PH 1140: OSCILLATIONS AND WAVES

Study Guide #2: Describing and Adding Vibrations

Introduction:

French begins the discussion of simple harmonic motion in Chapter 1, pages 4-7.
The motion of a particle is described by the Physical Standard Form: x(t) = Acos(wt + f).
However, French uses the sine function, but you have to learn to cope with different notations.
The coefficient A tells you how large how far from equilibrium the maximum displacement is and is called the amplitude.
The constant w, called the angular frequency, determines how often the particle moves back and forth per second.
This can be seen as follows:
The time T is the time for one complete cycle, i.e. the time needed for the argument of the cosine function to increase by 2p and is called the period.
This clearly requires that wT = 2p, or T = 2 p/w.
This in turn gives you the frequency f, which is the number of oscillations per second (or Hertz, Hz):
f = 1/T = w/2p.

Finding the meaning of the phase shift (sometimes called the initial phase angle or phase displacement) f is a bit tricky.
It determines just when the displacement takes its maximum value, or its minimum value, or just when the displacement is zero, etc.
For instance, the maxima occur when the argument of the cosine is a integral multiple of 2p:

wt + f = n·2 p

Solving for t we find

t = n[(2p/w) - (f/w) = [n - (f/2p)]T

This shows that the time interval between successive maxima is T, as expected, and that the precise timing of the maxima depends on the phase angle f.
Note that if f = 0 then a f = 2p lead to exactly the same times.

Do enough exercises to become thoroughly familiar with the relations discussed in the preceding two paragraphs.

Vibrations are so common in science and engineering that they seldom come alone.
Interference and diffraction in acoustics and optics describe the effect of several vibrations arriving at a detector at the same time
(based on the Principle of Superposition)
The sounds produced by musical instruments can be described as sums of sinusoidal vibrations, and so can the motion of electrons in an antenna.
Similar sums occur in quantum mechanics and in the design of steel structures.
Summing vibrations is a technique that all of you will have to master.

We look in particular at three different cases:

• Two or more vibrations along the same line and with the same frequency,
• Two vibrations along the same line with slightly different frequencies, and
• Two vibrations with the same frequency, along two perpendicular lines.

In each case, the phase angle of the vibrations will turn out to be critically important.

These cases are all treated in French, Chapter 2:

Case 1 is solved by means of phasors: vectors in the complex plane.
Each vibration is represented by a phasor; the vector sum of these phasors is the phasor for the sum vibration.

Case 2 leads to beats:
Two vibrations with slightly different frequencies add up to a single vibration with an amplitude that varies at the difference frequency.
Come to class; I'll show you a treatment that is much easier than the one French uses.

Case 3 leads to interesting geometrical patterns.
If the x AND y-coordinates of a particle are sinusoidal functions of time and the frequencies are the same, the particle describes an elliptical orbit.
For a particular set of amplitudes and phases, this can degenerate into a straight line or a circle.
When the frequencies are not the same, the orbit can take all sorts of interesting shapes, called Lissajous figures.

Objectives:

After studying this material we expect you to be able to

• Explain the meaning of amplitude, phase angle, angular frequency, frequency, and period, and be able to use the relations among the last three quantities.
• Represent vibrations by phasors in the complex plane.
• Use phasors and trigonometry to calculate the amplitude and the phase of the sum of two vibrations along the same line and with the same frequency.
• Generalize this summation to more than two vibrations.
• Find the beat frequency and the maximum and minimum amplitude of the sum of two vibrations with slightly differing frequencies.
• Plot the path traversed by a point carrying out two vibrations along perpendicular lines when the frequencies are the same, although the amplitude and phase may differ.

Practice Problem Set #2:

Once you have figured out how to solve a problem, write a fair copy of the solution in your problem notebook (see Hints).
Remember to write your solutions in English; mere list of numbers and formulas become unreadable after a very short time, even a few days.
Additional problems may be found in French.

1. The displacement of an object suspended from a spring is given by:
   
   x(t) = 0.20·cos(8·t - 0.3)
   
   
   1. Plot x as a function of t for the first second. (Remember to use radians!
   2. What is the amplitude of this vibration?
   3. What is the absolute (total) phase angle at time 0, 0.1, and 0.2 sec?
   4. What is the angular frequency?
   5. What is the frequency?
   6. When does x take its maximum value for the first time?
   7. A second oscillator carries out exactly the same motion, but everything happens 0.1 sec later.
      How should the equation above be changed to describe the motion of the second oscillator?
   8. Use the modified equation to find the time at which the second oscillator reaches its maximum displacement for the first time.
   9. Compare the result of part (h) with the result of part (f). Are they related as you would expect?
   
   
2. The displacement of a small mass on a stiff spring is given, in millimeters, by:
   
   x(t) = 0.20·cos(600·t + f)
   
   Determine the phase angle f for the following cases, and sketch for each case the phasor:
   
   
   1. The displacement has its maximum value at t = 0.
   2. The displacement has its maximum value at t = 0.001 sec.
   3. The displacement has its maximum value at t = 0.002 sec.
   4. The displacement has its maximum value at t = 0.003 sec.
   5. The displacement is zero at time t = 0, and the speed is in the direction of positive x.
   6. The displacement is zero at time t = 0, and the speed is in the direction of negative x.
   
   
3. Write down expressions for the velocity and the acceleration as a function of time for the vibration given by:
   
   x(t) = 0.25·cos(200·t - 0.3)
   
   Then
   
   
   1. Make properly labeled plots of x(t), v(t), and a(t).
   2. If x is expressed in millimeters, how does the peak value of the acceleration compare with the acceleration of gravity?
   3. How would you represent x(t), v(t), and a(t) in complex notation?
   4. How can you picture these quantities in the complex plane, i.e. as phasors?
   5. Draw, by hand, a graph of 3cos(wt) and a graph of 4sin(wt) for one period, starting at t = 0.
   6. Sketch the sum of these two graphs as accurately as you can.
   7. Sketch the two phasors that represent these vibrations.
   8. Calculate the amplitude and phase of the sum, and show that this result corresponds to that obtained graphically.
      
   
   
4. Determine the amplitude and phase of the following vibrations:
   
   
   1. 5cos(wt + p/4) + 5cos(wt -p/4),
   2. 4cos(wt) + 3sin(wt),
   3. 5cos(wt + p/2) + 5cos(wt -p/2),
   4. 5cos(wt) + 4cos(wt- p).
   5. 15cos(wt + p/6) + 8cos(wt + 2p/3).
   
   Draw for each of these cases a phasor diagram before you do any calculations.
   After you have completed the calculations, substitute a few values for wt to see if your answers are correct.
   In particular make sure that you get the signs right in case (b).
   
   
5. Make up more problems in the style of the previous one, till you feel comfortable doing these calculations.
   Always verify your results with numerical checks.
   
   
6. A particle is subjected to three vibrations , all at the same frequency, and all along the x-axis.
   The first vibration has an amplitude of 2 cm.
   The second vibration has an amplitude of 4 cm and a phase retardation of 60 degrees relative to the first.
   The third vibration has an amplitude of 3 cm and a phase retardation of 90 degrees relative to the second.
   Find the total amplitude, and the retardation of the sum relative to the first vibration.
   
   
7. A particle is simultaneously subjected to five harmonic vibrations along the x-axis, all with the same frequency and amplitude.
   The phases, relative to the first vibration, are 60, 120, 180, and 240 degrees.
   Find the sum of these five vibrations graphically, using phasors.
   
   
8. Two vibrations along the same line are described by the equations:
   
   x₁(t) = 3.5·cos(4.0p·t)
   x₂(t) = 3.0·cos(4.5p·t)
   
   
   1. Find the frequencies of these two vibrations, and find the beat frequency of the sum vibration.
   2. Find the maximum and minimum value of the amplitude of the sum vibration.
   3. Check your work by plotting the sum vibration using Maple, Mathcad, or a similar program.
   4. Change the phase of the second vibration by 180 degrees, and repeat parts (a), (b), and (c).
   5. Explain why the maxima and the minima have interchanged when they occur.
   
   
9. A particle moves in the (x, y) plane as follows:
   
   x(t) = 3 cos(wt)
   y(t) = 4 cos(wt - f)
   
   Sketch the orbit of the particle for f = 0, 45 degrees, +90 degrees, -90 degrees, and 180 degrees.
   Is there any difference between the +90 degree and -90 degree case?
   
   

Homework Set #2:

Give terse but readable accounts of the methods you use to solve the problems.

1. This problem concerns the sum of the three vibrations:
   
   10.5·cos(wt + p/2)
   
   2.66·sin(wt)
   
   1.55·cos(wt + 3p/4)
   
   
   1. Determine the sum first graphically, using phasors.
   2. Use the graphical representation to write the sum in the form Acos( wt + f).
   3. Substitute wt = 1 and wt = 10 to verify that your answer gives the correct result.
   
   
2. Draw a phasor diagram that represents the following sum:
   
                5
       å cos(wt + m·2p/5)
             m = 1
   
   Determine the value of the sum, and substitute wt = 0.25 to verify that your result is correct.
   
   
3. A vibration 0.03·cos(2755·t) is added to the vibration 0.05·cos(2765·t).
   Describe in a few words the resulting beat phenomenon.
   
   
4. A small object vibrates in two perpendicular directions at the same time.
   The amplitudes and frequencies of these two perpendicular vibrations are the same, but they have a phase difference of 45 degrees.
   Sketch the path traced out by the object.
   
   

FOOTNOTES:

Last Modified: 27 October 2001, GSI