Indian Institute of Technology Hyderabad Department of Electrical Engineering

EE3220 – Control Engineering Assignment 01 – (Mathematical Modeling)

Submission Deadline: None Key Learning from the Assignment:

• Modeling of Mechanical systems • Modeling of Electrical systems • Transfer function

Instructions: RN = last two digits of your roll number.

1. For the mechanical system shown below, a. Derive a mathematical model relating forcing function F(t), displacement x1(t) and x2(t). b. Obtain transfer functions: X1(s)/F(s) and X2(s)/F(s). c. Draw an equivalent force-voltage analogous system.

M1

M2

f k1

k2F

x2

2. For the mechanical system shown below, a. Derive a mathematical model relating forcing functions F1(t) and F2(t), and displacement

variables x1(t) and x2(t). b. Obtain all possible transfer functions.

f1

k1

k2 f2M1

k3

M2

F 1 F 2

x1 x2

3. For the mechanical system shown below,

a. Is this system equivalent to the system shown in Q. 2 above? Why? Prove your answer. b. Will the dynamics of the system change if location of the spring (k2) and damper (f2) are

exchanged? Why? Prove your answer. c. Draw a force-current analogous system.

f1

k1

k2 f2M1

k3

M2

F 1 F 2

x1 x2

4. For the mechanical system shown below,

a. Derive a mathematical model relating forcing functions F1(t) and F2(t), and displacement variables x1(t) and x2(t).

b. Obtain all possible transfer functions.

F 1

x1

x2

M1

M2

f

k1

k2

F 2

5. For the spring loaded pendulum shown below,

a. Derive a mathematical model for this system. Assume the following: (i) Spring force acting on pendulum is zero when the pendulum is vertical, i.e. θ = 0. (ii) Angle of oscillations θ is small. (iii) Length of the pendulum is l and springs are connected to pendulum at a distance a. (iv) No friction.

b. How will the dynamics be affected if spring constants are not identical? Show it.

m

k k

θ

l

a

6. For the mechanical system shown below,

a. Derive a mathematical model for this system. Displacements of mass M1, M2 and M3 are denoted as x1, x2 and x3. Also, M1 = M2 = M3 = 0.5RN kg, k1 = 0.5RN N/m, k2 = 0.7RN N/m, k3 = RN N/m, f1 = 1 N-s/m, f2 = RN N-s/m and f3 = 1.1RN N-s/m.

b. Obtain transfer functions for all displacement variables to input force.

M1 M2

M3

k1

k3

f1

f2

k2

f3

F

F rictionless

7. For the mechanical system shown below, a. Derive a mathematical model for this system. Also, J1 = 1 kg-m2, f1 = f2 = 1 N-m-s/rad, f3 =

1 N-s/m, k1 = 1 N/m, M = 1 kg, radius of ideal 1:1 gear = 2 m and moment of inertia of idea 1:1 gear = 1 kg-m2.

b. Find transfer function G(s) = X(s)/T(s)

J 1

T (t)

M

N1=10RN

N2=20NR f1

N3=30NR

N4=60NRIdeal 1:1 gear

 f2

 f3k1

x

8. For the mechanical system shown below,

a. Derive a mathematical model for this system. The gears have inertia and bearing friction as shown.

b. Obtain transfer function G(s) = θ(s)/T(s). T (t)

N1

J 1, f1N3

J 3, f3

J 5, f5

N2

J 2, f2N4

J 4, f4

θ(t)

9. For the mechanical system shown below,

a. Derive a mathematical model for this system. The gears are inertia less. J1 = 3 kg-m2, J2 = 200 kg-m2 and f1 = 1000 N-m-s/rad.

b. Obtain transfer function G(s) = θ(s)/T(s).

250 N‐m/rad

T (t)

θ(t)

N1=10RN

N2=100RN

N3=50

J 2

J 1 f1

10. For the mechanical system shown below, obtain transfer function G(s) = θ2(s)/θ1(s).

J 5 J 6

T (t) θ1(t)

θ2(t)N1

J 1, f1N2

J 2, f1N3

J 3N4

J 4, f1 f1

 f1

k2

k1

11. An electrical network is shown below. Derive differential equation for the system. Also find

transfer functions: VR4(s)/Vi(s), VC1/ Vi(s), VC2/ Vi(s), VL2/ Vi(s), I2(s)/ Vi(s) and I3(s)/ Vi(s).

R 2

R 1 R 3

L 2

R 4C 2

C 1

+‐

L 1

vi i2i1

i3

12. Derive differential equation for the op-amp circuit shown below. Also find transfer functions

Vo(s)/ Vi(s), VR3(s)/ Vi(s). vi vo+

‐C 1

R 1R 2

R 3

13. Derive differential equation for the op-amp circuit shown below. Also find transfer functions

Vo(s)/ Vi(s), VR3(s)/ Vi(s).

+

‐vi

vo

R1

R2

C

R3

14. Derive differential equation for the op-amp circuit shown below. Also find transfer functions

Vo(s)/ Vi(s), VR(s)/ Vi(s).

+

‐

vi2R

R5R

C

vo

5R

C

15. Derive a mathematical model for a vehicle towing a trailer through a spring-damper coupling hitch. Where, Mass of the trailer (M), spring constant of the hitch (k), viscous damping coefficient of the hitch (f1), viscous friction coefficient between the trailer and road (f2), displacement of the trailer (x1), force applied by the vehicle (F). Also obtain transfer function X1(s)/F(s). Derive differential equations of the system when one more trailer having same parameters as the first one is connected to the first trailer through identical hitch. Obtain transfer function X2(s)/F(s) if displacement of second trailer is x2. Draw force-current analogous system for the two-trailer system