J b T = us(t )

ω

Chapter 5

System Response andSimulation: Problems

Figure 5.1: A rotor supported by bearings, driven by torque.

1. Figure 5.1 depicts a rotor of inertia J supported by bearings with damping b, and

driven by a torque T . Obtain the shaft angular velocity ω = ω(t) due to a unit step

input in torque T = 1 N m us(t). Assume zero initial conditions and parameter

values J = 2 N m s2 and b = 4 N m s. Also estimate the system’s time constant.

2. During starting of an automobile, a switch engages the 12 volt battery to the DC

starter motor, which cranks the engine. The starter motor’s electrical resistance R =0.01 ohm, and the motor’s current to torque constant K = 1 N m A−1. The engine

presents friction (damping b = 5 N m s) and inertia (J = 5 N m s2) to the starter

motor. Obtain the system equation(s), and determine the response for a typical start.

3. Determine the time varying fluid volume v = v(t) in an initially empty tank of

uniform cross section area A = 200 m2, due to a step increase in pipe pressure

P (t) = 10us(t) kPa at the mouth of a short pipe leading into the bottom of the

tank. The pipe’s resistance Rp = 100 Pa s m−3. The fluid is water, with density

ρ = 1000 kg m−3.

1

2CHAPTER 5. SYSTEM RESPONSE AND SIMULATION: PROBLEMS

4. Figure ?? of problem ?? of chapter ?? depicts a seismograph for measuring ground

motions x(t). From the state equations obtained from that problem, determine the

response z(t) of the seismograph’s mass M for step and sinsusoidal ground motions

x(t).

5. Given the ordinary differential equation

x + 3x + 9x = 4us(t), t > 0 (5.1)

with initial conditions x(0) = 49 and x(0) = 0. Determine the damping ratio

and damped natural frequency. Determine the system eigenvalue(s). Determine the

complete solution to the initial value problem.

6. Given the system of state equations

x = −3x + 2y + 2e−tus(t)

y = −x − y − e−tus(t)

valid for t > 0 with initial conditions x(0) = 0 and y(0) = 0. Determine the

system eigenvalue(s). Determine the general homogeneous solution and express it

in terms of real functions. Determine a particular solution. Determine the complete

solution.

7. Given the system of state equations:

x1 + 6x1 = 0

x2 + 2x1 + 7x2 − 2x3 = 0

x3 + x1 + 2x2 + x3 = 2e−t

(5.2)

Determine all eigenvalues, at least one eigenvector, and a particular solution.

8. Given the system of equations

[xp

]=

[2h − 1 −32h/3 −1

] [xp

]+

[10

]us(t), t > 0.

with initial conditions x(0) = 1 and p(0) = 0. For what value(s) of h ≤ 0will the transient response oscillate? HINT: Inspect the characteristic equation. For

h = 1, give expressions for the natural frequency, the damping ratio, and the

damped natural frequency.

y

θ

w

0Se 1

I:mt ire1

TF:w/2

I:Mcar

0Se 1

I:mtire

TF:w/2

C:kspring

1 I:Jcar

1

R:bshock C:kspring1

R:bshock

3

9. Given

[x1

x2

]=

[−3 21 −2

] [x1

x2

]+

[−22

]us(t), t > 0.

with initial conditions [x1(0), x2(0)]T = [0, 1]T . Determine the system eigen-

value(s), and if it applies, determine the damping ratio and damped natural frequency.

Finally, determine the complete solution.

10. Determine all eigenvalues and one eigenvector associated with the system of state

equations in matrix form

x1

x2

x3

=

−6 0 0−2 −5 0−1 1 −4

x1

x2

x3

+

1061

e−12t cos t.

Figure 5.2: Front end suspension system, for problem 11.

11. Shown in figure 5.2 is the front end suspension system of an automobile. The car

body can translate vertically along y and rotate about θ, rendering two degrees of

freedom (DOF) for motions of the car body. From the bond graph in figure 5.2,

extract the state equations. From the homogeneous solution, determine the four

modes of response.

12. Augment problem 11 to include the tires. Replace the effort sources in the bond

graph of figure 5.2 with groups of elements that model the stiffness and damping

of the tires, excited by flow sources that model the vertical motions imposed on the

tires by bumps on the road. Then extract the system of state equations, and perform

a complete modal analysis.

vf

f lat

Mk

Mg

µs

v

µd

vd

µ(v)

4CHAPTER 5. SYSTEM RESPONSE AND SIMULATION: PROBLEMS

13. A system x− 2bx + (b2 − 1)x = us(t) has zero initial conditions. Determine

the range of values of parameter b for which the system is stable.

14. Determine all eigenvalues associated with the system of state equations

x1

x2

x3

=

h −h 10 0 1−3 −2 −1

x1

x2

x3

+

2−80.6

e−12tus(t), t > 0.

Here h is a parameter to be set by the designer. For what values of parameter his the system stable? Will the system transient response oscillate? If so, at what

frequency will it oscillate?

Figure 5.3: Slider with friction, for problem 15.

15. In figure 5.3, a flat moving at speed vf slides against a block of mass M , attached

to a wall by spring k. The block’s weight pressing against the flat induces friction

which opposes sliding. Formulate a bond graph for the system of figure 5.3 and

extract the state equations. For the friction constitutive law, use coulomb friction—

friction force proportional to normal force via coefficient of friction µ. Approximate

by the dashed line in figure 5.3 the local dependence µ = µ(v) on sliding velocity

v shown by the solid curve. Show that if friction force diminishes with sliding

velocity, i.e., if the local slope of friction force versus sliding velocityis negative, unstable “stick-slip” vibrations can result. Stick-slip causes the nasty

“squealing” sounds of brakes, or fingernails running against blackboards. HINT:

Find the equation of the dashed straight line, and determine the eigenvalues.

16. In problem 15, the block in figure 5.3 can also vibrate normal to the direction

of sliding. These motions perturb the normal force, and thus the friction force.

Incorporate these effects. Construct a second bond graph for normal motions of the

block, and include block weight Mg, mass inertia M , and contact compression

between block and flat. Similar to problem ??, view the block and flat as locally

-3 -2 -1 1 2 3

-80

-60

-40

-20

20

40

Magnitude MdB

Phase φ (degrees)

10 10 1010 1010ω

5

curved bodies with contact force N = Cδ3/2 nonlinearly dependent on contact

compression δ. Derived the coupled set of equations, and estimate the eigenvalues.

Figure 5.4: Bode chart for problem 17.

17. Estimate the forced response of a system (whose Bode plot is given in figure 5.4) to

a sinusoidal excitation of frequency 10,000 radians/second and amplitude 100.