damped harmonic motion - physics department ucc · z 0 z z j m b a ... driven damped harmonic...

PY1054Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland

ROINN NA FISICEDepartment of Physics

Damped Harmonic Motion

PY1054

Special Topics in Physics

1



Driven Damped Harmonic Motion

2

What if we apply a harmonic force?: ti

h BeF

2

2

dt

xdm

dt

dxbkxFF h The total force is then:

Assume a solution of the form: tiAex




3

http://www.cabrillo.edu/~jmccullough/Applets/Flash/Fluids,%20Oscil

lations%20and%20Waves/DrivenSHM.swf

The amplitude of the oscillations…

im

BA

22

0 22222

0

m

BA

m

b

http://www.cabrillo.edu/~jmccullough/Applets/Flash/Fluids, Oscillations and Waves/DrivenSHM.swf



Driven Damped revised

4

What if we apply a harmonic force?:ti

h BeF

2

2

dt

xdm

dt


Assume a solution of the form: tiAex

tiAeidt

dx

tiAedt

xd 2

2

2




5

But, the imaginary part must be zero, so:

cossin22

0

22

0

1tan




6

Take a look at the phase:

𝜙 = tan−1 −𝛾𝜔

𝜔02 − 𝜔2

= −tan−1𝛾𝜔

𝜔02 − 𝜔2

Which fit into: 𝑥 = 𝐴𝑒𝑖 𝜔𝑡+𝜙

We can also note that if: 𝑥 = 𝐴𝑒𝑖 𝜔𝑡−𝜙

𝜙 = tan−1𝛾𝜔

𝜔02 − 𝜔2

Both signs possible, depending on initial assumptions




7

Thus:

The amplitude can be calculated :

22222

0

m

BA as before

0m

BA When: 0

𝐵2 = 𝑚2𝐴2 𝜔02 − 𝜔2 2 + 𝜔2𝛾2




8

What if we apply a harmonic force?: tBFh sin

2

2

dt

xdm

dt


Assume a solution of the form: tDtCx cossin

tDtCdt

dx sincos

tDtCdt

xd cossin 22

2

2

222 DCA

Amplitude:




9

tDmtCmtDtCb

tDtCktB

cossinsincos

cossinsin

22

tCmtDbtkCtB sinsinsinsin 2

tDmtCbtkD coscoscos 2

Separate into sine & cosine terms:

sine: DCmCDm

bC

m

kmB

22

0

2

cosine: DDm

kC 22

0

2




10

DCmB 22

0CD

22

0

CmB

22

0

2222

0

22222

0

22

0

22

0

2222

0

m

B

m

BC

222 DCA But we need:




11

2

222

0

22

1 C

222 DCA

2

222

0

22222

0 C

222222

0

2

222

0

22

m

BC




12

So, finally

Can also calculate phase w.r.t. B, from C and D

𝐴 =𝐵

𝑚 𝜔02 − 𝜔2 2 + 𝜔2𝛾2

Some algebra to simplify, first:

A =𝐵

𝑚𝜔02 1 −

𝜔𝜔0

2 2

+𝜔𝜔0

2 𝛾2

𝜔02




13

Then set 𝐵 = 𝑚𝜔02, →

𝐵

𝑚𝜔02 = 1

𝐴 =1

1 −𝜔𝜔0

2 2

+𝜔𝜔0

2 𝛾2

𝜔02

When: 𝜔 → 𝜔0 𝐴 𝜔 = 𝜔0 =1

𝛾2

𝜔02

=𝜔0

𝛾




14

The Q factor 𝑄 =𝜔0

𝛾, →

𝛾2

𝜔02 =

1

𝑄2

𝐴 =1

1 −𝜔𝜔0

2 2

+𝜔𝜔0

2 1𝑄2



The amplitude vs. frequency

15

0

A



The Phase

16

22

0

1tan

If: 0 2

m

b0where:

0

22

0

022

0

2

0 022

0

0



The Phase

17

22

0

1tan

If: 0 2

m

b0where:

0

22

0

022

0

2

0 022

0



The phase vs. frequency

18

0



Power absorbed

19

The oscillator will absorb power from the driving force, which results in the increased oscillations:

𝑃 = 𝐹ℎ𝑣

𝐹ℎ = 𝐵 sin 𝜔𝑡 𝑥 = 𝐴 sin 𝜔𝑡 + 𝜙

𝑣 =𝑑𝑥

𝑑𝑡= 𝜔𝐴 cos 𝜔𝑡 + 𝜙

𝑃 = 𝐴𝐵𝜔 sin 𝜔𝑡 cos 𝜔𝑡 + 𝜙

= 𝐴𝐵𝜔 sin 𝜔𝑡 cos 𝜔𝑡 cos𝜙 − sin 𝜔𝑡 sin𝜙

= 𝐴𝐵𝜔 sin 𝜔𝑡 cos 𝜔𝑡 cos𝜙 − sin2 𝜔𝑡 sin𝜙



Power absorbed

20

Consider the time averages:

sin 𝜔𝑡 cos 𝜔𝑡 =1

𝑇න0

𝑇

sin 𝜔𝑡 cos 𝜔𝑡 𝑑𝑡

𝑥 = sin(𝜔𝑡) ,𝑑𝑥

𝑑𝑡= 𝜔 cos 𝜔𝑡 ,→

𝑑𝑥

𝜔= cos 𝜔𝑡 𝑑𝑡

sin 𝜔𝑡 cos 𝜔𝑡 =1

𝑇න0

𝑇

𝑥 𝑑𝑥 =𝑥2

2ቚ0

𝑇=sin2 𝜔𝑡

2ቚ0

𝑇= 0



Power absorbed

21

Consider the time averages:

sin2 𝜔𝑡 =1

𝑇න0

𝑇

sin2 𝜔𝑡 𝑑𝑡 =1

2𝑇න0

𝑇

1 − cos 2𝜔𝑡 𝑑𝑡

=1

2𝑇𝑥 −

cos 2𝜔𝑡

2𝜔0

𝑇

=1

2

Thus the power absorbed:

𝑃 = 𝐴𝐵𝜔 sin 𝜔𝑡 cos 𝜔𝑡 cos𝜙 − sin2 𝜔𝑡 sin𝜙

=1

2𝐴𝐵𝜔 sin𝜙 Referred to in assignment



The power vs. frequency

22

0

P Δ𝜔0

𝑄 =𝜔0

𝛾=

𝜔0

Δ𝜔0



How long does it take?

23

Look at the following web site, and examine how long it takes for the oscillations to increase with small damping:

http://www.cabrillo.edu/~jmccullough/Applets/Flash/Fluids,%20Oscil

lations%20and%20Waves/DrivenSHM.swf

The picture on the next page gives an idea, about why Q oscillations are required, where:

𝑄 =𝜔0

𝛾=

𝜔0

Δ𝜔0

http://www.cabrillo.edu/~jmccullough/Applets/Flash/Fluids, Oscillations and Waves/DrivenSHM.swf



The oscillator takes time to charge

24

damped harmonic motion - physics department ucc · z 0 z z j m b a ... driven damped harmonic...

Documents