Transcript The Simple Harmonic Oscillator - FROM MTSU Capone

Syllabus overview
• No text. Because no one has written one for
the spread of topics that we will cover.
• MATLAB. There will be a hands-on
component where we use MATLAB
programming language to create, analyze,
manipulate sounds and signals. Probably 1
class per week (in computer lab at end of
hall WPS211); typically Fridays.
Some good books
• Fundamentals of Acoustics by Kinsler, Frey,
Coppens, and Sanders (3rd ed.),
• Science of Musical Sounds by Sundberg
• Science of Musical Sounds by Pierce
• Sound System Engineering by Davis & Davis
• Mathematics: A musical Offering by David
Benson. (online version available)
• The Science of Sound by Rossing, Moore, Wheeler
Grading
• Participation is key!
• Attempt all the work that is assigned.
• Ask for help if you have trouble with the
homework.
• If you make a good faith effort, don’t miss
quizzes, hand in all homework on time, etc.
you should end up with an A or a B.
Web page
• Lecture Powerpoints are on the web, as are
homeworks, and (after the due date) the
solutions.
• MATLAB exercises are also on the web page
http://physics.mtsu.edu/~wroberts/Phys3000home.htm
Objectives
• Physical understanding of acoustics effects and
how that can translate to quantitative
measurements and predictions.
• Understanding of digital signals and spectral
analysis allows you to manipulate signals without
understanding the detailed underlying
mathematics. I want you to become comfortable
with a quantitative approach to acoustics.
Areas of emphasis
•
•
•
•
The basics of vibrations and waves
Room and auditorium acoustics
Modeling and simulation of acoustics effects
Digital signal analysis
– Filtering
– Correlation and convolution
– Forensic acoustics examples
The Simple Harmonic Oscillator
… good vibrations…
The Beach Boys
Simple Harmonic Oscillator (SHO)
• SHO is the most simple, and hence the most
fundamental, form of vibrating system.
• SHO is also a great starting point to
understand more complex vibrations and
waves because the math is easy. (Honest!)
• As part of our study of SHOs we will have to
explore a bunch of physics concepts such as:
Force, acceleration, velocity, speed,
amplitude, phase…
Ingredients for SHO
• A mass (that is subject to)
• A linear restoring force
– We have some terms to define and understand
•
•
•
•
Mass
Force
Linear
Restoring
Mass
• Boy, this sounds like the easy one to start
with; but you’ll be amazed at how
confusing it can get!
• Gravitational mass and inertial mass. Say
what!
• What is the difference between mass and
weight?
Force and vectors
•
•
•
•
What does a force do to an object?
Why is the idea of vectors important?
What is a vector?
What is the difference between acceleration,
velocity, and speed?
• Acceleration, velocity, and calculus…aargh
Calculus review?
• What does a derivative mean in mathematical
terms?
• Example:
y  A sin(t )
dy
 A cos(t )
dx
Sin and Cos curves
y
t
Position versus time graph-what
does the slope mean?
Position along x-axis (meters)
10
8
6
4
2
0
-2 0
5
10
-4
-6
-8
Time (seconds)
15
Velocity versus time graph—what does
slope mean?
velocity (meters per second)
6
5
4
3
2
1
0
-1 0
5
10
-2
time (seconds)
15
20
Summarize
• Position (a vector quantity)
• Velocity (slope of position versus time graph)
• Acceleration (slope of velocity versus time
graph). Same as the second derivative of
position versus time.
• Key: If I know the math function that relates
position to time I can find the functions for
velocity and acceleration.
Digital representation of functions
• The math you learn in calculus refers to
continuous variables. When we model,
synthesize, and analyze signals we will be
using a digital representation.
• Example: y=cos(t)
• Decisions: Sampling rate and number of
bits of digitization.
Newton’s Second Law
• Relation between force mass and acceleration
F  ma
Apply Newton’s second law to mass on
a spring
• Linear restoring force—one that gets larger as the
displacement from equilibrium is increased
• For a spring the force is
Fsp  Kx
• K is the spring constant measured in Newtons per
meter.
• x and F are vectors for position and force—the
minus sign is important! Which direction does the
force point?
• Newton's second law
F  ma
• Substitute spring force relation
 K x  ma
• Write acceleration as second derivative of
position versus time
2
d x
 Kx  m 2
dt
Final result
2
d x
K


x
2
dt
m
•Every example of simple harmonic oscillation
can be written in this same basic form.
•This version is for a mass on a spring with K
and m being spring constant and mass.
Solution
• The solution to the SHO equation is always of the
form
x  A sin(wt )
• To show that this function is really a solution
differentiate and substitute into formula.
• Note: A and w are constants; x, t are variables. w is
determined by the physical properties of the
oscillator (e.g. k and m for a spring)
Dust off those old calculus skills
• First differential
dx
 Aw cos( wt )
dt
• Second differential
2
d x
2
  Aw sin(wt )
2
dt
Put it all together
• Substitute parts into the equation
K
 Aw sin(wt )   A sin(wt )
m
2
• Conclusion (after cancellations)
K
w 
m
2
General form of SHO
2
d x
2


w
x
2
dt
Why is this solution useful?
• We can predict the location of the mass at
any time.
x  A sin(wt )
• We can calculate the velocity at any time.
v
dx
 Aw cos( wt )
dt
• We can calculate the acceleration at any
time.
d 2x
a
dt 2
  Aw 2 sin(wt )
Example
• What is the amplitude, A?
• How can we find the angular frequency, w?
• At which point in the oscillation is the
velocity a maximum? What is the value of
this maximum velocity?
• At which point in the oscillation is the
acceleration a maximum? Value of amax?
One other item: phase
• The solution as written is not complete. The
simple sine solution implies that the
oscillator always is at x=0 at t=0. We could
use the solution x=Acos(wt) but that means
that the oscillator is at x=A at t=0. The
general solution has another component –
PHASE ANGLE f
x  A sin(wt  f )
Example
• To find the phase angle look at where the
mass starts out at the beginning of the
oscillation, i.e. at t=0.
• Spring stretched to –A and released.
• Spring stretched to +A and released
• Mass moving fast through x=0 at t=0.
Worked example
• A mass on a spring oscillates 50 times per
second. The amplitude of the oscillation is 1
mm. At the beginning of the motion (t=0)
the mass is at the maximum amplitude
position (+1 mm) (a) What is the angular
frequency of the oscillator? (b) What is the
period of the oscillator? (c) Write the
equation of motion of the oscillator
including the phase.
What is the phase here?
Helmholtz Resonator
• Trapped air acts
as a spring
• Air in the neck
acts as the
mass.
vs
f 
2
A
Vl
(vs is the speed of sound)
Helmholtz resonator II
•
•
•
•
•
Where is the air oscillation the largest?
Why does the sound die away? Damping
Real length l versus effective length l’.
End correction 0.85 x radius of opening.
Example guitar 1.7 x r.
SHO : relation to circular motion
• Picture that makes SHO a little bit clearer.
Complex exponential notation
• Complex exponential notation is the more
common way of writing the solution of
simple harmonic motion or of wave
phenomena.
• Two necessary concepts:
– Series representation of ex, sin(x) and cos(x)
– Square root of -1 = i
Exponential function
• Very common relation in nature
• Number used for natural logarithms
• Defined (for our purposes) by the infinite
series
2
3
4
x
x
x
e  1  x     ...
2! 3! 4!
x
x
e
has a simple derivative
d x
x
e e
dx
d ax
ax
e  ae
dx
Sin and cos can be described by
infinite series
• Sin(x)
3
5
7
x
x
x
sin(x)  x     ...
3! 5! 7!
• Cos(x)
2
4
6
x
x
x
cos(x)  1     ...
2! 4! 6!
Imaginary numbers
• Concept of √-1 = i
• i2 = -1, i3 = -i, i4 = ?
• Not a “real” number—called an imaginary
number.
• Cannot add real and imaginary numbers—
must keep separate. Example 3+4i
• Argand diagram—plot real numbers on the
x-axis and imaginary numbers on the y-axis.
Argand diagram
6
5
4
3
2
1
0
-3
-2
-1
-1 0
-2
-3
-4
1
2
3
4
Two ways of writing complex
numbers
• 3+4i = 5[cos(0.93) + i sin(0.93)]
Can we put sin and cos series
together to get ex series? Not if x is
real. But with i…
2
3
4
x
x
x
e  1  x     ...
2! 3! 4!
x
3
5
7
x
x
x
sin(x)  x     ...
3! 5! 7!
x2 x4 x6
cos(x)  1     ...
2! 4! 6!
ix
e
2
series
3
4
5
(ix) (ix) (ix) (ix)
e  1  ix 



 ...
2!
3!
4!
5!
2
4
3
5
x
x
x
x
ix
e  (1    ...)  i( x    ...)
2! 4!
3! 5!
ix
e  cos(x)  i sin(x)
ix
Complex exponential solution for
simple harmonic oscillator
i (wt f )
y  Ae
 A[cos(wt  f )  i sin(wt  f )]
• Note: We only take the real part of the solution (or
the imaginary part).
• Complex exponential is just a sine or cosine
function in disguise!
• Why use this? Math with exponential functions is
much easier than combining sines and cosines.
Relation to circular motion.
• Simple harmonic motion is equivalent to
circular motion in the Argand plane.
Reality is the projection of this circular
motion onto the real axis.