Physics for EE (5EPD0)
Module V-a – Vibrations
March 13, 2025
Rob Mestrom
/e
Some small changes
Based on how the course is going so far, I will make two small
changes
▶ No more mentimeter wordclouds
⇒ exam-level MC question instead
▶ Brain links: think – pair – share
⇒ follow up with asking a random student to share a
thought
2
Mentimeter quiz: astronauts
Two astronauts push each other as they float freely in the zero-gravity
environment of space. The mass of astronaut A is larger than the mass of
astronaut B. Which of the astronauts will get the largest momentum
(magnitude) and which of the two gets the largest kinetic energy?
Please vote in mentimeter!
3
Module V
Contents
▶ Periodic motion (Y&F, Chapter 14)
▶ Mechanical waves (Y&F, Chapter 15)
4
Pendulum video
5
Examples of oscillations
www.ibphysicshelp.com
6
Oscillations
Describing oscillatory motion
7
Oscillatory motion
Characteristics of oscillatory motion
▶ Amplitude A: maximum displacement from
equilibrium
▶ Period T : time for the motion to repeat itself
▶ Frequency f : # oscillations per unit of time
same period T same amplitude A
1
f=
T
[f ] = Hertz (Hz) = cycles/s (= 1/s)
Heinrich Hertz
(1857-1894)
8
Example: oscillating ruler
Consider an oscillating ruler, which does 28 cycles in 10 s, and moves a total
distance of 8.0 cm. What are the amplitude, period, and frequency of this
oscillatory motion?
Amplitude = 8.0 cm / 2 = 4.0 cm
10 s
T= = 0.36 s/cycle
28 cycles
1 28 cycles
f= = = 2.8 Hz
T 10 s
A, T, f do not specify an oscillation completely
What is the physics involved? Equation of motion?
9
Frequency and period
Units of frequency
1 1
f= , and T= Frequency
103 Hz = 1 kHz
Period
1 ms
T f 106 Hz = 1 MHz 1 μs
109 Hz = 1 GHz 1 ns
10
Oscillatory motion – the essence
Mass spring-system, no friction
▶ Spring exerts no force if x = 0
▶ Displacement to the right
⇒ spring force to the left
⇒acceleration to the left
▶ Displacement to the left
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⇒ spring force to the right
⇒acceleration to the right
Restoring force is essential
11
Oscillatory motion – the essence (cont’d)
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12
Oscillatory motion – restoring force
▶ If the restoring force depends
linearly on displacement
F = −kx
⇒motion is harmonic, with 1
oscillation frequency
▶ Simple harmonic motion
(shm)
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13
Oscillatory motion – restoring force (non-ideal)
▶ In practice: restoring force
deviates from linear
▶ Linear spring force usually
good approximation for small
displacements
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14
Brain link: think – pair – share
Oscillatory motion
▶ Recall most important concepts from the topic and write
them down
▶ Pair up with a neighbour student, and explain (share) these
concepts
15
Oscillations
Simple harmonic motion (SHM)
16
Simple harmonic motion (SHM) – math
Characteristics
▶ amplitude A
▶ period T [s]
Mathematical description
▶ x(t) = A cos(ωt + ϕ)
▶ angular frequency ω = 2πf = 2πT in [rad/s]
▶ frequency f =
T in [1/s] or [Hz]
1
▶ phase angle ϕ with x(0) = A cos(ϕ) (initial value)
17
Quiz: SHM
x(t) = A cos(ωt + ϕ)
Which parameter varies in figures
A, B and C on the right?
(a) A: A; B: ω; C: ϕ
(b) A: ω; B: ϕ ; C: A
(c) A: ω; B: A; C: ϕ ⇐ !!
(d) A: ϕ ; B: A; C: ω
18
SHM – position, velocity, acceleration
Position
x(t) = A cos(ωt + ϕ)
Velocity
dx(t)
v(t) = = −ωA sin(ωt + ϕ)
dt
Acceleration
d2 x(t)
a(t) = 2
= −ω2 A cos(ωt + ϕ)
dt
= −ω2 x(t)
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19
SHM – position, velocity, acceleration (cont’d)
Position
x(t) = A cos(ωt + ϕ)
Velocity
dx(t)
v(t) = = −ωA sin(ωt + ϕ)
dt
Acceleration
d2 x(t)
a(t) = 2
= −ω2 A cos(ωt + ϕ)
dt
= −ω2 x(t)
20
SHM – differential equation
Relation between position and acceleration
d2 x(t)
a(t) = 2
= −ω2 x(t)
dt
A harmonic oscillation is apparently described by a
second-order differential equation
d2 x(t)
2
+ ω2 x(t) = 0
dt
How do we obtain x(t) = A cos(ωt + ϕ) as a solution for this?
21
SHM – differential equation: solution
d2 x
Differential equation: + ω2 x = 0 (1)
dt2
d2 x
Start with a solution: x(t) = C exp(λt), which gives: dt2 = Cλ2 exp(λt)
Substitute into (1) to find: Cλ2 exp(λt) + ω2 C exp(λt) = 0
The characteristic equation: λ2 + ω2 = 0 has two roots: λ1,2 = ±jω
Solution: x(t) = C1 exp(jω) + C2 exp(−jω)
= C1 {cos(ωt) + j sin(ωt)} + C2 {cos(ωt) − j sin(ωt)}
= (C1 + C2 ) cos(ωt) + j(C1 − C2 ) sin(ωt)
| {z } | {z }
B1 B2
Result: x(t) = B1 cos(ωt) + B2 sin(ωt) Æ
or: x(t) = A cos(ωt + ϕ), with A = B21 + B22 , and ϕ = arctan (B2 /B1 )
22
Brain link: think – pair – share
Simple harmonic motion
▶ Recall most important concepts from the topic and write
them down
▶ Pair up with a neighbour student, and explain (share) these
concepts
23
Quiz: acceleration graph
This is an ax − t graph for an object in simple harmonic motion. At which of
the following times does the object have the most negative displacement x?
(a) t = 0.10 s
(b) t = 0.15 s ⇐ !!
(c) t = 0.20 s
(d) t = 0.25 s
(e) Two of the above are tied for
the most negative displacement Copyright © 2020 Pearson Education Ltd. All Rights Reserved
The largest positive acceleration comes from the largest positive restoring
force, so at the most negative displacement x
24
Simple harmonic motion
What’s up from the physics side?
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SHM – mass-spring system (re-visit)
Newton’s second law
X
Fx = ma
d2 x
−kx = m 2
dt
d2 x k
This gives + x=0
dt2 m
d2 x
General form: 2
+ ω2 x = 0
dt q
Angular frequency: ω = 2πf = mk
26
Examples: tuning forks and weighing in space
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27
Quiz: Two oscillating systems
The figure shows two identical masses attached to two identical springs and
resting on a horizontal, frictionless surface. Spring 1 is stretched to 5 cm,
spring 2 to 10 cm, and the masses are released at the same time.
Which mass reaches the equilibrium position first?
(a) Mass 1 2
1
(b) Mass 2
(c) Both at the same time ⇐ !!
See next slide
28
Quiz: Two oscillating systems – answer
2
Because k and m are the same, 1
the systems have the same os-
cillation period. So they must
return to equilibrium at the
same time.
The frequency and period of
SHM are independent of the
amplitude
29
SHM – simple pendulum
Newton’s second law
X d2 s
Fs = mas ⇒ −mg sin θ = m 2
dt
2
d s d θ
2
s = Lθ ⇒ as = 2 = L 2
dt dt
θ small ⇒ sin θ ≈ θ
d2 θ d2 θ g
This gives mL 2 + mgθ = 0 ⇒ + θ =0
dt dt2 L
q
g
Angular frequency: ω = 2πf = L
Independent of mass!
30
SHM – simple pendulum – large amplitude oscillations
We used sin θ ≈ θ
How much of an error do we
make with this assumption?
(note: graph used ϕ0 for θ )
v
2π tL
T0 = = 2π
ω g
1 ϕ 1 3 2 ϕ
0 0
T = T0 1 + 2 sin2 + 2 sin4 + ...
2 2 2 4 2
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Brain link: think – pair – share
Physics of oscillations
▶ Recall most important concepts from the topic and write
them down
▶ Pair up with a neighbour student, and explain (share) these
concepts
32
Quiz: height of a lighthouse
A visitor of a lighthouse wants to determine the height
of the tower. She only has a rope, without any measure-
ment scale. Can she measure the height?
(a) No
(b) Yes ⇐ !!
She makes a simple pendulum, which she hangs down
the center of the spiral staircase in the tower. The period
of oscillation is 9.40 s.
v
tL
T = 2π ⇒ L =height
g
T2g 9.402 · 9.81
L= = = 21.9 m
4π2 4π2
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Example: SHM – mechanical watch
Torsion spring: moment τz = −κθ
(torsion constant κ)
Inertia I
(= rotational equivalent of mass)
Newton’s second law for rotation
d2 θ d2 θ κ
−κθ = I 2 ⇒ + θ =0
dt dt2 I Copyright © 2020 Pearson Education Ltd. All Rights Reserved
Æκ
Angular frequency: ω = 2πf = I
34
Example: SHM – fluid in a tube
Mass depends on water column of
length ℓ: m = ρAℓ
(A is cross-section, ρ is density)
Restoring force: F = −2xρAg
Newton’s second law
d2 x d2 x 2g Length
−2xρAg = ρAℓ 2 ⇒ + x=0
dt dt2 ℓq
2g
Angular frequency: ω = 2πf = ℓ
35
Example: SHM – buoyancy
Buoyancy gives for depth L:
Mg = ρALg
(A is cross-section, ρ is density)
Restoring force: F = −y(ρAg)
Newton’s second law
d2 y d2 y g
−y(ρAg) = ρAL 2 ⇒ + y=0
dt dt2 L q
g
Angular frequency: ω = 2πf = L
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Example: SHM – oscillations of molecules
▶ Van der Waals force between two
atoms: Lennard-Jones potential U
▶ Around the equilibrium: potential
U ≈ quadratic function in r
▶ For that case: interaction force
varies linearly with r.
▶ F⃗ = −∇U
⃗ towards equilibrium:
restoring force
▶ molecules oscillate harmonically Copyright © 2020 Pearson Education Ltd. All Rights Reserved
37
Example: SHM – friction oscillator
Video courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State
From Engineering Dynamics (2015), Meriam, Kraige, Bolton, 8th edition
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Simple harmonic motion – energy
Only conservative forces:
∆K + ∆U = 0
K + U is constant
(mechanical energy)
Use x(t) = A cos(ωt + ϕ)
Total energy
1 1 1
E = K + U = mv2x + kx2 = kA2
2 2 2
Umax = 12 kA2 (vx = 0 ⇒ K = 0)
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Damped oscillations
Some dissipation is always present
40
New element: damper
The force of a damper is proportional to the velocity, but
opposite to the direction
41
Quiz: Damper
The force produced by a damper . . .
(a) is always conservative
(b) is always non-conservative ⇐ !!
(c) depends on the application
Fdamper = −bv. Since v is always in the direction of motion, the force is always
against the direction of the path taken. Hence W < 0 for a closed path.
42
SHM – mass-spring-damper system
Newton’s second law
X
Fx = ma
d2 x
−kx − bv = m 2
dt
2
d x dx
This gives m 2 + b + kx = 0
dt dt
Try solution x(t) = A cos(ωt + ϕ):
−mω2 A cos(ωt + ϕ) − bωA sin(ωt + ϕ) + kA cos(ωt + ϕ) = 0
Only works when b = 0 ⇒ use complex exponentials
43
Undamped and damped oscillation
Undamped oscillation
↑ ↑
→ →
What does a damped oscillation look like?
What is the function for the amplitude?
44
Damped oscillation
Differential equation for damped oscillation:
d2 x b dx k
+ + x=0
dt2 m dt m
−b
With solution x(t) = A exp 2m t cos(ω′ t + ϕ)
Characteristics:
▶ Decaying amplitude due to damping: A exp
−b
2m t
r
▶ Damped angular frequency: ω =
′ b 2
ω20 − 2m
Ç
▶ Angular frequency of undamped oscillation: ω0 = k
m
45
Damped oscillation (cont’d)
Solution depends on damping ratio:
2
−b
x(t) = A exp 2m t cos(ω′ t + ϕ), with ω′2 = k b
m − 4m 2
With stronger damping
▶ amplitude decreases more
rapidly
▶ oscillation period T increases
(T0 is period without damping)
p
▶ critical damping for b = 2 km
46
Brain link: think – pair – share
Damped oscillation
▶ Recall most important concepts from the topic and write
them down
▶ Pair up with a neighbour student, and explain (share) these
concepts
47
Quiz: acceleration graph
The force on a damped oscillator is Fx = −kx − bvx . During its motion, the
oscillator loses mechanical energy most rapidly . . .
(a) when it is at maximum positive displacement
(b) when it is at maximum negative displacement
(c) when it is passing through the equilibrium position ⇐ !!
(d) when it is at either maximum positive of maximum negative
displacement
(e) Misleading question: the oscillator loses mechanical energy at the same
rate throughout the motion
There is only energy loss through damping. Velocity (and thus the
dissipation through damping) is highest when the oscillator is passing
through the equilibrium position
48
Example of a damper: car shock absorber
49
Example: extremely bad suspension
50
Energy in damped oscillations
Damping force is non-conservative
⇒ mechanical energy of the system decreases continuously
1 1
E = mv2x + kx2
2 2
dE dvx dx
Rate of change in E: = mvx + kx = vx (max + kx)
dt dt dt
Use −kx − bvx = max ⇒ kx + max = −bvx
dE
This yields: = vx (−bvx ) = −bv2x
dt
⇒ energy dissipation in the damper
51
Forced oscillations
External forcing on an oscillator
52
Forced oscillations
External force Fext (t) present:
forced oscillations
Assume Fext (t) is harmonic
with driving frequency ωd :
Fext (t) = Fmax cos(ωd t)
d2 x dx
This gives (Newton): m 2 + b + kx = Fmax cos(ωd t)
dt dt
Response depends on damping of the system and the
difference between ωd and ω′
53
Forced oscillations (cont’d)
Differential equation:
d2 x dx
m 2 + b + kx = Fmax cos(ωd t)
dt dt
Fmax
A= q
(k − mω2d )2 + bω2d
Fmax
or A= q
m (ω20 − ω2d )2 + (b/m)2 ω2d
k
where ω20 =
m
Maximum amplitude for ωd ≈ ω0 : resonance
54
Brain link: think – pair – share
Forced oscillations, resonance
▶ Recall most important concepts from the topic and write
them down
▶ Pair up with a neighbour student, and explain (share) these
concepts
55
Example: forced ocsillator: RLC circuit
Well-known example from Circuits course:
d2 q dq 1
L 2 +R + q=V
dt dt C
56
Example: resonating glass
57
Example: Tacoma Narrows bridge
Collapsed on November 7, 1940: YouTube video
58
Recap
To remember
▶ Parameters to describe oscillations
▶ Physics underlying oscillatory motion (restoring force!)
▶ Simple harmonic motion (SHM)
• Mathematical description
• Energy stored in SHM
▶ Damped oscillations
▶ Forced oscillations
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