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Lesson 1

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Lesson 1 - Oscillations
• Harmonic Motion
Circular Motion
• Simple Harmonic
Oscillators
– Linear Horizontal/Vertical
Mass-Spring Systems
• Energy of Simple
Harmonic Motion
Math Prereqs
d
dпЃ±
d
dпЃ±
sin пЃ± пЂЅ
co s пЃ±
cos пЃ± пЂЅ
пЂ­ sin пЃ±
2пЃ°
2пЃ°
пѓІ cos пЃ± d пЃ± пЂЅ пѓІ sin пЃ± d пЃ± пЂЅ
0
1
2пЃ°
0
0
2пЃ°
пѓІ cos
0
2
пЃ±dпЃ± пЂЅ
1
2пЃ°
2пЃ°
пѓІ sin
0
2
пЃ±dпЃ± пЂЅ
1
2
Identities
sin пЃ± пЂ« cos пЃ± пЂЅ 1
2
2
cos пЂЁ пЃ± п‚± пЃ¦ пЂ© пЂЅ cos пЃ± cos пЃ¦
cos пЃ± пЂ« cos пЃ¦ пЂЅ 2 cos
пЃ±пЂ«пЃ¦
sin пЃ± sin пЃ¦
sin
2
cos пЃ± пЂЅ
2
1
2
e
п‚± iпЃ±
пЂ«
1
cos 2 пЃ±
2
пЂЅ cos пЃ± п‚± i sin пЃ±
пЃ±пЂ­пЃ¦
2
Math Prereqs
пЂЅ " T im e A verag e "
f пЂЁtпЂ© п‚є
1
T
f пЂЁ t пЂ© dt
пѓІ
T
0
Example:
1
1 пѓ©1 1
1
пѓ¶
пѓ¦ 2пЃ° пѓ¶пѓ№
2 пѓ¦ 2пЃ°
2 пѓ¦ 2пЃ° пѓ¶
t пѓ· dt пЂЅ пѓІ пѓЄ пЂ« cos пѓ§ 2
t пѓ· пѓє dt пЂЅ
cos пѓ§
t пѓ· пЂЅ пѓІ cos пѓ§
T 0
T 0 пѓ«2 2
2
пѓЁ T пѓё
пѓЁ T пѓёпѓ»
пѓЁ T пѓё
T
T
Harmonic
Relation to circular motion
x пЂЅ A co s пЂЁ пЃ± пЂ« пЃ¦ пЂ© пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
пЃ·пЂЅ
2пЃ°
T
Horizontal mass-spring
 F  ma
Frictionless
Fs пЂЅ пЂ­ kx
Hooke’s Law:
2
пЂ­ kx пЂЅ m block
2
d x
dt
2
пЂ«
k
m block
d x
dt
2
x пЂЅ0
Solutions to differential equations
• Guess a solution
• Plug the guess into the differential equation
– You will have to take a derivative or two
• Check to see if your solution works.
• Determine if there are any restrictions (required
conditions).
• If the guess works, your guess is a solution, but it
might not be the only one.
• Look at your constants and evaluate them using
initial conditions or boundary conditions.
Our guess
x пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
Definitions
x пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
• Amplitude - (A) Maximum value of the displacement (radius of
circular motion). Determined by initial displacement and velocity.
• Angular Frequency (Velocity) -  Time rate of change
of the phase.
• Period - (T) Time for a particle/system to complete one cycle.
• Frequency - (f) The number of cycles or oscillations completed in
a period of time
• Phase - t   Time varying argument of the trigonometric
function.
• Phase Constant -  Initial value of the phase. Determined by
initial displacement and velocity.
The restriction on the solution
k
пЃ· пЂЅ
2
m block
f пЂЅ
T пЂЅ
пЃ·
2пЃ°
2пЃ°
пЃ·
1
k
2пЃ°
m block
пЂЅ 2пЃ°
m block
пЂЅ
k
The constant – phase angle
x пЂЁt пЂЅ 0пЂ© пЂЅ A
пЃ¦пЂЅ0
v пЂЁt пЂЅ 0пЂ© пЂЅ 0
v пЂЅ пЂ­ A пЃ· sin пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
x пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
a пЂЅ пЂ­ A пЃ· co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
2
x пЂЁt пЂЅ 0пЂ© пЂЅ 0
v пЂЁ t пЂЅ 0пЂ© пЂЅ v0
пЃ¦пЂЅ
пЃ°
2
Energy in the SHO
E пЂЅ
1
mv пЂ«
2
2
vпЂЅп‚±
1
kx пЂЅ
2
2
k
m
1
2
пЂЁA пЂ­ x
2
2
пЂ©
kA
2
Average Energy in the SHO
x пЂЅ A co s пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
U пЂЅ
1
k x
2
пЂЅ
2
dx
dt
K пЂЅ
2
m v
2
kA
2
cos
2
vпЂЅ
1
1
пЂЅ
1
2
пЂЁпЃ·t пЂ« пЃ¦пЂ© пЂЅ
1
kA
2
4
пЂЅ пЂ­ A пЃ· sin пЂЁ пЃ· t пЂ« пЃ¦ пЂ©
mпЃ· A
2
2
2
sin
2
пЂЁпЃ·t пЂ« пЃ¦ пЂ© пЂЅ
K пЂЅ U
1
4
mпЃ· A пЂЅ
2
2
1
4
kA
2
Example
• A mass of 200 grams is connected to a light spring that has
a spring constant (k) of 5.0 N/m and is free to oscillate on a
horizontal, frictionless surface. If the mass is displaced 5.0
cm from the rest position and released from rest find:
• a) the period of its motion,
• b) the maximum speed and
• c) the maximum acceleration of the mass.
• d) the total energy
• e) the average kinetic energy
• f) the average potential energy
Damped Oscillations
“Dashpot”
Fdam ping пЂЅ пЂ­ bv
пЂ­ kx пЂ­ b
dx
пЂЅ ma
dt
2
Equation of Motion
Solution
m
d x
dt
2
x пЂЅ Ae
пЂ«b
dx
пЂ« kx пЂЅ 0
dt
пЂ­ пЃЎt
cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
x пЂЅ Ae
vпЂЅ
dx
пЂ­ пЃЎt
пЂЅ пЂ­ Ae
пЂ­ пЃЎt
пЂЅ пЂ­Ae
пЂ­ пЃЎt
dt
cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« A пЂЁ пЂ­ пЃЎ пЂ© e
aпЂЅ
dt
2
пЂЅ пЂ­ Ae
пЂЅ Ae
2
d x
dt
2
Ae
пЂ­ пЃЎt
пЂ«
пЂ­ пЃЎt
пЂ­ пЃЎt
cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
пѓ©пѓ« пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« пЃЎ cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пѓ№пѓ»
2
d x
пЂ­ пЃЎt
пЃ· п‚ў cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« A пЃЎ e
2
пЃ» 2 пЃЎ пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« пѓ©пѓ« пЃЎ
b dx
k
пЂ«
m dt
2
пЂ­ пЃЎt
пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« A пЃЎ e
пЂ­ пЃЎt
пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« A пЃЎ e
2
пЂ­ пЃЎt
cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
пЃЅ
2
пЂ­ пЃ· п‚ў пѓ№пѓ» cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ©
xпЂЅ0
m
пЃ» 2 пЃЎ пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« пѓ©пѓ« пЃЎ
пЂ­
Ae
пЃЎ пЂЅ
b
t
2m
2
пЃЅ
b
k
2
пЂ­ пЃЎt
пЂ­ пЃЎt
пЂ­ пЃ· п‚ў пѓ№пѓ» cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ­
Ae
A e cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂЅ 0
пѓ©пѓ« пЃ· п‚ў sin пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ« пЃЎ cos пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пѓ№пѓ» пЂ«
m
m
пѓ¬пѓ©
b
пѓ№
пЃ· п‚ў пѓє s in пЂЁ пЃ· п‚ў t пЂ« пЃ¦ пЂ© пЂ«
пѓ­ пѓЄ 2 пЃЎ пЃ·п‚ў пЂ­
m пѓ»
пѓ®пѓ«
b
2m
b
k пѓ№
пѓј
пѓ© 2
п‚ў2
п‚ў
пѓЄ пЃЎ пЂ­ пЃ· пЂ­ m пЃЎ пЂ« m пѓє cos пЂЁ пЃ· t пЂ« пЃ¦ пЂ© пѓЅ пЂЅ 0
пѓ«
пѓ»
пѓѕ
2
пѓ¦ b пѓ¶
2
пЂ­пѓ§
пѓ· пЂ­ пЃ·п‚ў пЂЅ 0
m пѓЁ 2m пѓё
k
пЃ·п‚ў пЂЅ
пѓ¦ b пѓ¶
пЂ­пѓ§
пѓ·
m пѓЁ 2m пѓё
k
2
Damped frequency oscillation
пЃЎ пЂЅ
b
2m
k
пЃ·п‚ў пЂЅ
m
пЂ­
b
2
4m
2
b п‚і 4mk
2
B - Critical damping (=)
C - Over damped (>)
Giancoli 14-55
• A 750 g block oscillates on the end of a spring
whose force constant is k = 56.0 N/m. The mass
moves in a fluid which offers a resistive force F =
-bv where b = 0.162 N-s/m.
– What is the period of the motion? What if there had
been no damping?
– What is the fractional decrease in amplitude per cycle?
– Write the displacement as a function of time if at t = 0,
x = 0; and at t = 1.00 s, x = 0.120 m.
Forced vibrations
Fext пЂЅ F0 cos пЃ· t
пЂ­ kx пЂ­ b
dt
2
m
dx
d x
dt
2
пЂ«b
dx
dt
пЂ« F0 cos пЃ· t пЂЅ m a
пЂ« kx пЂЅ F0 cos пЃ· t
x пЂЅ A 0 sin пЂЁ пЃ· t пЂ« пЃ¦ 0 пЂ©
Resonance
x пЂЅ A 0 sin пЂЁ пЃ· t пЂ« пЃ¦ 0 пЂ©
k
пЃ·0 пЂЅ
Natural frequency
m
F0
A0 пЂЅ
m
пЂЁпЃ·
2
пЂ­пЃ·
2
0
пЂ©
2
b пЃ·
2
пЂ«
m
2
пѓ¦ m пЂЁ пЃ· 2 пЂ­ пЃ· 02 пЂ© пѓ¶
пЂ­1
пѓ·
пЃ¦ 0 пЂЅ tan пѓ§
пѓ§
пѓ·
bпЃ·
пѓЁ
пѓё
2
Quality (Q) value
• Q describes the sharpness of
the resonance peak
• Low damping give a large Q
• High damping gives a small Q
• Q is inversely related to the
fraction width of the resonance
peak at the half max amplitude
point.
Q пЂЅ
mпЃ·0
b
пЃ„пЃ·
пЃ·0
пЂЅ
1
Q
пЃ„пЃ·
Tacoma Narrows Bridge
Tacoma Narrows Bridge (short clip)
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