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2.4. Forced Oscillations and Exponential Response
Solution Page
Linear Constant Coefficient Equations
dy
d 2y
dy
D ay C f .t/
Second order A
C Cy D f .t/
CB
2
dt
dt
dt
dN y
dy
Nth order AN
C CA1 CA0 y D .AN D N C CA0 /y D P .D/y D f .t/
dt
dt N
First order
Null solutions yn have f .t/ D 0
Substitute y D e st to find the N exponents s
d st
.e / D ae st
dt
Second order As 2 C Bs C C D 0
s D a and yn D ce at
First order
Nth order
yn D c1 e s1 t C c2 e s2 t
yn D c1 e s1 t C C cN e sN t
P .s/ D 0
Exponential response to f .t/ D e ct
Step response for c D 0
Look for y D Ye ct
e ct
1
has Y D
c a
c a
e ct
Second order Y .Ac 2 C Bc C C /e ct D e ct yp D
D Y e ct
Ac 2 C Bc C C
te ct
e ct
Nth order
YP .c/e ct D e ct
yp D
or
when P .c/ D 0
P .c/
P 0 .c/
First order
d
Y e ct
dt
aY e ct D e ct
yp D
Fundamental solution g.t/ D Impulse response when f .t/ D ı.t/
First order
g.t/ D e at
e s1 t e s2 t
A.s1 s2 /
sin !n t
g.t/ D
A!n
starting from g.0/ D 1
Second order g.t/ D
starting from g.0/ D 0 and g 0 .0/ D 1=A
Undamped
sin !d t
underdamped g.t/ D e Z!n t
A!d
Nth order
g.t/ D yn .t/
g.0/ D g 0 .0/ D : : : D 0; g .N
1/
.0/ D 1=AN
Very particular solution for each driving function f .t/ : zero initial conditions on yvp
Zt
Multiply input at every time s
y.t/ D
g.t s/ f .s/ ds
by the growth factor over t s
0
Undetermined coefficients
Variation of parameters
Solution by Laplace transform
Solution by convolution
Direct solution for special f .t/ in Section 2.6
yp .t/ comes from yn .t/ in Section 2.6
Transfer function D transform of g.t/ in Section 2.7
y.t/ D g.t/ f .t/ in Section 8.6