Example: Prove that the angular frequency of a vertical spring with a spring constant k and a hanging mass m is still given by k ω = m

Physics 106 Lecture 12 Oscillations – II SJ 7th Ed.: Chap 15.4, Read only 15.6 & 15.7

• Recap: SHM using phasors (uniform circular motion) • Phys ica l pendul um exampl e • Damped harmonic oscillations • Forced oscillations and resonance. • Resonance examples and discussion – music – structural and mechanical engineering – waves • Sample problems • Oscillations summary chart

1 Goals

Oscillations in the presence of damping

Oscillations in the presence of oscillating external force

Damped Oscillations

• Non-conservative forces may be present – Friction is a common nonconservative force – No longer an ideal system (such as those dealt with so far) • The mechanical energy of the system diminishes in time, motion is said to be damped • The motion of the system can be decaying oscillations if the damping is “weak”.

• If damping is “strong”, motion may die away without oscillating. • Still no driving force, once system has been started

2 Add Damping: Emech not constant, oscillations not simple • Spring oscillator as before, but with dissipative force

Fdamp neglect such as the system in the figure, with vane moving in fluid. gravity

Fdamp viscous drag force, proportional to velocity

Fdamp = −bv

• Previous force equation gets a new, damping force term d2x(t) dx(t) Fnet = m = − kx()(t) − b dt2 dt

new term d2x(t) b dx(t) k + = − x(t) dt2 m dt m

Solution for Damped oscillator equation

new term d2x(t) b dx(t) k + = − x(t) dt2 m dt m

bt − 2 Solution: 2m k b x(t) = xme cos(ω't + φ) ω'≡ − modified m 2 oscillations exponentially altered ω’ can be real 4m decaying envelope frequency or imaginary k ω = : natural frequency 0 m

22 ωω'(/2)≡−0 bm

• Recover undamped solution for b Æ 0

3 Damped physical systems can be of three types

bt − 2 Solution: 2m k b x(t) = xme cos(ω't + φ) ω'≡ − damped m 4m2 oscillations

UdUnder damped : small bk< 2 m bk2 < , for which ω is positive. 4mm2

Critically damped: bkm= 2 2 b k 2 ≈ ≡ ω0 for which ω'≈ 0 4m2 m 2 Overdamped: b k 2 > ≡ ω0 for which ω' is imaginary 4m2 m Math Review: cos(ix )==+ cosh( x ) ( exx e− ) / 2 sin(ix )==− sinh( x ) ( exx e− ) / 2 cos(ix+ y )=− cos( ix )cos( y ) sin( ix )sin( y )

Types of Damping, cont.

a) an underdamped oscillator b) a critically damped oscillator c) an overdamped oscillator

For critically damped and overdamped oscillators there is no periodic motion and the angular frequency ω has a different meaning

4 b2 k << ≡ ω2 Weakly damped oscillator : 4m2 m 0

bt − kb2 − 2m ω ' ≡≈ω0 xt()= xem cos(ω0 t+ϕ ) m 4m2

bt - 2m Xm = xm()(t) ≈ xme slow decay of amplitude envelope

≈ cos(ω0t + φ) small fractional change in amplitude during one complete cycle

b2 k << ≡ ω2 Weakly damped oscillator : 4m2 m 0 bt kb2 − ω ' ≡≈− ω0 xt()= xe2m cos(ω t+ϕ ) m 4m2 m 0 bt - Amplitude : X = A 2m m xm(t) ≈ xme

small fractional slow decay change in amplitude of amplitude envelope during one complete cycle

≈ cos(ω0t + φ)

Velocity with weak damping: find derivative bt − d 2m maximum velocity v(t) = x(t) ≈ vme sin(ω't + φ) dt vm = − ω0xm

altered exponentially frequency ~ ω decaying envelope 0

5 Mechanical energy decays exponentially in an “weakly damped” oscillator (small b)

E = K(t) + U(t) = 1 mv2(t) + 1 kx2(t) mech 2 2

bt − 2m xt()=+ xem cos(ω0 t ϕ ) Velocity with weak damping: find derivative bt − d 2m maximum velocity v(t) = x(t) ≈ vme sin(ω't + φ) dt vm = − ω0xm

altered exponentially frequency ~ ω decaying envelope 0

bt − ⎛⎞b 2m xem ⎜⎟−+cos(ω0 tϕ ) term is negligible, because b is small.. ⎝⎠2m

Mechanical energy decays exponentially in an “weakly damped” oscillator (small b)

E = K(t) + U(t) = 1 mv2(t) + 1 kx2(t) mech 2 2 Substitute previous solutions: bt bt − − 2m 2m x(t) = xme cos(ω't + φ) v(t) ≈ −ω0xme sin(ω't + φ)

E = 1 mω2x2 e−bt / msin2(ω't + φ) mech 2 0 m + 1 kx2 e−bt / mcos2(ω't + φ) 2 m As always: cos2(x) + sin2(x) = 1

2 k Also: ω ≡ 0 m ∴ E (t) = 1 kx2 e−bt / m mech 2 m

exponential decay at twice Initial mechanical energy the rate of amplitude decay

6 Damped physical systems can be of three types

bt − 2 Solution: 2m k b x(t) = xme cos(ω't + φ) ω'≡ − damped m 2 exponentially altered ω’ can be real 4m oscillations decaying envelope frequency or imaginary

2 Underdamped: b k 2 << ≡ ω0 for which ω'≈ ω0 4m2 m ƒ The restoring force is large compared to the damping force. ƒ The system oscillates with decaying amplitude 2 Critically damped: b k 2 ≈ ≡ ω0 for which ω'≈ 0 4m2 m ƒ The restoring force and damping force are comparable in effect. ƒ The system can not oscillate; the amplitude dies away exponentially 2 Overdamped: b k 2 > ≡ ω0 for which ω' is imaginary 4m2 m ƒ The damping force is much stronger than the restoring force. ƒ The amplitude dies away as a modified exponential ƒ Note: Cos( ix ) = Cosh( x )

Forced (Driven) Oscillations and Resonance ƒ An external driving force starts oscillations in a stationary system ƒ The amplitude remains constant (or grows) if the energy input per cycle exactly equals (or exceeds) the energy loss from damping

ƒ Eventually, Edriving = Elost and a steady-state condition is reached ƒ Oscillations then continue with constant amplitude

ƒ Oscillations are at the driving frequency ωD

FD(t) = F0 cos(ωDt + φ')

FD(t) Oscillating driving force applied to a dampe d osc illat or

7 Equation for Forced (Driven) Oscillations k ω = natural frequency ω = 0 0 m ωD = driving frequency of external force External driving force function:

FD(t) = F0 cos(ωDt + φ') dx() t d2 x () t FFt==( ) -b - k x(t) m net D dt dt 2 FD(t)

Solution for Forced (Driven) Oscillations dx() t d2 x () t FFt==( ) -b - k x(t) m net D dt dt 2

FD(t) = F0 cos(ωDt + φ')

Solution (steady state solution):

x(t) = Acos(ωDt + φ)

F0 /m where A = 2 2 2 bωD 2 (ωD − ω0 ) + ( ) m FD(t)

The system always oscillat es at the driving frequency ωD in steady-state The amplitude A depends on how k ω ω ω = close D is to natural frequency 0 0 m “resonance”

8 Amplitude of the driven oscillations: F /m A = 0 bω (ω2 − ω2 )2 + ( D )2 D 0 m ƒ The largest amplitude oscillations occur at or resonance

near RESONANCE (ωD ~

ω0) As damping becomes weaker Æ resonance sharpens & amplitude at resonance increases.

Resonance

ƒ At resonance, the applied force is in phase with the velocity and the power Fov transferred to the oscillator is a maximum. ƒ The amplit ud e of resonant oscillations can become enormous when the damping is weak, storing enormous amounts of energy

Applications: • buildings driven by earthquakes • bridges under wind load • all kinds of radio devices, microwave • other numerous applications

9 Forced resonant torsional oscillations due to wind - Tacoma Narrows Bridge

Roadway collapse - Tacoma Narrows Bridge

10 Twisting bridge at resonance frequency

Breaking glass with voice

Another Breaking glass with voice

Yet another breaking glass with voice

11