ENGR 1990 Engineering Mathematics
Equations Sheet #8 – Differential Equations for a Spring-Mass-Damper System
Equilibrium
Position
k
1. Differential Equation of Motion
mx + cx + kx = f (t )
2. Free Response ( f (t ) = 0 )
x
(a) Characteristic Equation: ms + cs + k = 0
(b) Form of Solution depends on type of roots. Coefficients found by
applying initial conditions.
2
Case
1
2
3
Type of
Roots
Type of
Motion
Real,
unequal
Real,
equal
Overdamped
Critically
damped
Complex
conjugates
Underdamped
c
m
f (t )
Form of Solution
x(t ) = Ae s1t + Be s2t
x(t ) = Ae st + B t e st
− c
t
x(t ) = e ( 2 m) [ A sin(ωd t ) + B cos(ωd t )]
ωd =
k
m
− ( 2cm )
2
3. Forced Response ( f (t ) ≠ 0 )
(a) Solution is the sum of the homogeneous and particular solutions, x(t ) = xH (t ) + xP (t ) .
(b) Homogeneous solution has the form of the free response.
(c) Particular solutions have the forms:
Form* of xP (t )
f (t )
constant
linear
quadratic
exponential
sine or cosine
exponential/ sine or
cosine product
*
B tn
a0
( B1t + B0 ) t n
a1t + a0
a eβ t
(B t + B t + B )t
(B e )t
a sin (ω t ) or a cos (ω t )
⎡⎣ B1 sin (ω t ) + B2 cos (ω t ) ⎤⎦ t n
a2t 2 + a1t + a0
2
2
1
βt
n
0
n
1
a e β t sin (ω t )
or a e cos (ω t )
βt
e β t ⎡⎣ B1 sin (ω t ) + B2 cos (ω t ) ⎤⎦ t n
The exponent n is the smallest, non-negative integer so every term in xP (t ) is different from
every term in xH (t ) . That is, n = 0 unless the same type of term appears in xH (t ) .
(d) Coefficients found by applying initial conditions.
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