l 08 oscillation - wordpress.com

Ver 1.0 © Chua Kah Hean xmphysics 1

XMLECTURE

08 OSCILLATION NO DEFINITIONS. JUST PHYSICS.

8.1 Simple Harmonic Motion ............................................................................................................. 2

8.1.1 SHM in time-domain ............................................................................................................. 3

8.1.2 SHM in the x-domain ............................................................................................................ 4

8.1.3 The Half-Amplitude Problems ............................................................................................... 6

8.2 Restoring Force .......................................................................................................................... 7

8.2.1 Natural Frequency ................................................................................................................ 9

8.3 Energy of Oscillation ................................................................................................................. 11

8.3.1 SHM Energy in time-domain ............................................................................................... 13

8.3.2 SHM Energy in the x-domain .............................................................................................. 15

8.4 Damped Oscillations ................................................................................................................. 16

8.4.1 Practical Applications of Damping ...................................................................................... 19

8.5 Forced Oscillation ..................................................................................................................... 20

8.5.1 Resonance Curve ............................................................................................................... 22

8.5.2 Effect of Damping on Resonance ....................................................................................... 23

8.A Angular Frequency vs Angular Velocity .................................................................................... 25

8.B More Demonstrations ............................................................................................................... 26

Online resources are provided at https://xmphysics.com/osc

https://xmphysics.com/osc


8.1 Simple Harmonic Motion

An oscillation such as a pendulum is fun to watch. When displaced and released, the pendulum will

go into a periodic to-and-fro motion about its equilibrium position. It’s always rushing towards the

equilibrium position, but overshoots every time, and is doomed to repeat this motion for eternity (until

its energy is totally sapped by resistive forces).

The pendulum (at small amplitude) and the spring-mass system are two examples of simple harmonic

motions (SHM). SHM is “simple” in the sense that its motion can be described by one single sinusoidal

function.

see animation at xmphysics.com

Specifically, the displacement x of a SHM with amplitude x0 and period T can be described by the

equation

0

2sinxx t

T

Math buffs may recognise the 2

T

term as the scaling factor that makes one cycle occupy one

period T. In physics, this 2

T

term is given the name angular frequency ω.

22 f

T

The angular frequency is a rather abstract concept which takes time to fully understand. For the time

being, it is sufficient to think of it simply as the frequency of the oscillation multiplied by 2 .

displacement x

time t

x0

0

-x0

T 2T

http://www.xmphysics.com/8.1



8.1.1 SHM in time-domain

We can understand loads about SHM from just a few lines of math.

Let’s begin1 with 0 sinx x t

Since dx

vdt

, it follows that 0 cosv x t

Since dv

adt

, it follows that 2

0 sina x t


Just by looking at these three equations, we can make the following deductions about SHM.

1. Displacement, velocity and acceleration all vary sinusoidally with time. Math buffs would take

pleasure to point out that the time-derivative of a sinusoid is also a sinusoid, hence this beautiful

outcome.

2. Acceleration always leads velocity by a quarter cycle, and velocity in turn leads displacement by

a quarter cycle. Hence acceleration reaches its maximum value a quarter period before velocity

reaches its maximum value a quarter period before displacement reaches its maximum value.

3. By inspecting 0 cosv x t , we can tell that maximum velocity is given by

max 0v x

4. By inspecting 2

0 sina x t , we can tell that maximum acceleration is given by

2

max 0a x

1 We wrote 0 sinx x t because we imagined the oscillation to start from 0x . If we imagined the oscillation

to start from 0x x , we would have written down 0 cosx x t . We can use any starting position. It will not

affect the analysis.

x a

t

T/4

v

T/4 T/4 T/4

http://www.xmphysics.com/8.1.1



8.1.2 SHM in the x-domain

In this section, we will obtain the a-x and v-x equations which we can use to calculate the acceleration

and velocity of a SHM at each displacement.

We begin by writing down the three time-domain equations.

0

0

2

0

sin

cos

sin

x x t

v x t

a x t

Recognizing that 0 sinx t can be replaced by x, we can rewrite the a-t equation as

2a x


As the mathematics has shown, the acceleration of a SHM is directly proportional in magnitude but

opposite in sign to the displacement. The a-x graph is thus a straight line passing through the origin,

with the gradient corresponding to 2 .

To obtain the v-x equation, we rewrite the x-t and v-t equations as

0

sinx

tx

0

cosv

tx

x0

-x0 x

a

-ω2x0

ω2x0




Using the trigonometic identity 2 2sin cos 1x x , we deftly get rid of the t terms.

2 2

2 2

0 0

2 2 2 2 2

0

2 2

0

sin cos 1

( ) ( ) 1

t t

x v

x x

x v x

v x x

Mathematics aficionados would recognize the first line as the equation of an ellipse, with intercepts at

0x x and 0v x .


Note also that the oscillation progresses in the clockwise direction along the elliptical graph. This is

obvious if you realize that for the top half of the graph, the velocity is positive, so the displacement

should progress towards the right (to become more positive). Likewise, for the bottom half of the graph,

the velocity is negative, so the displacement should progress towards the left (to become more

negative).

+ωx0

-ωx0

-x0 +x

0

x

v



8.1.3 The Half-Amplitude Problems

watch video explanation at xmphysics.com

Very often, exam questions involve the calculation of time taken for a SHM to progress from the

extreme position ( 0x x ) to the half-amplitude position ( 00.5x x ). And students would start thinking,

hmm, it takes 4

T to go from the extreme position to the equilibrium position right? (right) So it takes

8

T for it to go from the extreme position to the mid-point position right? (wrong).

Wrong. Wrong. Wrong. Because SHM is not a constant speed motion. It travels more slowly near the

extreme positions and more quickly near the equilibrium positions. As such, it takes 8

T to go from

extreme to mid-point, but 8

T to go from mid-point to equilibrium position.

You can calculate the exact time taken using 0cosx x t or similar equations. But for the half-

amplitude position, there is an easier method! I call it the “by inspection” method. If you can recall

simple trigonometric values such as cos60 0.5 and sin30 0.5 , you can “see” the t values for

when the oscillation is at the half-amplitude points. You can then “see” that it takes

60 1

360 6T T

to travel between the extreme and the half-amplitude position, and

30 1

360 12T T

to travel between the equilibrium and half-amplitude position.

t

T

x

x0

-x0

-0.5x0

0.5x0

60° 30° 30° 60° 60° 30° 30° 60°




8.2 Restoring Force

Now that we have gone through the “kinematics” of SHM, let’s work out the “dynamics” behind SHM.

We begin by writing down the x-t, v-t and a-t equations for SHM.

0

0

2

0

sin

cos

sin

x x t

v x t

a x t

Since 0 sinx t is displacement x, we can rewrite acceleration as

2a x

If we multiply the mass of the oscillating body m to both sides, we get

2

netma F m x

From this equation, we attain the revelation that for a mass to undergo SHM, it must be experiencing

a net force that is (1) proportional to its displacement and (2) opposite in direction to its displacement.

Since the net force is always directed towards the equilibrium position, it is often called the restoring

force.

2

netF m x

x0

-x0 x

Fnet

-mω2x0

mω2x0



At this point you may be wondering, where on earth can we encounter a restoring force that varies

proportionately with displacement? Actually, such a restoring force is quite common. Take for example

the elastic force of a spring that obeys Hooke’s Law.


As illustrated in the diagram above, a mass is hung on a spring. In (b), the mass is at the equilibrium

position, where the downward weight mg is balanced by the upward tension force T of the spring. The

restoring force is zero at the equilibrium position.

In (c), the mass is displaced downward by x, so the spring is further extended by x. This results in the

tension increasing by kx, where k is the spring constant. Since T is larger than mg by kx, the restoring

force is kx and upward.

Similarly, in (d), the mass is displaced upward by x. Compared to the equilibrium position, the

extension is now shorter by x. This implies that T is smaller than mg by kx. The restoring force is thus

kx and downward.

In one equation

netF kx

That’s why when the mass is displaced from the equilibrium position and released, it goes into simple

harmonic motion. Neat.

equilibrium position

mg

(a)

T

Fnet

T

mg

T

mg

Fnet

x

unstretched

(b) (c) (d)



8.2.1 Natural Frequency

Say we have two identical pendulums of exactly the same length. However, one is released further

away from the equilibrium position, thus having a larger amplitude. Which pendulum should have a

larger period of oscillation?

watch video at xmphysics.com

Surprisingly (to many people), the period of oscillation of a SHM is independent of its amplitude. Each

time a SHM system is displaced from its equilibrium position and released, it oscillates at its own

particular frequency called the natural frequency.

Take for example the vertical spring-mass system. We note that at any

displacement x, the weight and tension are unbalanced by the amount

kx, meaning kx is the net force.

netF kx

ma kx

ka x

m

Comparing this with the SHM equation 2a x , we can deduce that

2 k

m

Hence, the natural frequency is 1

2n

kf

m and the natural period is 2n

mT

k .

This 2 k

m relationship can explain quite a bit. It explains why a “stiff” system has a high natural

frequency: a stiff system has a high k value, so the restoring force at every displacement is stronger.

It also explains why a heavy system has a low natural frequency: a heavy system has a large inertia

m, so the acceleration at every displacement is smaller. This “stiffness-to-inertia ratio” is a useful idea

that can provide us with a solid qualitative feel of the natural frequency of most oscillating systems.

m

k

video at xmphysics.com





Having said that, this “stiffness to inertia ratio” can be misleading when

applied to the pendulum. The mass of a pendulum does not affect its

natural frequency! Why? Because a larger mass results in both a

larger restoring force and a larger inertia. So the effect of a larger mass

cancels itself out (just like all object free fall at g regardless of mass).

However, the “stiffness” of a pendulum does depend on the

acceleration of free fall g, and its “inertia” does increase with length L.

The natural frequency of a pendulum thus turns out to be2 1

2n

gf

L .

Now that you understand the phenomenon of natural frequency, you will have a deeper satisfaction

watching these wonderfully intricate patterns (called Lissajous Figures) produced by two

simultaneous oscillations.



2 Be aware that this formula is valid only for small amplitude oscillations. Its derivation is not so straight forward

and is not required by the H2 syllabus.

L g

video at xmphysics.com





8.3 Energy of Oscillation

After we’re done with the kinematics and dynamics, it’s time to approach SHM from the energy

perspective.

An oscillation obviously has energy. Let’s begin by asking, where did this energy come from?

To produce an oscillation with amplitude x0, the system must first be displaced from its equilibrium

position to 0x x . This requires an external force to pull or push against the restoring force. The

work done by this external force corresponds to the (total) energy of oscillation, TE.

Recall that 2a x and the restoring force is 2

netF m x . Since the external force must match

the restoring force at every position, 2

extF m x at every position.


Since the area under the F-x graph is work done, we can derive the formula for TE to be

2 2 2

0 0 0

1 1( )( )

2 2TE x m x m x

-x0

x0

mω2x

0

-mω2x

0

x

Fext




During the oscillation, the restoring force does work continuously, resulting in continuous energy

conversion between kinetic energy KE and potential energy PE. From the principle of conservation of

energy, we know that at anytime and at anywhere,

TE KE PE

But at the extreme positions, KE is zero and PE is at its maximum. So

2 2

max 0

1

2PE TE m x .

Conversely, at the equilibrium position, PE is zero and KE is at its maximum. So

2 2

max 0

1

2KE TE m x

Just for fun, let’s write everything in one line

2 2

max max 0

1

2TE KE PE KE PE m x

Personally, I “memorise” this formula by remembering that max 0v x . So

2 2 2 2

max max max 0 0

1 1 1( )

2 2 2TE PE KE mv m x m x


8.3.1 SHM Energy in time-domain

Assuming that 0 cosv x t , we can write

2 2 2

0

1cos

2KE m x t

Since maxTE KE , we know

2 2

0

1

2TE m x

Since PE TE KE , we can work out that

2 2 2 2 2

0 0

2 2 2

0

2 2 2

0

1 1cos

2 2

1(1 cos )

2

1sin

2

PE m x m x t

m x t

m x t

Let’s now see the equations on a graph.


Be aware that because 21

2KE mv , positive and negative v give the same KE. So the KE graph is

the same whether it is the positive or negative half cycle of v. Math geeks will point out that using the

double-angle trigonometric identity, we can write 2 2

0

1cos

2mv t as 2

0

1 1 cos2( )

2 2

tmv

, which

explains why the KE-t graph is a raised sinusoid with frequency 2f.

t

T

KE

v




Similarly, there are two PE cycles in one oscillation cycle because PE is maximum at both the extreme

positions, and zero whenever the oscillation crosses the equilibrium position.


TE, being constant, is of course a horizontal line.


PE

t

T

x

PE KE

t

T

E

TE




8.3.2 SHM Energy in the x-domain

From 2 2

0v x x , we obtain

2 2 2

0

1( )

2KE m x x

Since maxTE KE at 0x ,

2 2

0

1

2TE m x

Since PE TE KE , we can work out

2 2 2 2 2

0 0

2 2

1 1( )

2 2

1

2

PE m x m x x

m x

A few things to note.

1. Both KE and PE are quadratic graphs and are mirror image of each other

2. The graphs intersect at 1

2E TE , 0

2

xx .


-x0 x

0

x

E

TE

KE

PE




8.4 Damped Oscillations

If an oscillation is completely undamped, the oscillation will go on forever. In

practice however, all oscillations must encounter some damping forces (e.g. air

resistance, friction). Damping saps the oscillation’s energy continuously, causing

its amplitude to decrease continuously over time.

see video at xmphysics.com

The behavior of damped oscillations has been studied extensively, in particular for the case of the

damping force being proportional to the velocity. It turns out that there are three distinct outcomes

depending on whether the oscillation experiences light damping, heavy damping or critical damping.

Light Damping

If the oscillation is subjected to light damping (a.k.a. underdamping), the amplitude of the oscillation

decreases gradually over time. In fact, the displacement-time graph of a damped oscillation is

approximately enveloped by an exponential decay function. The higher the amount of damping, the

faster the rate of decay and the steeper the exponential decay function.

Damping also has an effect on the period of oscillation. It causes the oscillation to have longer periods

than if they were undamped. However, the lengthening of the period under very light damping can be

negligibly small.




Heavy Damping

If the oscillation is subjected

to heavy damping (a.k.a.

overdamping), the oscillator

returns slowly to the

equilibrium position without

ever going past the

equilibrium position. In that

sense, heavy damping

totally stops the system from

oscillating.

Critical Damping

Critical damping is the

transition point between

light and heavy damping.

Critical damping sends a

displaced oscillator back to

the equilibrium position in

the shortest amount of time.

Because of this, critical

damping is the design

objective in many

engineering applications

(e.g. shock absorbers, car

suspension systems etc.).


This pictures below illustrate oscillations at different amounts of damping:

see images at xmphysics.com

For a real life demonstration, check out the following video:





8.4.1 Practical Applications of Damping

We actually encounter a lot of oscillatory systems in our daily lives. Some of these oscillations are

nuisances and should be damped and brought to rest quickly, before they drive us crazy.

The car or bus we travel in are basically inverted spring-mass systems. The suspension system that

the car sits on is set into oscillations every time the car hits a bump, or drives over an uneven surface.

Shock absorbers are designed to be critically damped so that the oscillations dissipate away in the

shortest amount of time.


Buildings are also inverted pendulums. They may look rigid to our eyes, but they are actually elastic

and bendable. They too can be set into dangerous oscillation by strong gusts of wind, or ground

tremors. For this reason, skyscrapers are equipped with devices to damp these oscillations.






8.5 Forced Oscillation

Do you know that there are two ways to get an oscillation going?

The open and candid way is to displace the oscillator from its

equilibrium position and then release it. The oscillator will oscillate at

its natural frequency. This is called a free oscillation. The covert and

surreptitious way is to exert a small but continuous external periodic

driving force to the oscillator. In this case, the oscillator will be forced

to oscillate at the frequency of the driving force. This is called a

forced oscillation.


For a free oscillation, the energy is transferred to the oscillator in “one lump payment” at the beginning

when the oscillator was displaced. The amplitude of oscillation starts off at a maximum and decreases

gradually over time due to damping.

For a forced oscillation, however, energy is continuously transferred to the oscillator by the periodic

driving force. The amplitude of the forced oscillation starts at zero, and increases over time, and

stabilizes at a final amplitude when the rate of input of energy from the driver is matched by the rate

of loss of energy to the surrounding due to damping.


The efficacy of energy transfer from the driver to the oscillator is dependent on how close the driving

frequency is to the natural frequency of the oscillatory system. When there is a large mismatch, the

energy transfer from the driver to the oscillator is inefficient, and the forced oscillation will only attain

a low amplitude.

time

displacement Final amplitude of forced oscillation





Resonance occurs when the driving frequency matches the natural frequency of the oscillator. At

resonance, transfer of energy from the driver to the oscillator is at its most efficient, allowing for

maximum amplitude to be attained.


Many teachers are very fond of the swing analogy, that you must

push the swing at the correct timing if you want the swing to go

higher and higher. Obviously, if you push the swing when it is coming

back towards you, you’re being destructive to the amplitude building

process. So in a similar (but not exact) fashion, to achieve resonance

the periodic driving force must be “synchronized” to the natural

frequency of the oscillator, to ensure only positive (and no negative)

work is done all the time to the oscillator.

Resonance is a common occurrence around us. But because the driving force can be surreptitiously

small, many people may not realize know what’s happening.

see demonstration at xmphysics.com




8.5.1 Resonance Curve

Consider an oscillatory system with natural frequency f0. Let’s couple this

system to a driver of varying frequency f but fixed amplitude A0. If we plot

on a graph how the amplitude of the forced oscillation A varies with the

driver’s frequency f, we obtain the so-called resonance curve.

see applet at xmphysics.com

The main features of the resonance curve are:

1. Amplitude A peaks at 0f f .

When the driver is driving at the natural frequency of the oscillator, it’s like a match made in heaven.

The energy transfer from the driver to the oscillator is at its most efficient. In fact, the oscillator is

able to accumulate the energy transferred and reaches a maximum amplitude that is many times

larger than the amplitude of the driver.

2. As f approaches 0, A approaches A0.

Put it this way. When a slow driver meets a quick oscillator, the oscillator is so “nimble” it tracks

the motion of the driver exactly. So the oscillator has the same amplitude of oscillation as the

driver.

3. As f approaches infinity, A approaches zero.

When a quick driver meets a slow oscillator, the oscillator is so “retarded” it simply cannot respond

in time. The driving forces changes too rapidly for the oscillator and its amplitude is stuck at zero.

A

f

f0

A0

driver

oscillator

A0

A

f




8.5.2 Effect of Damping on Resonance

Resonance can be a nuisance at times. In such situations, damping forces are actually very desirable.

Remember that in a forced oscillation, the final amplitude reached is the amplitude at which the rate

of gain of energy (from the driver) is matched by the rate of loss of energy (to the surrounding). With

higher damping, the rate of energy loss at each amplitude is higher. This results in a lower resonance

amplitude.


Now let’s look at the effect of damping on our resonance curves.


0

5

10

15

20

25

30

35

40

45

50

0 0.5 1 1.5 2

Damping Ratio:

--- 0.01

--- 0.05

--- 0.10

A

A0

f f0



http://www.xmphysics.com/osc


Notice that whenever damping is increased, the entire resonance curve is lower not just at the

resonance frequency, but at every frequency (except 0f ). On the other hand, decreased damping

lowers the rate of energy loss to resistive forces, and results in a higher resonance curve. In theory,

at zero damping, the resonance amplitude should reach infinity. In practice, however, there are

physical constraints. For examples, a pendulum cannot swing more than 90°, a spring cannot be

compressed to zero length, or the oscillator cannot withstand the physical stress and breaks down

when the oscillation is too vigorous.

Do you remember that damping lengthens the period of oscillation? Similarly, damping causes

resonance to occur at a frequency slightly lower than the oscillator’s natural frequency. So the

resonance peaks will shift towards lower frequency as damping increases. But again, the shift is not

significant at very light damping conditions.

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3

Damping Ratio:

--- 0.05

--- 0.10

--- 0.25

--- 0.50

--- 0.75

A A0

f f0


8.A Angular Frequency vs Angular Velocity

The angular frequency is a rather abstract concept which deserves some discussion.

22 f

T

At the most basic level, from 2 f , you should appreciate that the angular frequency ω is simply

the frequency of the oscillation multiplied by 2 .

At a more abstract level, from 0 sinx x t , we can

think of t as the phase angle of the oscillation.

So ω is the “velocity” at which the phase of the

oscillation progresses. Watch this demonstration

where 17 pendulums with different ω are put to a

race.


The angular frequency ω in SHM is actually very similar in concept to the angular velocity ω in circular

motion. (This explains why they are both given the symbol ω.)

For oscillations, 2d

dt T

is the rate of change of phase angle.

For circular motion, 2d

dt T

is the rate of change of angular displacement.

In fact, many teachers are very fond of pointing out that a uniform circular motion with angular velocity

ω, when viewed “sideway”, is reduced to a simple harmonic motion with the same angular frequency

ω.


θ

θ

angular velocity ω

angular frequency ω

time

circular motion shm

http://www.xmphysics.com/8.A

http://www.xmphysics.com/8.A

http://www.xmphysics.com/osc


8.B More Demonstrations

These demonstrations are arguably beyond the H2 syllabus. Nevertheless, they are fun and they may

help you understand the H2 syllabus.

watch videos at xmphysics.com

Barton’s Pendulum

The Coupled Pendulum

Tuning Fork Resonance

The Coupled Inverted Pendulum

http://www.xmphysics.com/8.B

http://www.xmphysics.com/8.B

l 08 oscillation - wordpress.com

Documents