Vibrations: Simple Harmonic Motion

1
VibrationsSimple Harmonic Motion
Object moving back and forth repeatedly over same
path about an equilibrium position is said to be
vibrating or oscillating Small oscillations of an
object about an equilibrium position very often
can be characterized by specific relations
between its acceleration, velocity, displacement
and energy which describe the oscillation
independently of the precise nature of the motion
(child on swing, electrical currents, sound, etc)
After this lecture you should know about Simple
harmonic motion pendulums and springs Elastic
potential energy Damped and forced oscillations
resonance
2
Example Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed
  position.
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality. (often called spring
  constant or force constant)

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Example Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed
  position.
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality. (often called spring
  constant or force constant)

relaxed position
FX -kx gt 0
x
x lt 0
x0
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Example Springs

• Hookes Law The force exerted by a spring is
  proportional to the distance the spring is
  stretched or compressed from its relaxed
  position.
• FX -k x Where x is the displacement from
  the relaxed position and k is the constant
  of proportionality. (often called spring
  constant or force constant)

relaxed position
FX - kx lt 0
x
x gt 0
x0
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Springs and Simple Harmonic Motion
F-kx Fma ma -kx a -kx/m x -ma/k
Note Acceleration not constant but function of x
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Springs and Simple Harmonic Motion II

• Assume the object is initially pulled to a
  distance A and released from rest
• As the object moves toward the equilibrium
  position, F and a decrease, but v increases
• At x 0, F and a are zero, but v is a maximum
• The objects momentum causes it to overshoot the
  equilibrium position
• The force and acceleration start to increase in
  the opposite direction and velocity decrease
• The motion momentarily comes to a stop at x - A
  
• It then accelerates back toward the equilibrium
  position
• The motion continues indefinitely

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Simple Harmonic Motion

• Motion that occurs when the net force along the
  direction of motion obeys Hookes Law
• The force is proportional to the displacement and
  always directed toward the equilibrium position
• F -kx
• Equations of motion relate amplitude, period and
  frequency in a characteristic way
• Amplitude, A Maximum position relative to the
  equilibrium position
• In the absence of friction, an object in simple
  harmonic motion will oscillate between the
  positions x A

• Period, T Time that it takes to complete one
  complete cycle of motion
• From x A to x - A and back to x A
• Frequency, ƒ Number of complete cycles per unit
  time
• ƒ 1 / T, f 1/s hertz
• Frequency is the reciprocal of the period

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Simple Harmonic Motion and Uniform Circular
Motion
What does moving in a circle have to do with
moving back forth in a straight line ??

• A ball is attached to the rim of a turntable of
  radius A
• The focus is on the shadow that the ball casts on
  the screen
• When the turntable rotates with a constant
  angular speed, the shadow moves in simple
  harmonic motion

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Simple Harmonic Motion and Uniform Circular
Motion
What does moving in a circle have to do with
moving back forth in a straight line ??
x
x
1
1
R
2
2
8
8
f
R
3
3
7
f
y
0
7
4
-R
4
6
6
5
5
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Simple Harmonic Motion and Uniform Circular
Motion
What does moving in a circle have to do with
moving back forth in a straight line ??
x
v -vt sin F
1
2
8
f
f
f
vt
vt
y
y
7
x
4
6
1
a -ar cos F
5
2
8
f
ar
y
f
ar
7
4
6
5
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Simple Harmonic Motion and Uniform Circular
Motion
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Simple Harmonic Motion Equations of Motion
x(t) Acos(?t) v(t) -A?sin(?t) a(t)
-A?2cos(?t)
Maximum value
xmax A vmax A? amax A?2
Angular frequency ? 2pf 2p/T ? rad/s
T
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Reminder The Radian

• Radian Measuring a distance via an angle or vice
  versa
• The radian can be defined as the arc length s
  along a circle divided by the radius r
• Comparing degrees and radians
• Converting from degrees to radians

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Example 1A

• A mass on a spring oscillates back forth with
  simple harmonic motion of amplitude A. A plot of
  displacement (x) versus time (t) is shown below.
  At what points during its oscillation is the
  speed of the block biggest?
• 1. When x A or -A (i.e. maximum displacement)
• 2. When x 0 (i.e. zero displacement)
• 3. The speed of the mass is constant

xmax A vmax ?A
x(t) Acos(?t)v(t) -A?sin(?t)
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Example 1B

• A mass on a spring oscillates back forth with
  simple harmonic motion of amplitude A. A plot of
  displacement (x) versus time (t) is shown below.
  At what points during its oscillation is the
  magnitude of the acceleration of the block
  biggest?
• 1. When x A or -A (i.e. maximum displacement)
• 2. When x 0 (i.e. zero displacement)
• 3. The acceleration of the mass is constant

x(t) Acos(?t)v(t) -A?sin(?t)a(t)
-A?2cos(?t)
xmax Avmax ?Aamax ?2A
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Period and (angular) frequency of a spring
xmax A vmax ?A amax ?2A
a(t) -?2x(t)
F ma -m ?2x F -kx
?2 k/m
Conclusion for mass on spring,
f is called characteristic frequency of the
spring
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Example 2

• If the amplitude of the oscillation (same block
  and same spring) was doubled, how would the
  period of the oscillation change? (The period is
  the time it takes to make one complete
  oscillation)
• 1. The period of the oscillation would double.2.
  The period of the oscillation would be halved3.
  The period of the oscillation would stay the same

x
2A
t
-2A
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Work done when compressing a spring

• Force exerted to compress a spring is
  proportional to the amount of compression or
  extension.

Work of the spring
k - is the force constant (or spring constant)
Units "Nm-1" or "kgs-2"
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Elastic Potential Energy of a Spring
Where x is measured fromthe equilibrium position
US
Analogy marble in a bowl U ? K ? P ? K ?
P Height of bowl proportional to x2
x
0
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Same thing for a vertical spring
y
y0
Where y is measured fromthe equilibrium position
US
y
0
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In either case...
Etotal 1/2 Mv2 1/2 kx2 constant
kinetic energy elastic potential energy
Kmax Mv2max /2 M?2A2 /2 kA2 /2 Umax kA2
/2 Etotal kA2 /2
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Conservation of Energy
Total Energy (1/2) mv2 (kinetic energy) mgh
(gravitational potential energy) (1/2) kx2
(spring potential energy)
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Example 3

• In Case 1 a mass on a spring oscillates back and
  forth. In Case 2, the mass is doubled but the
  spring and the amplitude of the oscillation is
  the same as in Case 1. In which case is the
  maximum kinetic energy of the mass the biggest?
• 1. Case 12. Case 23. Same

KEmax PEmax kA2/2 Etotal Since A and k are
the same the maximum kinetic energy will also be
the same.
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Remark to example 3

• The same would be true for vertical springs...

P 1/2k y2
Y0
P 1/2k y2
Y0
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Small Oscillations of a Pendulum

• The simple pendulum is another example of simple
  harmonic motion
• The force is the component of the weight tangent
  to the path of motion
• Ft - m g sin ?

In general, the motion of a pendulum is not
simple harmonic However, for small angles, it
becomes simple harmonic In general, angles lt 15
are small enough sin ? ? Ft - m g ? This
force obeys Hookes Law
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Small Oscillations of a Pendulum

• For small oscillation, period does not depend
  on
• mass
• amplitude

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Example 4

• A pendulum is hanging vertically from the ceiling
  of an elevator. Initially the elevator is at
  rest and the period of the pendulum is T. Now
  the elevator accelerates upward. The period of
  the pendulum will now be
• 1. greater than T
• 2. equal to T
• 3. less than T

Effective g is larger when accelerating
upward (you feel heavier)
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Example 5

• Suppose a grandfather clock (a simple pendulum)
  runs slow. In order to make it run on time you
  should
• 1. Make the pendulum shorter
• 2. Make the pendulum longer

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Physical Pendulum

• A physical pendulum can be made from an object of
  any shape
• The center of mass oscillates along a circular
  arc
• The period of a physical pendulum is
• I is the objects moment of inertia
• m is the objects mass
• For a simple pendulum, I mL2 and the equation
  becomes that of the simple pendulum as seen before

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Damped Oscillations

• Only ideal systems oscillate indefinitely
• In real systems, friction retards the motion
• Friction reduces the total energy of the system
  and the oscillation is said to be damped
• Example Shock absorber

With a low viscosity fluid, the vibrating motion
is preserved, but the amplitude of vibration
decreases This is known as underdamped
oscillation
Amplitude A(t) e-Gt
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More Types of Damping

• With a higher viscosity, the object returns
  rapidly to equilibrium after it is released and
  does not oscillate
• The system is said to be critically damped
• With an even higher viscosity, the piston returns
  to equilibrium without passing through the
  equilibrium position, but the time required is
  longer
• This is said to be over damped

Plot a under damped Plot b critically
damped Plot c over damped
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Forced oscillations and resonance
A system with a driving force will force a
vibration at its frequency When the frequency of
the driving force equals the natural frequency of
the system, the system is said to be in resonance
Example Pendulum A is set in motion The others
begin to vibrate due to the vibrations in the
flexible beam Pendulum C oscillates at the
greatest amplitude since its length, and
therefore frequency, matches that of A
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Forced oscillations and resonance
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Simple Harmonic MotionSummary
x(t) Acos(?t) v(t) -A?sin(?t) a(t)
-A?2cos(?t)
x(t) Asin(?t) v(t) A?cos(?t) a(t)
-A?2sin(?t)
OR
Period T (seconds per cycle) Frequency f
1/T (cycles per second) Angular frequency ?
2pf 2p/T (radians per second)
xmax A vmax A? amax A?2