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 or without damping). Assume a time dependent force f (t ) is applied to the system. The ODE governing displacement becomes m d 2 x dt 2 = -β dx dt - kx + f (t ), β ≥ 0. Divide out m and let F (t )= f (t )/m to obtain the nonhomogeneous equation d 2 x dt 2 + 2λ dx dt + ω 2 x = F (t ) () October 30, 2013 1 / 26
19
Embed
Oct. 30, 2013 Math 2306 sec 005 Fall 2013facultyweb.kennesaw.edu/lritter/Oct30_2306f.pdfOct. 30, 2013 Math 2306 sec 005 Fall 2013 5.1.3: Driven Motion We can consider the application
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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)
() October 30, 2013 1 / 26
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!
() October 30, 2013 2 / 26
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.
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.)
() October 30, 2013 4 / 26
() October 30, 2013 5 / 26
5.1.4: Series Circuit Analog
Figure: Kirchhoff’s Law: The charge q on the capacitor satisfiesLq′′ + Rq′ + 1
C q = E(t).
() October 30, 2013 6 / 26
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.
() October 30, 2013 7 / 26
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.
() October 30, 2013 8 / 26
() October 30, 2013 9 / 26
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 .
() October 30, 2013 11 / 26
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 .
() October 30, 2013 12 / 26
Find the Laplace transform of f (t) = 1
() October 30, 2013 13 / 26
() October 30, 2013 14 / 26
Find the Laplace transform of f (t) = t
() October 30, 2013 15 / 26
() October 30, 2013 16 / 26
Find the Laplace transform of f (t) = eat , a 6= 0
() October 30, 2013 17 / 26
() October 30, 2013 18 / 26
Find the Laplace transform of f (t) = sin t —2
2The following is immensely useful!∫eαt sin(βt) dt =