LECTURE 16: FORCED OSCILLATIONS AND RESONANCE
MINGFENG ZHAO
October 10, 2014
Undetermined Coefficients:
Let a, b and c be constants, consider the equation:
ay 00 + by 0 + cy = f (x).
Let pn (x) and p̃n (x) be polynomials with degree n, a particular solution yp (x) to ay 00 + by 0 + cy = f (x) can be taken as:
f (x)
yp (x)
pn (x)emx cos(kx) + p̃n (x)emx sin(kx)
xα [qn (x)emx cos(kx) + q̃n (x)emx sin(kx)]
where
• α is one of 0, 1 and 2 (α is the multiplicity of m + ki as the solutions to ar2 + br + c = 0):
– If m + ki is not a root of ar2 + br + c = 0, then α = 0.
– If m + ki is a root of ar2 + br + c = 0 and b2 − 4ac 6= 0, then α = 1.
– If m + ki is a root of ar2 + br + c = 0 and b2 − 4ac = 0, then α = 2.
• qn (x) and q̃n (x) are undetermined polynomials with degree n.
Variation of Parameters:
To find a particular solution to the nonhomogeneous equation y 00 + p(x)y 0 + q(x)y = f (x):
I. Find a fundamental set of solutions y1 (x) and y2 (x) to the homogeneous equation y 00 + p(x)y 0 + q(x)y = 0.
II. Let yp (x) = u1 (x)y1 (x) + u2 (x)y2 (x) be a particular solution to y 00 + p(x)y 0 + q(x)y = f (x)
III. Compute yp0 (x), we get
yp0 (x)
= u01 (x)y1 (x) + u02 (x)y2 (x)
+u1 (x)y10 (x) + u2 (x)y20 (x)
Take
u01 (x)y1 (x) + u2 (x)y2 (x) = 0.
1
2
MINGFENG ZHAO
Then
yp0 (x)
= u1 (x)y10 (x) + u2 (x)y20 (x)
yp00 (x)
= u01 (x)y10 (x) + u1 (x)y100 (x) + u02 (x)y20 (x) + u2 (x)y200 (x).
IV. Plug yp (x), yp0 (x) and yp00 (x) into y 00 + p(x)y 0 + q(x)y = f (x), we get
u01 (x)y10 (x) + u02 (x)y20 (x) = f (x).
V. Solve u01 (x) and u02 (x) from the system:


 u01 (x)y1 (x) + u02 (x)y2 (x) = 0

 u0 (x)y 0 (x) + u0 (x)y 0 (x) = f (x).
1
1
2
2
Then
u01 (x) =
−y2 (x)f (x)
,
W (y1 , y2 )
and u02 (x) =
y1 (x)f (x)
.
W (y1 , y2 )
VI. Solve u1 (x) and u2 (x), then
Z
u1 (x) =
−y2 (x)f (x)
dx,
W (y1 , y2 )
Z
and u2 (x) =
y1 (x)f (x)
dx.
W (y1 , y2 )
VII. Write down the solution:
Z
yp (x) = −y1 (x)
y2 (x)f (x)
dx + y2 (x)
W (y1 , y2 )
Z
y1 (x)f (x)
dx .
W (y1 , y2 )
where
y1 (x) y2 (x)
W (y1 , y2 ) = y10 (x) y20 (x)
= y1 (x)y20 (x) − y10 (x)y2 (x).
Forced oscillations and resonance
Mass-Spring System:
Let x(t) be the displacement of the mass, then
mx00 + cx0 + kx = F (t) .
We are interested in periodic forcing, that is, F (t) = F0 cos(ωt). So we have
mx00 + cx0 + kx = F0 cos(ωt) .
LECTURE 16: FORCED OSCILLATIONS AND RESONANCE
Figure 1. Mass-Spring System
Rewrite the equation, we have
x00 + 2px0 + ω02 x =
F0
cos(ωt) ,
m
where
c
≥ 0,
p=
2m
r
and ω0 =
k
> 0.
m
Undamped forced motion and resonance
The differential equation for the undamped forced motion (c = 0) is:
x00 + ω02 x =
F0
cos(ωt).
m
The characteristic equation of x00 + ω02 x = 0 is:
r2 + ω02 = 0.
Solve r2 + ω02 = 0, we get
r1 = ω0 i,
and r2 = −ω0 i.
The general solution to x00 + ω02 x = 0 is:
x(t) = A cos(ω0 t) + B sin(ω0 t) .
Since the nonhomogeneous function is
F0
cos(ωt), we have two cases:
m
3
4
MINGFENG ZHAO
I. If ω 6= ω0 , let xp (t) = D cos(ωt) + E sin(ωt) be a particular solution to x00 + ω02 x =
x0p (t)
= −Dω sin(ωt) + Eω cos(ωt)
x00p (t)
= −Dω 2 cos(ωt) − Eω 2 sin(ωt)
x00 + ω02 x
F0
cos(ωt), then
m
= −Dω 2 cos(ωt) − Eω 2 sin(ωt) + Dω02 cos(ωt)
= D[ω02 − ω 2 ] cos(ωt) − Eω 2 sin(ωt)
F0
cos(ωt).
m
=
So we get
D=
F0
,
− ω2 ]
m[ω02
So a particular solution to x00 + ω02 x =
and E = 0.
F0
cos(ωt) can be:
m
xp (t) =
F0
cos(ωt) .
m[ω02 − ω 2 ]
Example 1. Take 0.5x00 + 8x = 10 cos(πt), x(0) = 0, x0 (0) = 0.
Then
r
m = 0.5,
ω0 =
√
8
= 16 = 4,
0.5
F = 10,
and ω = π.
The general solution to 0.5x00 + 8x = 10 cos(πt) is:
x(t) = A cos(4t) + B sin(4t) +
10
20
cos(πt) = A cos(4t) + B sin(4t) +
cos(πt).
2
2
0.5(4 − π )
16 − π 2
Then
x0 (t) = −4A sin(4t) + 4B cos(4t) −
20π
sin(πt).
16 − π 2
Since x(0) = 0 and x0 (0) = 0, then
A+
20
= 0,
16 − π 2
and
4B = 0.
So we get
A=−
20
,
16 − π 2
and B = 0.
So the solution to 0.5x00 + 8x = 10 cos(πt), x(0) = 0, x0 (0) = 0 is:
x(t) =
20
[cos(πt) − cos(4t)] .
16 − π 2
LECTURE 16: FORCED OSCILLATIONS AND RESONANCE
Figure 2. Graph of
5
20
[cos(πt) − cos(4t)]
16 − π 2
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