Student of the Week. Introductory Video: Simple Harmonic Motion Simple Harmonic Motion.
Lesson 1 - Oscillations Harmonic Motion Circular Motion Simple Harmonic Oscillators –Linear -...
-
Upload
rosaline-green -
Category
Documents
-
view
222 -
download
0
Transcript of Lesson 1 - Oscillations Harmonic Motion Circular Motion Simple Harmonic Oscillators –Linear -...
Lesson 1 - Oscillations
• Harmonic Motion Circular Motion
• Simple Harmonic Oscillators– Linear -
Horizontal/Vertical Mass-Spring Systems
• Energy of Simple Harmonic Motion
Math Prereqs
dcos
d
dsin
d
cos
sin
2 2
0 0
cos d sin d
0
2 22 2
0 0
1 1cos d sin d
2 2
1
2
Identities
cos cos 2cos sin2 2
2 2sin cos 1
cos cos cos sin sin
2 1 1cos cos 2
2 2
ie cos i sin
Math Prereqs
T
0
1f t f t dt
T
"Time Average"
2 2cos t
T
T T2
0 0
1 2 1 1 1 2 1cos t dt cos 2 t dt
T T T 2 2 T 2
Example:
Harmonic
Relation to circular motion
x A cos A cos t
2
T
Horizontal mass-spring
F ma
Hooke’s Law: sF kx
2
block 2
d xkx m
dt
2
2block
d x kx 0
dt m
Frictionless
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 A cos t
Definitions
• 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.
x A cos t
The restriction on the solution
2
block
k
m
block
1 kf
2 2 m
blockm2T 2
k
The constant – phase angle x t 0 A v t 0 0 0
x t 0 0 0v t 0 v 2
x A cos t v A sin t
2a A cos t
Energy in the SHO
2 2 21 1 1E mv kx kA
2 2 2
2 2kv A x
m
Average Energy in the SHO
2 2 2 21 1 1U k x kA cos t kA
2 2 4
2 2 2 2 2 2 21 1 1 1K m v m A sin t m A kA
2 2 4 4
x A cos t
dxv A sin t
dt
K U
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
“Dashpot”
dampingF bv
dxkx b ma
dt
2
2
d x dxm b kx 0
dt dt
tx Ae cos t
Equation of Motion
Solution
Damped Oscillations
tx Ae cos t
t tdxv Ae sin t A e cos t
dt
2
t 2 t t 2 t2
d xa Ae cos t A e sin t A e sin t A e cos t
dt
t 2 2Ae 2 sin t cos t
tAe sin t cos t
2
2
d x b dx kx 0
dt m dt m
t t2 2 tb kAe Ae Ae 0
m mcos t cos t cos2 sin tt sin t
t 2 2
b
2mb k
cAe 0b
2 s oin tm
s tm m
22k b
0m 2m
2k b
m 2m
b
2m
Damped frequency oscillation
2
2
k b
m 4m
2b 4mk
B - Critical damping (=)C - Over damped (>)
b
2m
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
ext 0F F cos t 0
dxkx b F cos t ma
dt
2
02
d x dxm b kx F cos t
dt dt
0 0x A sin t
Resonance
0
k
m Natural frequency
0 0x A sin t
0
0 2 222 2
0 2
FA
bm
m
2 201
0
mtan
b
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.
0mQ
b
0
1
Q
Tacoma Narrows Bridge
Tacoma Narrows Bridge (short clip)