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.

() October 30, 2013 3 / 26

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 =

eαt

α2 + β2 (α sin(βt)− β cos(βt)) .

() October 30, 2013 19 / 26