Lesson 1 - Oscillations

• Harmonic Motion Circular Motion

• Simple Harmonic Oscillators– Linear -

Horizontal/Vertical Mass-Spring Systems

• Energy of Simple Harmonic Motion

Math Prereqs

dcos

d

dsin

d

cos

sin

2 2

0 0

cos d sin d

0

2 22 2

0 0

1 1cos d sin d

2 2

1

2

Identities

cos cos 2cos sin2 2

2 2sin cos 1

cos cos cos sin sin

2 1 1cos cos 2

2 2

ie cos i sin

Math Prereqs

T

0

1f t f t dt

T

"Time Average"

2 2cos t

T

T T2

0 0

1 2 1 1 1 2 1cos t dt cos 2 t dt

T T T 2 2 T 2

Example:

Harmonic

Relation to circular motion

x A cos A cos t

2

T

Horizontal mass-spring

F ma

Hooke’s Law: sF kx

2

block 2

d xkx m

dt

2

2block

d x kx 0

dt m

Frictionless

Solutions to differential equations

• Guess a solution• Plug the guess into the differential equation

– You will have to take a derivative or two• Check to see if your solution works. • Determine if there are any restrictions (required

conditions).• If the guess works, your guess is a solution, but it

might not be the only one.• Look at your constants and evaluate them using

initial conditions or boundary conditions.

Our guess

x A cos t

Definitions

• Amplitude - (A) Maximum value of the displacement (radius of circular motion). Determined by initial displacement and velocity.

• Angular Frequency (Velocity) - Time rate of change of the phase.

• Period - (T) Time for a particle/system to complete one cycle.

• Frequency - (f) The number of cycles or oscillations completed in a period of time

• Phase - t Time varying argument of the trigonometric function.

• Phase Constant - Initial value of the phase. Determined by initial displacement and velocity.

x A cos t

The restriction on the solution

2

block

k

m

block

1 kf

2 2 m

blockm2T 2

k

The constant – phase angle x t 0 A v t 0 0 0

x t 0 0 0v t 0 v 2

x A cos t v A sin t

2a A cos t

Energy in the SHO

2 2 21 1 1E mv kx kA

2 2 2

2 2kv A x

m

Average Energy in the SHO

2 2 2 21 1 1U k x kA cos t kA

2 2 4

2 2 2 2 2 2 21 1 1 1K m v m A sin t m A kA

2 2 4 4

x A cos t

dxv A sin t

dt

K U

Example

• A mass of 200 grams is connected to a light spring that has a spring constant (k) of 5.0 N/m and is free to oscillate on a horizontal, frictionless surface. If the mass is displaced 5.0 cm from the rest position and released from rest find:

• a) the period of its motion, • b) the maximum speed and • c) the maximum acceleration of the mass.• d) the total energy• e) the average kinetic energy• f) the average potential energy

“Dashpot”

dampingF bv

dxkx b ma

dt

2

2

d x dxm b kx 0

dt dt

tx Ae cos t

Equation of Motion

Solution

Damped Oscillations

tx Ae cos t

t tdxv Ae sin t A e cos t

dt

2

t 2 t t 2 t2

d xa Ae cos t A e sin t A e sin t A e cos t

dt

t 2 2Ae 2 sin t cos t

tAe sin t cos t

2

2

d x b dx kx 0

dt m dt m

t t2 2 tb kAe Ae Ae 0

m mcos t cos t cos2 sin tt sin t

t 2 2

b

2mb k

cAe 0b

2 s oin tm

s tm m

22k b

0m 2m

2k b

m 2m

b

2m

Damped frequency oscillation

2

2

k b

m 4m

2b 4mk

B - Critical damping (=)C - Over damped (>)

b

2m

Giancoli 14-55

• A 750 g block oscillates on the end of a spring whose force constant is k = 56.0 N/m. The mass moves in a fluid which offers a resistive force F = -bv where b = 0.162 N-s/m. – What is the period of the motion? What if there had

been no damping?

– What is the fractional decrease in amplitude per cycle?

– Write the displacement as a function of time if at t = 0, x = 0; and at t = 1.00 s, x = 0.120 m.

Forced vibrations

ext 0F F cos t 0

dxkx b F cos t ma

dt

2

02

d x dxm b kx F cos t

dt dt

0 0x A sin t

Resonance

0

k

m Natural frequency

0 0x A sin t

0

0 2 222 2

0 2

FA

bm

m

2 201

0

mtan

b

Quality (Q) value

• Q describes the sharpness of the resonance peak

• Low damping give a large Q• High damping gives a small Q• Q is inversely related to the

fraction width of the resonance peak at the half max amplitude point.

0mQ

b

0

1

Q

Tacoma Narrows Bridge

Tacoma Narrows Bridge (short clip)