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Mechanical Engineering
Vibrations and Resonance
Introduction
An important phenomenon in many areas of engineering is that of resonance. If you have
a mechanical system which is vibrating then it is possible that the natural frequency of
the system may match up or come close to the frequency driving the system. This
occurrence is called resonance. Sometimes it is advantageous, sometimes it is a serious
problem. The goal of this project is to use elementary calculus and physics to see the
basics behind it and to look at a couple of occurrences of it.
What kinds of uses are there for resonance, you ask? Resonance is an important concept
in how AM radio works, valve train design in internal combustion engines, tuning a
piano, bridge design, Magnetic Resonance Imaging (MRI), microwave ovens, and air
intake systems on automobiles, to name but a few.
Background needed: introductory Physics (1110 or 1111) and Calculus I. Ability to
plot in Maple is also useful.
Basic Physics and Math of Mass-Spring Systems
A good understanding of resonance can be gained by studying forces mass spring
systems. We will do this in two stages:
A. Free vibrations.
The basis here is Newton's Second Law, that the product of mass and acceleration is
equal to the sum of the external forces on the mass;
ma 

F
Let's assume that we have a coordinate system with the origin placed where the mass is at
rest and that x denotes the displacement from rest, t time.
What external forces are there? The spring is acting in the opposite direction to
displacement and may behave according to Hooke's Law, that the force exerted by the
spring is proportional to the distance stretched from equilibrium. This translates into
F = - kx
where k is the "spring constant".
Damping may be involved. This is true in shock absorbers (or McPherson struts) for cars.
A model of this situation is that the force is proportional to the velocity:
F = -c dx/dt
where c is the "damping coefficient". The minus sign indicates that the force is in the
opposite direction of travel, as it should be (it is trying to slow you down!).
Putting this together and using Newton's 2nd, we have
md2x/dt2 = -kx -c dx/dt
or
md2x/dt2 + c dx/dt +kx = 0
This is a second order differential equation, the kind of thing you will study extensively
in MA 2051, Differential Equations, among other places.
Mechanical engineers will sometimes ignore the damping term. In such cases they offer a
couple of possible explanations for this. One is that the damping is simply too small and
can be left out. The other is that the time frame they are interested in is short enough to
see no effects of damping. We will do that here with the general comment that
performing mathematical modeling is more than just generating equations and solving
them – it also involves simplification and making judgments.
We will ignore the damping term here, so our problem is now
md2x/dt2
+kx = 0
A differential equation such as this also involves "initial conditions" or it is ambiguously
stated. In this case we will assume that we pull or push the mass away from its resting
place and let go of it with no initial velocity given. This translates into two conditions,
that
x(0) = x0 (the starting point) and dx/dt(0) = 0 (no velocity at t=0)
Let's take a minute and summarize things. Our job is to find a function x(t) which
satisfies 3 conditions:
md2x/dt2 +kx = 0, x(0) = x0 (given), dx/dt(0)=0
This function should then describe the behavior of the mass attached to the spring.
A Solution
We will simplify the differential equation a little and offer a possible solution. If we
divide through by m we have
d2x/dt2
+(k/m)x
=0
*
Now k and m are both positive so we define w02 to be equal to k/m. (This is supposed
to be the Greek letter "omega". I have used a w instead as Microsoft Greek letters do not
always show up correctly on browsers). The equation now becomes
d2x/dt2
+ w02 x = 0
Problem 1 : show that the functions Sin(w0t) and
the differential equation *.
Cos(w0t)
are both solutions to
Problem 2: show that for arbitrary constants A and B that
ASin(w0t) +
BCos(w0t)
is also a solution to *.
Remark: A common complaint at this point is "I don't know how to do this. I didn't take
differential equations yet. ". This only requires a knowledge of Calculus I. The problem
is often that people don't know what it means to show something is a solution. They think
they are being asked to derive it. All you have to do is substitute the proposed solution(s)
into the differential equation * and show that they work – in other words that the equation
is satisfied. If so then by definition you have a solution. How one gets it is totally another
matter for another course.
So…take a second derivative, do some algebra and see if they work!
Problem 3 For the solution ASin(w0t) + BCos(w0t)
apply the initial conditions and determine what A and B must be.
Problem 4: What is the Principle of Superposition? What does it have to do with
Problem 2??
Problem 5: Referring back to your answer to Problem 3, what does the graph of your
solution look like? What is the frequency of your solution?
B. Forced Vibrations
So far we have looked at what a mass spring system does if stretched out and released.
No external forces.
Now introduce periodic external forces to the situation. The camshaft in your valvetrain
is such a force. The vibrations from the crankshaft are as well.
To keep things simple yet still follow the basic concept, we will assume the external force
has a single frequency, denoted by w (omega again) which is different from w0 above
and appears as
F0 cos(wt)
so F0 is its amplitude and w is its frequency.
Problem 6. Why should the differential equation now appear as
d2x/dt2
+ w02 x = F0 cos(wt)
?
hint: refer to Newton's 2nd
Problem 7. Show that a solution to this problem is given by
- F0
cos(wt)
m( w2 – w02)
(this is literally called a particular solution).
Problem 8. Show that a more general solution to this problem is given by
ASin(w0t) +
BCos(w0t) - F0
cos(wt)
m( w2 – w02)
Problem 9 Let's assume the mass is simply initially at rest with no initial velocity and
that it only begins moving due to the external force. Then the initial conditions are now
x(0) = 0 and x'(0) = 0
Find values for A and B so that the solution from Problem 8 satisfies these condition.
Write down the solution from Problem 8 when you are finished – this is the solution to
the initial value problem and describes how the mass will behave now acted on by the
external vibration.
You can see visually how this works by going to the following Java applet:
http://www.walter-fendt.de/ph14e/resonance.htm
In particular, take the value of w to be near to w0 and report on what you saw happen as
compared to w being far from w0. ( you might try 2.1 and 2 as compared to 4 and 2,
respectively.
The phenomenon of resonance occurs when w approaches w0. First plot
this solution with Maple or Matlab using the following sample data:
F0 = 1
m=1
w0 = 2 and w = 2.1
and graph the solution for t between 0 and 20 Pi.
Problem 10 Finally we get to resonance. This is when w = w0. The trouble with
our solution, as shown in Problem 8 would have you dividing by 0 – not
algebraically very desirable! Your job for the moment is to show that the
solution to
d2x/dt2
+ w02 x = F0 cos(w0t)
x(0) = 0 x'(0) = 0
is given by
x(t) = t sin(w0t)/(2w0)
Problem 11 plot this for the same time frame and describe how you see the
amplitude behaving. What is happening to the total energy of the mass spring
system over time?
Problem 12 We now wish to relate this to a couple of actual situations. Please
watch the video "Passage to Resonance". Write a 1-2 page section for your
project summarizing what you saw and how it relates to the theoretical
development of resonance that you have worked on.
Problem 13 (bonus points!!) Watch the video "Goalposts" and see if you can
offer an explanation of why the goalposts on the football field sometimes oscillate
back and forth.