View
230
Download
0
Category
Preview:
Citation preview
13. Oscillatory Motion
Oscillatory Motion
3
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.
4
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
5
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
6
Simple Harmonic Motion
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
7
Simple Harmonic Motion
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
8
Simple Harmonic Motion
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
9
Simple Harmonic Motion
.
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
10
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
11
The equation
can be solved if we set
T = 2, that is, if we set
Simple Harmonic MotionFrequency and Period
.
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
12
For simple harmonic motion
of the mass-spring system,
we can write
Simple Harmonic MotionFrequency and Period
.
12
mT
f k
13
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
14
Simple Harmonic MotionPosition, Velocity, Acceleration
cos( )Ax t
sin( )Av t
2 cos( )Aa t
Position
Velocity
Acceleration
15
cos( )Ax t
sin( )Av t
2 cos( )Aa t
Simple Harmonic MotionPosition, Velocity, Acceleration
Applications of SHM
17
At equilibriumupward force of spring = weight of block
Gravity changes onlythe equilibrium position
Vertical Mass-Spring System
18
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
19
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?
20
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
21
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
23
Energy in Simple Harmonic Motion
cos( )Ax t
sin( )Av t
2 cos( )Aa t
Position
Velocity
Acceleration
24
Energy in Simple Harmonic Motion
Kinetic Energy
2
2
22
1
21
sin ( )2
A
K mv
m t
25
Energy in Simple Harmonic Motion
Potential Energy
2
2 2
1
21
cos ( )2
A
U x
k t
k
26
Energy in Simple Harmonic Motion
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:
27
Energy in Simple Harmonic Motion
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
29
Damped Harmonic Motion
Non-conservative forces, such as friction, cause the amplitude of oscillation to decrease.
cos( )Ax t
30
Damped Harmonic Motion
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
31
The solution of the differential equation
is of the form
For simplicity, we take x = A at t = 0, then = 0.
Damped Harmonic Motion
2
2
dxkx b
dt dm
d x
t
/( ) cos( )tx t Ae t
32
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
Damped Harmonic Motion
/( ) cos( )tx t Ae t
02
0
2 /
1 1/( )
m b
0 /k m
33
The larger the damping constant b the shorter the damping time . There are 3 damping regimes:
(a) Underdamped
(b) Critically damped
(c) Overdamped
Damped Harmonic Motion
0 0/ 2( ) cos( 1 1/( ) )tx t Ae t
34
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
35
Example – Bad Shocks
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
36
Example – Bad Shocks
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
38
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
39
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
40
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
41
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.
Recommended