DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Note
Homework Problems with an X beside them
are ONLY for MAE 524 students.
Grading Policy

Neat work must be submitted. Points are taken off for poorly drawn
diagrams and sloppy work.

No late homework is accepted.
Warning
Homework is to be done individually. General discussion is encouraged, but you
must do all of the work yourself.
Violations (generally referred to me by
classmates) are documented and sent to the NCSU division of student affairs for
processing academic misconduct.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 1
Consider the following single degree-of-freedom system:
This system is a switch that rotates about point O in the x-y plane. It consists of a uniform
bar, connected to a spring and a damper. The spring is unstretched when the bar’s angle is
0, as shown.
Note: The switch will be acted on by an applied force F in a later homework problem.
For now, let F = 0.
(a) Draw a free body diagram of the system when it’s rotated an arbitrary angle .
(b) Develop a general expression for the moment acting on the bar by the spring.
(c) Derive the general nonlinear differential equation governing the motion of the
system for an arbitrarily large angle.
(d) Derive the general nonlinear algebraic equation governing the static equilibrium
of the system for an arbitrarily large angle.
(e) Find all of the system's equilibrium positions.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 2
Consider the system in Homework No. 1 and specifically the nonlinear differential
equation of motion that you derived in part (c).
(a)
Write the nonlinear differential equation of motion in the general form
f
f
and
about each of
  f ( ,). Find the stability derivatives


its equilibrium positions.
(b)
For the neighborhood around each each equilibrium position (k = 1, 2,
… ), let  (t )   (t )   0( k ) where  0( k ) is the k-th equilibrium angle
( 0   0(1) ) and write down the associated linear approximation of f in
terms of  (t ) and  (t ).
(c)
Write down the linear differential equation governing the motion of the
system in the neighborhood of each equilibrium position.
(d)
Find a general form of the solution of each linear differential equation.
(e)
State which equilibrium positions are stable and which are unstable.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 3
Consider again the system in Homework No. 1. Assume that the system was initially at
rest, and that the initial angle was 10○ larger than the first equilibrium angle – which
places the system in the neighborhood of the first equilibrium position. Determine the
response of the system (t) in the following ways:
(a)
Use the linear differential equations you found in Homework No. 2 to find
the analytical solution of (t). Using MATLAB, plot (t) showing it for
about 6 oscillations.
(b)
Rewrite the nonlinear differential equation you found in Homework No. 1
in part (c) as two first-order nonlinear differential equations (state
equations).
(c)
Use the Euler Method to numerically integrate the state equations you
found in part (b) to obtain the numerical solution of (t). Using MATLAB,
write a short program to do the integration (do not use a “canned” code”)
and plot (t) for about 6 oscillations. Use a step size of about T = 0.005Tf.
(d)
Describe what accounts for the differences between the responses obtained
in (a) and (c)?
(e)X
Find (t) numerically by solving the nonlinear state equations using the
Second-order Runga Kutta Method. Use the same step size as in (c). Write
a program in MATLAB to do this and plot your results.
(f)X
Find (t) numerically by solving the linear state equations using the
Second-order Runga Kutta Method. Use the same step size as in (c). Write
a program in MATLAB to do this and plot your results.
(g)X
Describe what accounts for the differences between the responses obtained
in (a), (c), (e), and (f)?
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 4
Consider again the system in Homework No. 1. The system is now acted on by an applied
motor moment M = aF. Two moments will be compared – a finite pulse f1 and an
instantaneous impulse f2. When the time of the pulse is “short” enough, the response of
the system acted on by the finite pulse should look like the response of the system acted
on by the impulse. The question arises how short a time is “short.”
The short-time pulse f1 and the impulse f2 are given by:
0  t  T0
A ,
f1   0
,
T0  t  
0,
f 2  A0T0 (t )
where T0 is the period of the pulse (not to be confused with step size of period). Let T0 =
Tf/10 and A0 = 1 lb. Notice for both functions that the integral over time of the force is the
same, that is
T0
0
T
f1 (t )dt  0 0 f 2 (t )dt  A0T0 .
(a)
Draw a free body diagram of the system over again. This time let the angle
of the bar be small in the diagram and remember that sin() =  and cos(
) = 1.
(b)
Using this free body diagram, find the linear differential equation that
describes the motion of the system. (It will turn out to be the same as the
linear differential equation you found in part (d) of Homework No. 1.)
(c)
Find the response (t) of the system subject to each of the two forcing
functions assuming that the system is initially at rest. Using MATLAB,
plot each response for about 6 oscillations.
(d)
Look at the two responses and comment on the nature of the
approximation.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 5
Consider again the system in Homework No. 4. Assume now that the system is
subject to a function f3 which is a square wave, as shown below.
Mathematically, the square wave can be written as
 A0 ,
f3  
  A0 ,
2( r  1)T0  t  (2r  1)T0
(2r  1)T0  t  2rT0
( r  1,2,...)
in which T0 denotes a half-period. (Use the same values of T0 and A0 here that you used in
Homework No. 4).
(a)
Find the response of the system by representing the forcing function f3 as a
Fourier Series. Plot the response for about 6 oscillations.
(b)
Find the response of the system by representing the forcing function f3 as a
series of step functions. Again, plot the response for about 6 oscillations.
(c)X
Find the response using the convolution integral.
(d)X
Find the response by numerical integration using the Second-Order RungaKutta Method.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 6
Consider the 1 dof system shown below.
This homework problem is about the effectiveness of open-loop tracking. The interest lies
in designing an open-loop tracking controller to move the collar from rest position A (x(0)
= 0) to rest position B (x(0) = L) in time T. Let m = 100 slug, T = 10 sec, and L = 20 ft.
Assume that the kinetic friction coefficient between the collar and the guide is 0.1.
(a)
Find the desired path of the collar.
(b)
Find the associated open-loop tracking force.
(c)
The next four parts evaluate the sensitivity of the response to parameter
errors: You’ll now assume that the actual parameters are slightly different
than the ones you postulated to get the expressions in (a) and (b).
Specifically you’ll change the mass and the initial conditions but not
change the tracking force. The changed parameters represent the actual
system and the force is the actual force (based on postulated parameters).
First, plot the nominal case, in which the postulated parameters are the
same as the actual parameters.
(d)
Let m = m/5, x(0) = 0 and plot the response.
(e)
Let m = - m/5, x(0) = 0 and plot the response.
(f)
Let m = 0, x(0) = x(T)/10 and plot the response.
(g)
Let m = 0, v(0) = x(T)/(10T) and plot the response.
(h)
Describe the differences between the responses.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 7
Consider again the system in Homework No. 1. Assume now that the system is subject to
a linear state (proportional-derivative) feedback control force f = fC . Design the feedback
controller to reduce the peak-overshoot by a factor of 2 and to dampen 90% of the motion
in 6 oscillations.
(a)
Determine the controller’s control gains.
(b)
Determine the response of the uncontrolled system and the control system
by the Euler method. Plot both over about 6 oscillations. Let (0) =  and
d (0)/dt = 0.1 rad/s.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 8
Consider again the system in Homework No. 1. Use a PID (proportional-integralderivative) feedback control force fC to control the motion of the system. The feedback
controller is designed to reduce peak-overshoot by a factor of 2, and to dampen 90% of
the motion, including the errors associated with a “bias” acting on the system, in 4 natural
periods. A natural period is the period of the uncontrolled system. A bias is a constant
force acting on the system that causes its equilibrium position, if uncontrolled, to be nonzero. You’ll need to assume, in addition to the controller, that a bias force acts on the
system. Select a bias force fB that increases the equilibrium position of the uncontrolled
system by 20○.
(a)
Determine the bias force fB.
(b)
Find the control gains g, h, and i.
(c)
Determine the response by the Euler method. Plot the uncontrolled and
controlled responses over about 6 oscillations. Let (0) = d (0)/dt = 0.
Note: In order to solve this problem, you’ll need to introduce the state
variables
t
x   ( s)ds, x   , x   .
1
0
2
3
The corresponding initial conditions are then
0
x1 (0)  0  ( s)ds  0, x2 (0)   (0),
(d)X
x3 (0)  (0).
Assume that there is a time delay Td associated with the electronics.
Examine the effect of the time delay on the response of the system and
comment on your observations.
Note: The time delay can be introduced in the computer program by letting
it equal to a multiple p of the step size T, i.e., let Td = pT. You will need to
store the values of the calculated force over the period of the time delay (p
values). The initial p values of the tracking force are zero.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 9
Consider the two degree-of-freedom (2dof) mass-spring system shown below. The spring
is unstretched when A = A0 and B = B0.
(a)
Draw a free body diagram of the system assuming small angles relative to
equilibrium.
(b)
Derive the linear equations governing the motion of the system.
(c)
Determine the system's mass and stiffness matrices.
(d)
Calculate the system natural frequencies of oscillation, and natural modes
of vibration
(e)
Assume that the system is initially at rest with the springs unstretched (in
the positions shown). Plot the response of the each degree of freedom as a
function of time.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 10
Consider again the system in Homework No. 9. The performance of the system needs to
be modified as follows: 90% of the motion needs to be damped out in 4 fundamental
periods. Assume that collocated motors at the pivot point can apply control moments MA
(positive clockwise) and MB (positive counter-clockwise).
(a) Design a full-dimensional PD control system to control the system. Determine
the control gain matrices G and H.
(b) Write out the equations in the state space.
(c) Using Euler integration, find and plot both the uncontrolled response and the
controlled response (the uncontrolled response was found in Homework No.
9).
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 11
Consider again the system in Homework No. 9. The performance of the system needs to
be modified as follows: 90% of the motion needs to be damped out in 4 fundamental
periods – now including the steady-state error due to the spring compression.
(a)
Determine the control gain matrices G, H, and I.
(b)
Write out the equations in the state space.
(c)
Using Euler integration, find and plot both the uncontrolled response and
the controlled response.
(d)X
Compare the forces (control effort) in Homework Nos. 10 and 11.
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DYNAMICS AND CONTROL OF MECHATRONIC SYSTEMS
Homework No. 12
Consider again the system in Homework No. 9. The performance of the system needs to
be modified as follows: 90% of the motion needs to be damped out in 4 fundamental
periods.
(a)
Assume that there is no sensor to measure B. Design an observer to
estimate it.
(b)
Find the control gains of the observer.
(c)
Write out the state equations for the controlled system and the observer.
Use the control gains you found in Homework No. 10.
(d)
Using Euler integration, find and plot the controlled response.
(e)
Compare the control response in Homework Nos. 10 and 12.
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