Oct. 30, 2013 Math 2306 sec 005 Fall 2013
5.1.3: Driven Motion
We can consider the application of an external driving force (with orwithout damping). Assume a time dependent force f (t) is applied tothe system. The ODE governing displacement becomes
md2xdt2 = −βdx
dt− kx + f (t), β ≥ 0.
Divide out m and let F (t) = f (t)/m to obtain the nonhomogeneousequation
d2xdt2 + 2λ
dxdt
+ ω2x = F (t)
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Forced Undamped Motion and Resonance
Consider the case F (t) = F0 cos(γt) or F (t) = F0 sin(γt), and λ = 0.Two cases arise
(1) γ 6= ω, and (2) γ = ω.
Case (1): x ′′ + ω2x = F0 sin(γt), x(0) = 0, x ′(0) = 0
x(t) =F0
ω2 − γ2
(sin(γt)− γ
ωsin(ωt)
)If γ ≈ ω, the amplitude of motion could be rather large!
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Pure Resonance
Case (2): x ′′ + ω2x = F0 sin(ωt), x(0) = 0, x ′(0) = 0
x(t) =F0
2ω2 sin(ωt)− F0
2ωt cos(ωt)
Note that the amplitude, α, of the second term is a function of t:
α(t) =F0t2ω
which grows without bound!
Forced Motion and Resonance Applet
Choose ”Elongation diagram” to see a plot of displacement. Try exciterfrequencies close to ω.
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ExampleA front loaded washing machine is mounted on a thick rubber pad thatacts like a spring. The weight W = mg (take g = 9.8m/sec2) of themachine depresses the pad 0.5 cm. When the rotor spins γ radiansper second, it exerts a vertical force F = F0 cos(γt) N on the machine.At what speed, in revolutions per minute, will resonance occur.(Neglect friction.)
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5.1.4: Series Circuit Analog
Figure: Kirchhoff’s Law: The charge q on the capacitor satisfiesLq′′ + Rq′ + 1
C q = E(t).
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LRC Series Circuit (Free Electrical Vibrations)
Ld2qdt2 + R
dqdt
+1C
q = 0
If the applied force E(t) = 0, then the electrical vibrations of thecircuit are said to be free. These are categorized as
overdamped if R2 − 4L/C > 0,critically damped if R2 − 4L/C = 0,underdamped if R2 − 4L/C < 0.
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ExampleAn LRC series circuit with no applied force has an inductance ofL = 2h and capacitance of C = 5× 10−3f. Determine the condition onthe resistor such that the electrical vibrations are
(a) Overdamped,
(b) Critically damped, or
(c) Underdamped.
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Section 7.1: The Laplace TransformIf f = f (s, t) is a function of two variables s and t , and we compute adefinite integral with respect to t ,∫ b
af (s, t)dt
we are left with a function of s alone.
Example: Compute the integral1∫ 4
0(2st+s2−t)dt
1The variable s is treated like a constant when integrating with respect tot—and visa versa.() October 30, 2013 10 / 26
Integral Transform
An integral transform is a mapping that assigns to a function f (t)another function F (s) via an integral of the form∫ b
aK (s, t)f (t)dt .
I The function K is called the kernel of the transformation.I The limits a and b may be finite or infinite.I The integral may be improper so that convergence/divergence
must be considered.I This transform is linear in the sense that∫ b
aK (s, t)(αf (t) + βg(t))dt = α
∫ b
aK (s, t)f (t)dt + β
∫ b
aK (s, t)g(t)dt .
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The Laplace Transform
Definition: Let f (t) be defined on [0,∞). The Laplace transform of f isdenoted and defined by
L {f (t)} =∫ ∞
0e−st f (t)dt = F (s).
The domain of the transformation F (s) is the set of all s such that theintegral is convergent.
Note: The kernel for the Laplace transform is K (s, t) = e−st .
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Find the Laplace transform of f (t) = 1
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Find the Laplace transform of f (t) = t
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Find the Laplace transform of f (t) = eat , a 6= 0
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Find the Laplace transform of f (t) = sin t —2
2The following is immensely useful!∫eαt sin(βt) dt =
eαt
α2 + β2 (α sin(βt)− β cos(βt)) .
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