1.10: Differential equations of motion
• Washing machine
(heaving motion)
Fig. 1.33
My + cy + ky = − Mg
W Mg
δ st = =
k
k
Mx + cx + kx = 0
See text for equations
y = x − δ st
for rocking motion
Problem 5.3
Derive equations of motion
Solution
• Free body diagram
1.11: Nature of excitations
• Harmonic excitation
F (t ) = A1 sin ωt + A2 cos ωt = A cos(ωt −ψ ) = Re Aeiωt
Amplitude: A = A + A
2
1
T=
2π
ω
2
2
phase angle
ψ = tan
−1
A1
A2
Periodic and Non-periodic excitations
• Periodic response can be analyzed with
the aid of Fourier series
• Non-periodic
Random Excitation
• Characterized their spectrum. Chapter 12,
not covered in the course.
• Figure 1.40
Linear system and superposition
• Linear spring mass
system
d 2 x1
dx1
m 2 +c
+ kx1 = F 1 (t )
dt
dt
d 2 x2
dx
m 2 + c 2 + kx2 = F 2 (t )
dt
dt
• Apply aF1+bF2 and
get
• What is c=2t2? Is this
still a linear system?
• What about kx3 term?
d 2 (ax1 + bx2 )
d (ax1 + bx2 )
m
c
+
+ k (ax1 + bx2 )
2
dt
dt
= aF 1 (t ) + bF2 (t )
Linear and Time invariant systems
• We will assume that system properties do
not change with time
• Differential equations with constant
coefficients
• Harmonic response to harmonic excitation
• Figure 1.43
1.13: Vibration about equilibrium
points
• Vibration are modeled by linearizing equations
of motion around a static equilibrium point.
• Allows determination whether equilibrium is
stable
• Typical equation and solution
x + ax + bx = 0
x = Ae st
2
a
⎛a⎞
s1,2 = − ± ⎜ ⎟ − b
2
⎝2⎠
x(t ) = A1e s1t + A2 e s2t
Types of response
• Decaying response
• What type of roots
does response
correspond to?
2
a
⎛a⎞
s1,2 = − ± ⎜ ⎟ − b
2
⎝2⎠
Constant and diverging response
• Fig. 1-49
What kind of roots?
Example 1.32
• Fig. 1.76 and solution
Reading assignment
Chapter 2
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