
- k2(y2 - Yı) (2)
where u(t) is the input function to the system. Take m, = 1 kg, m2 = 2 kg, c = 10 Ns/m and
k = 10 N/m, k2 = 20 N/m. Then, answer the following questions.
¥½(s)
a) Find the transfer functions G, (s) = 1 and G2(s) =
U(s)
U(s)
b) Use Laplace transform method and obtain the displacements in Laplace space (i.e. s domain)
Y,(s) and Y2(s) utilizing initial conditions y, (0) = 0, y2(0) = 1, ỷ, (0) = 1 and y2(0) = 0.
Use Cramer's rule for the solutions of Y, (s) and Y2(s).
c) Using Inverse Laplace transformation, obtain displacements in time space (i.e. time domain)
Y1(t) and y2(t) if,
c.l. u(t) = 8(t) (Impulse function)
(Ramp function starts from t=2)
c.2. u(t) = {"2, 2st<∞
0,
0st< 2
u(t)
t-2
O 2
d) Plot y, (t) and y2(t) in the interval t e [0,5]s together in the same plot that you found in
part c.1.
e) Plot y, (t) and y½(t) in the interval t e [0,5]s together in the same plot that you found in
part c.2.
Hint: For the partial fractions of lengthy functions, you may use Matlab's residue () function
b(s)
which is useful to obtain fractions as-
a(s)
s-Pi
s-P2
S-P3
S-Pn](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3dc0560-c656-48ca-8f0b-13994d41fd86%2F159b2d27-591e-4554-99b6-e91bc9c461fe%2Ftscftun.jpeg&w=3840&q=75)
Transcribed Image Text:4) Consider the following mass spring damper system:
m2
u(t)
Assuming that y, > y1, equation of motion (EOM) for this system can be obtained by applying
the Newton's second law:
= mỹ (i. e. RHS is positive)
Then, equations are written as:
m,y, = k2(y2 – Yı) – kıyı – cỷı
(1)
m2yz = u(t)
- k2(y2 - Yı) (2)
where u(t) is the input function to the system. Take m, = 1 kg, m2 = 2 kg, c = 10 Ns/m and
k = 10 N/m, k2 = 20 N/m. Then, answer the following questions.
¥½(s)
a) Find the transfer functions G, (s) = 1 and G2(s) =
U(s)
U(s)
b) Use Laplace transform method and obtain the displacements in Laplace space (i.e. s domain)
Y,(s) and Y2(s) utilizing initial conditions y, (0) = 0, y2(0) = 1, ỷ, (0) = 1 and y2(0) = 0.
Use Cramer's rule for the solutions of Y, (s) and Y2(s).
c) Using Inverse Laplace transformation, obtain displacements in time space (i.e. time domain)
Y1(t) and y2(t) if,
c.l. u(t) = 8(t) (Impulse function)
(Ramp function starts from t=2)
c.2. u(t) = {"2, 2st<∞
0,
0st< 2
u(t)
t-2
O 2
d) Plot y, (t) and y2(t) in the interval t e [0,5]s together in the same plot that you found in
part c.1.
e) Plot y, (t) and y½(t) in the interval t e [0,5]s together in the same plot that you found in
part c.2.
Hint: For the partial fractions of lengthy functions, you may use Matlab's residue () function
b(s)
which is useful to obtain fractions as-
a(s)
s-Pi
s-P2
S-P3
S-Pn
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