13. oscillatory motion. oscillatory motion 3 if one displaces a system from a position of stable...

13. Oscillatory Motion

Oscillatory Motion

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Oscillatory Motion

If one displaces a system from a position of stable equilibrium the system will move back and forth, that is, it will oscillate about the equilibrium position.

The maximum displacement fromthe equilibrium is called theamplitude, A.

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Oscillatory Motion

The time, T, to go through one

complete cycle is called the

period. Its inverse is called

frequency and is measured

in hertz (Hz).

1 Hz is one cycle per second.

1f

T

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Simple Harmonic Motion

For many systems, if the amplitude is small enough, the restoring force F satisfies Hook’s law.

The motion of such a system is called simple harmonic

motion (SHM)

F kxHook’s law

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As usual, we can compute the motion of the object using Newton’s 2nd law of motion, F = m a:

The solution of this differential equation

gives x as function of t.

2

2

dk

xx m

dt

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Suppose we start the system

displaced from equilibrium

and then release it. How will

the displacement x depend on

time, t ?

Let’s try a solution of the form

cosx tA

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Note that at t = 0, x = A.

A is also the amplitude. Why?

To find the value of we need

to verify that our tentative solution

is in fact a solution of the

equation of motion.

cosx tA

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.

22

2

cos

sin

cos

x t

dxt

dt

d xt

dt

A

A

A

therefore2

2

2

cos ( cos )

d xkx m

dt

k tA Am t

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We can get a solution

if we set k = m2, that is,

By definition, after a

period T later the motion

repeats, therefore:

Simple Harmonic MotionFrequency and Period

.

k

m

cos cos( )x A t TA t

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The equation

can be solved if we set

T = 2, that is, if we set


.

cos cos( )

cos cos sin sin

A t A t

A t A t

T

T T

22 f

T

is called the angular frequency

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For simple harmonic motion

of the mass-spring system,

we can write


.

12

mT

f k

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It is easy to show that

is a more general solution of the equation of motion.

The symbol is called the phase. It defines the

initial displacement

x = A cos

Simple Harmonic MotionPhase

.

cos( )Ax t

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Simple Harmonic MotionPosition, Velocity, Acceleration

cos( )Ax t

sin( )Av t

2 cos( )Aa t

Position

Velocity

Acceleration

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cos( )Ax t

sin( )Av t

2 cos( )Aa t

Simple Harmonic MotionPosition, Velocity, Acceleration

Applications of SHM

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At equilibriumupward force of spring = weight of block

Gravity changes onlythe equilibrium position

Vertical Mass-Spring System

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The Torsional Oscillator

A fiber with torsional constant provides a restoring torque:

The angular frequency depends on and the rotational inertia I:

I

Newton’s 2nd law for this device is

I

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The Pendulum

A simple pendulum consists of a point mass suspended from a massless string!

Newton’s 2nd law for such

a system is2

2sinL

dtg

dIm

I

The motion is not simple harmonic. Why?

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The Pendulum

If the amplitude of a pendulum is small enough, then we can write sin ≈ , in which case the motion becomes simple harmonic 2

2

dImg

tL

d

mgL

I This yields

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The Pendulum

For a point mass, m, a distance L from a pivot, the rotational inertia is I = mL2.

Therefore,

mgL g

I L

and2

2L

Tg

Energy in SHM

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Energy in Simple Harmonic Motion

cos( )Ax t

sin( )Av t

2 cos( )Aa t

Position

Velocity

Acceleration

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Kinetic Energy

2

2

22

1

21

sin ( )2

A

K mv

m t

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Potential Energy

2

2 2

1

21

cos ( )2

A

U x

k t

k

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Total Energy = Kinetic + Potential

2

2

222 21 1sin ( ) cos ( )

2 21

2

A A

A

E K U

m t k t

k

2m k

In the absence of non-conservative forces the total mechanical energy is constant

For a spring:

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In a simple harmonic oscillator theenergy oscillates back and forth between kinetic and potentialenergy, in such a way that the sumremains constant.

In reality, however, most systemsare affected by non-conservativeforces.

Damped Harmonic Motion

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Non-conservative forces, such as friction, cause the amplitude of oscillation to decrease.

cos( )Ax t

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In many systems, the non-conservative force (called the damping force) is approximately equal to

where b is a constant giving the damping strength and v is the velocity. The motion of such a mass-spring system is described by

bv

2

net

2

F

dxkx b

dt

a

d

m

xm

dt

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The solution of the differential equation

is of the form

For simplicity, we take x = A at t = 0, then = 0.


2

2

dxkx b

dt dm

d x

t

/( ) cos( )tx t Ae t

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If one plugs the solution

into Newton’s 2nd law, one will find

the damping time

and the angular frequency,

where

is the un-damped angular frequency


/( ) cos( )tx t Ae t

02

0

2 /

1 1/( )

m b

0 /k m

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The larger the damping constant b the shorter the damping time . There are 3 damping regimes:

(a) Underdamped

(b) Critically damped

(c) Overdamped


0 0/ 2( ) cos( 1 1/( ) )tx t Ae t

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Example – Bad Shocks

A car’s suspension can be

modeled as a damped

mass-spring system with

m = 1200 kg, k = 58 kN/m

and b = 230 kg/s. How

many oscillations does it

take for the amplitude of

the suspension to drop to half its initial value?http://static.howstuffworks.com/gif/car-suspension-1.gif

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First find out how long

it takes for the amplitude

to drop to half its initial

value:

= 2m/b = 10.43 s

exp(-t/) = ½

→ t = ln 2 = 7.23 s http://static.howstuffworks.com/gif/car-suspension-1.gif

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The period of oscillation isT = 2/

= 2/√(k/m – 1/2)= 0.904 s

Therefore, in 7.23 s, the shocks oscillate 7.23/0.904 ~ 8 times!

These are really bad shocks!

http://static.howstuffworks.com/gif/car-suspension-1.gif

Driven Oscillations

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Driven Oscillations

When an oscillatory system is acted upon by an external force we say that the system is driven.

Consider an external oscillatory force F = F0 cos(d t). Newton’s 2nd law for the system becomes

net

0 d

2

2cos

F

dxkx b F t

dt

a

d x

dt

m

m

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Driven Oscillations

Again, we try a solution of the

form x(t) = A cos(d t). When

this is plugged into the 2nd law,

we find that the amplitude has

the resonance form

2 22 2

0

0

( )( ) /

d

d d

Am b m

F

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Example – Resonance

November 7, 1940 – Tacoma Narrows Bridge Disaster. At about 11:00 am the Tacoma Narrows Bridge, near

Tacoma, Washington

collapsed after hitting

its resonant frequency.

The external driving

force was the wind.

http://www.enm.bris.ac.uk/anm/tacoma/tacnarr.mpg

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Summary

Systems that move in a periodic fashion are said to oscillate. If the restoring force on the system is proportional to the displacement, the motion will be simple harmonic.

The mass-spring system is a simple model that undergoes simple harmonic motion.

If the presence of non-conservative forces the system will undergo damped harmonic motion.

If driven, the system can exhibit resonant motion.

13. oscillatory motion. oscillatory motion 3 if one displaces a system from a position of stable...

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