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Mechanical vibrations: beats and resonance
Both mechanical vibrations and electric circuits can be accurately modeled by linear differential equations
with constant coefficients
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I have posted notes on the algebra and trig underlying some of the analysis
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(1)

where γ and k are nonnegative constants, and F = F (t) is the external force
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The damping coefficient γ might be due to friction, internal or external
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Often it is the least accurate approximation to the contributing forces, but is accurate enough for many applications to give robust qualitative
conclusions about many mechanical systems
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We then add sinusoidal forcing, where we first encounter
the phenomena of “beats” and “resonance”
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Undamped free vibrations
A free or unforced vibration is one where F = 0
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We say
that the system is perfectly elastic
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The general solution to the equation u + ku = 0 is
u = A cos(ω0 t) + B sin(ω0 t), where ω0 =

k/m,

(2)

and A and B are arbitrary constants
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This homogeneous solution can be rewritten in the form
u = R cos(ω0 t − δ), where R =

A2 + B 2 , R cos(δ) = A
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The phase shift is δ/ω0
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For our first
look at this situation we continue with the assumption that γ = 0
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Other initial conditions will merely effect the contribution from the
homogeneous solution, and hence not affect the conclusions
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= 1 2
2
|ω0 − ω 2 |
2

u=

(4)
(5)

[Check this!] The first sine factor is a lower frequencey “envelope” which modulates the higher-frequency
second sine factor
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As ω → ω0 the solution changes to
u=

F0 /m
t sin(ω0 t)
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Thus is the phenomenom called resonance
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We consider three cases, corresponding to increasing γ
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The underdamped case: if γ 2 < 4km then
u = R exp(−
µ=

1
2m

γ
t) cos(µt − δ), where
2m

(7)

4km − γ 2
1−

= ω0

(8)

γ2

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When γ is very small then
1−

µ ≈ ω0

γ2
4km


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The critically damped case: if γ 2 = 4km then
u = exp(−

γ
t)(At + B)
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The overerdamped case: if γ 2 > 4km then
u = c1 exp(r1 t) + c2 exp(r2 t), where r1 , r2 = −

1
γ
±
2m 2m

γ 2 − 4km
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(13)

t→+∞

Damped vibrations with periodic forcing
Now we consider the same forcing as above, but add the more realistic assumption that γ = 0
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∆ 0

(17)

[Check this!] The homogeneous solution uh is called the transient solution in this situation, while U is
called the steady state solution
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We observe the following facts:
R → F0 /k as ω → 0
R → 0 as ω → ∞

(18)
(19)

R is maximimzed where ∆ is minimized

(20)

γ2
γ2
d∆
2
2
= 0 when ω 2 = ω0 −
= ω0 1 −
2
dω
2m
2km
F0
Hence max R =

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(21)
(22)

Further reading
The application of linear differential equations to mechanical and electrical vibrations is covered in
sections 3
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9 of the text
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Reading quiz
1
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What is an underdamped free vibration?
3
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What is a overdamped free vibration?
5
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What is the quasifrequency?
7
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Under what situation do you encounter resonance?
9
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What is the steady state solution? Why is it called that?

Extra credit assignment 1: due Thursday, 3 August
1
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2
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Feel free to use Maple!

3



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