LECTURE 13: MECHANICAL VIBRATIONS
MINGFENG ZHAO
October 07, 2015
Mass-Spring System:
Figure 1. Mass-Spring System
Let x(t) be the displacement of the mass, then
mx00 + cx0 + kx = F (t) .
For free motion (that is, F (t) = 0), rewrite the equation, we have
x00 + 2px0 + ω02 = 0,
where
c
p=
,
2m
r
and ω0 =
k
.
m
Let x = ert be a solution to x00 + 2px0 + ω02 x = 0, then
x0 = rert ,
and x00 = r2 ert .
So we have
0
= x00 + 2px0 + ω02 x
= r2 ert + 2prert + ω02 ert
1
2
MINGFENG ZHAO
= ert [r2 + 2pr + ω02 ].
So r2 + 2pr + ω02 = 0, we get
r1,2 = −p ±
q
p2 − ω02 .
Then
I. Free undamped motion: c = 0 (that is, p = 0)
The general solution to x00 + ω02 x = 0 is:
x(t) = A cos(ω0 t) + B sin(ω0 t) .
II. Overdamping: c2 − 4km > 0 (that is, p2 − ω02 > 0)
The general solution to x00 + 2px0 + ω02 x = 0 is:
y(t) = Aer1 t + Ber2 t ,
where
r1 = −p −
q
p2 − ω02 < 0,
and r2 = −p +
q
p2 − ω02 < 0.
III. Critical damping: c2 − 4km = 0 (that is, p2 − ω02 = 0)
The general solution to x00 + 2px0 + ω02 x = 0 is:
y(t) = Ae−pt + Bte−pt .
IV. Underdamping: c2 − 4km < 0 (that is, p2 − ω02 < 0)
The general solution to x00 + 2px0 + ω02 x = 0 is:
y(t) = Ae−pt cos
q
q
ω02 − p2 t + Be−pt sin
ω02 − p2 t ,
Example 1. Suppose that m = 2 kg and k = 8 N/m. The whole mass and spring setup is sitting on a truck that was
traveling at 1 m/s. The truck crashes and hence stops. The mass was held in place 0.5 meters forward from the rest
position. During the crash the mass gets loose. That is, the mass is now moving forward at 1 m/s, while the other end
of the spring is held in place. The mass therefore starts oscillating. What’s is the frequency of the resulting oscillation
and what is the amplitude.
Let x(t) be the displacement of the mass at time t (the moving forward is in the positive direction), them the
differential equation is:
mx00 + kx = 2x00 + 8x = 0.
LECTURE 13: MECHANICAL VIBRATIONS
3
By the assumption, we have
x(0) = 0.5,
and x0 (0) = 1.
Rewrite the equation, we have
x00 + 4x = 0.
The characteristic equation of x00 + 4x = 0 is r2 + 4 = 0, then r1 = 2i and r2 = −2i, which implies that the general
solution to 2x00 + 8x = 0 is:
x(t) = A cos(2x) + B sin(2x).
Then
x0 (t) = −2A sin(2x) + 2B cos(2x).
Since x(0) = 0.5 and x0 (0) = 1, then
A = 0.5,
and
2B = 1.
A = 0.5,
and B = 0.5
Then
Therefore, the solution to 2x00 + 8x = 0, x(0) = 0.5, x0 (0) = 1 is:
√
π
2
cos 2x +
.
2
4
√
p
2
2
Hence the frequency is 2 =
Hz , and the amplitude is
0.52 + 0.52 =
.
2π
2
x(t) = 0.5 cos(2x) + 0.5 sin(2x) =
Example 2. Suppose you want to use a spring to weigh items. You place the mass on the spring and put it in motion.
You have two reference weight 1 kg and 2 kg to calibrate your setup. You put each in motion on your spring an measure
the frequency. For the 1 kg weight you measured 1.1 Hz, for the 2 kg weight you measured 0.8 Hz.
a) Find the spring constant k and the damping constant c.
b) Find a formula for the mass in terms of the frequency in Hz.
c) For an unknown object you measure 0.2 Hz, what is the mass of the object? Suppose that you know that the
mass of the unknown object is more than a kilogram.
The differential equation for the mass spring system is:
mx00 + cx0 + kx = 0.
Rewrite the equation, we have
x00 + 2px0 + ω02 x = 0,
4
MINGFENG ZHAO
where
c
p=
,
2m
r
and ω0 =
k
.
m
a) By the assumption, we know that the motion is under damped, and the frequency is
r
p
q
4ω02 − 4p2
1
1
k
1
c2
2
.
·
=
· ω0 − p2 =
·
−
2π
2
2π
2π
m 4m2
By the assumption, we know that
r
1
k
c2
= 1.1,
·
−
2π
1 4 · 12
and
1
·
2π
r
k
c2
= 0.8.
−
2 4 · 22
Then we have
k = 4π 2 · 1.35,
and c = 2π ·
√
0.56 .
b) Let f be the frequency corresponding to the weight m, then we have
r
1
4π 2 · 1.35 4π 2 · 0.56
·
−
= f.
2π
m
4m2
That is, we have
f2 =
1.35 0.14
− 2 .
m
m
c) If f = 0.2, then
1.35 0.14
− 2 = 0.22
m
m
That is, we have 0.04m2 − 1.35m + 0.14 = 0. Then
√
1.35 + 1.352 − 4 · 0.04 · 0.14
m=
≈ 33.645 .
2 · 0.04
Problems you can do:
Lebl’s Book [2]: All exercises on Page 68 and Page 69.
Braun’s Book [1]: Read all materials in Section 2.6 and Section 2.7.
References
[1] Martin Braun. Differential Equations and Their Applications: An Introduction to Applied Mathematics. Springer, 1992.
[2] Jiri Lebl. Notes on Diffy Qs: Differential Equations for Engineers. Createspace, 2014.
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C.
Canada V6T 1Z2
E-mail address: mingfeng@math.ubc.ca