The work of a curious fellow

Where that which goes out, comes back...

Extending our notion of what makes up the motion of a particle or system of particles, we should now consider that kind of motion where the object dithers about in the neighborhood of a point in the space of our reference frame. Such motion is called "vibration" or "vibratory motion". In fact we have already introduced motion of this sort with the spring and block thing we used to illustrate some of the ideas about work and energy. We will use that same mechanism to begin the study of vibration. In the Simple Harmonic Oscillator display the position and velocity of the block are plotted vs. time with time running down the page.

You may have noticed the label on the previous display was "Simple Harmonic Oscillator". This terminology comes from the form of the position vs. time plot. An oscillator in which the restoring force is proportional to the displacement vibrates with a particularly simple function of position vs. time. The function is x = A*cos(w*t + f) . The sine and cosine functions are called harmonic functions since they may be used to create any waveform, however complicated. The "simple" terminology just means that a single term of the sine or cosine function fully describes the motion. We will abbreviate the simple harmonic oscillator, SHO.

Many real dynamical systems behave very much like the SHO as long as the size of the vibratory motion is small compared to the size of the system. For that reason we will go into some additional detail on the SHO. The behavior of the block fits the basic description of vibration set forth above. It dithers about in the vicinity of the zero reference point. Because sooner or later I will confuse you by beginning to refer to vibrations as oscillations, let's agree at this point that these two terms are interchangeable. With that out of the way we can go on to discuss the other terms related to vibrations.

The maximum displacement from the neutral position in vibration is called the "amplitude" of the vibration. In the SHO modeled here that is .25 meters. The "state" of the system at any instant is defined by the position and velocity of all its moving parts. The state of the block in our example is constantly changing but because the motion is "periodic" the states are repeated in equal intervals of time. The time it takes for the moving block to return to the same state is called the "period" of the oscillation. The number of times the system revisits the same state in a unit of time is called the "frequency" of the oscillations. From these definitions you can see that frequency and period are reciprocals of each other, Period=1/Frequency.

Remember there are two ways of looking at the spring and block. One is based on the Hooke's law nature of the spring such that the force it exerts on the block is proportional to the block's displacement from zero. The other is to ignore the spring and just say that the potential energy of the system, for whatever reason is higher at other locations than it is at zero. It is a fundamental principle in dynamical systems, and all of nature for that matter, that systems tend to seek their least energy configuration. Mother Nature as it turns out is lazy. We will work for a bit with the first point of view but keep the other in mind.

The force from the spring on the block is f=-k*x, where k is the spring constant and x is the displacement. Note that we have dropped the vector notation since we are working here in 1 dimension. Then according to Newton's second law, m*x'' = f = -k*x . Here we have used the x'' notation to represent the acceleration, indicating that acceleration is the rate of change of the rate of change of position with respect to time, in other words, the second derivative of position with respect to time. The mass of the block is represented by m. The equation m*x''=-k*x may be rewritten as x'' = -k/m*x .

simple harmonic oscillator
sinusoidal second derivative

Whatever the function of time which is a solution to the differential equation above, its second derivative must be proportional to the negative of the function itself. If your memory is extremely good and you were paying strict attention, we have already found a function which meets that criteria, way back in the Rates of Change section of the course. Look at the Second Derivative of a Sinusoidal display as a reminder.

Notice that the sine function and its second derivative are negatives of one another. The first derivative of the sine is recognizable as a cosine function. The third derivative of the sine, being the second derivative of the first derivative, would be a negative cosine function meaning that the cosine also fits the requirements for being a solution to the original differential equation of motion. In fact any function of the form x = A*sin(w*t + f) or x = A*cos(w*t + f) , can be shown to be a solution to the equation of motion for the simple harmonic oscillator.

Arrrgh!The equation of motion we developed from Newton's law involves derivatives, in this case the second derivative of position with respect to time. To "solve" that equation means to come up with a function where x is the dependent variable and time is the independent variable, x=f(t), such that when we plug f(t) into the equation of motion in place of x, the equation holds true. The reason for doing this is that we want to predict the future and if we get x as a function of time we can do that just by plugging a future time into the function an calculating a future x.

The branch of mathematics that solves equations like our equation of motion is called "differential equations". Do not let the name turn you off. In fact there is only one way to solve differential equations. You guess at a solution and check to see if it works. There are some fancy guessing techniques (see the image below) that we do not need to study right now. We can guess just based on the curve of position vs. time displayed by the model.

Let's take the function x = A*cos(w*t + f) as a tentative solution to the equation of motion, x'' = -k/m*x . We could show by repeating the development which led up to the Second Derivative of the Sinusoidal display, that the first derivative of A*cos(w*t + f) is -w*A*sin(w*t + f) and the second derivative of A*cos(w*t + f) is -w2*A*cos(w*t + f). Substituting for x, the expression A*cos(w*t + f) and for x'', -w2*A*cos(w*t + f) we get -w2*A*cos(w*t + f) = -k/m * A*cos(w*t + f) .

If we choose the constant w such that w2=k/m then the equation of motion is satisfied, indicating that our tentative solution is in fact a solution as long as we choose w = (k/m)0.5.

The constants A and f remained undetermined in our analysis of A*cos(w*t + f) as a possible solution to the equation of motion. This means that any choice whatsoever for A and f still satisfy the equation so that a large variety of motions is possible for the oscillator. The parameter w is common to all allowed motions but A and f may differ among them. the parameters A and f we will see are determined by how the motion is started, the so called initial conditions.

devilish hardNow we are going to reason our way through the meaning of the parameters A, w and f. This is something that gets easier with experience. You have to have some intuition about the effect of parameters on functions, which only comes with looking at lots of functions.

Since the range of the cosine function is -1 to 1, it is clear that the symbol A is the amplitude of the SHO motion, the peak value that x may have either positive or negative is A. The parameter w is called the angular frequency. If the time t in x = A*cos(w*t + f) is increased by 2*p / w, the function becomes x = A*cos(w*(t + 2*p / w) + f) . This expands to x = A*cos(w*t + 2*p + f) . Adding 2*p to an angle just brings it back to its starting point so 2*p / w must be the period of the motion. T = 2*p / w , where T is the period. But since w2=k/m we have w = (k/m)0.5, so T = 2*p * (m/k)0.5 . The oscillatory period T is determined only by the mass and the force constant k. It is independent of amplitude.

The frequency, f, is the number of complete oscillations per unit time, f = 1/T = w / 2*p = 1/(2* p) * (k/m)0.5 . From this we see that f = w / 2*p or w = 2*p * f . The angular frequency w then differs from the frequency f by a factor of 2*p. It has dimensions of reciprocal time, the same as angular velocity, radians/second.

Use the SHO display to calculate the ratio of m/k used in that model. Determine the period from the plot. Then use T = 2*p * (m/k)0.5 to get the required ratio. You should find that m/k is about .5, remembering that picking of the period using the cursor is an approximation.

What about this constant, f, we have been carrying along through all this discussion? That quantity is called the phase angle of the motion. As you can see from its position in the function x = A*cos(w*t + f) , it plays the role of offsetting time from zero. If for example f = -p/2 then the function x = A*cos(w*t + f) = A*cos(w*t - p/2) = A*sin(w*t) so that displacement x is zero at t=0. If f=0 then x is A at t=0. Other initial conditions correspond to other phase angles. The amplitude A and the phase angle f are determined by the initial position and velocity of the particle.

guessing differential equation solutions
potential energy oscillator

Before exploring the effect of varying initial position and velocity, we should generalize our SHO model. Strictly speaking, the spring is not a necessary part of our system. In the SHO case the potential energy as a function of x was 1/2*k*x2 as we discovered in the section on Potential Energy and Fields , since that was the work done on the spring. Review the Work on Spring display to see the shape of the pe(x) curve. Let the display run long enough to fill in the curve.

Now if any mechanism, be it spring, electricity or whatever, creates a condition in space such that the potential energy as a function of displacement is proportional to the square of the displacement, an object placed in that space will behave like an SHO. To illustrate that concept run the Potential Energy SHO display. Notice that as you start the object in different positions with different velocities that the amplitude of the motion and the time when the object passes through the zero position take on different values.

To farther clarify the relationship between initial conditions and the parameters A and f we will dispense with the physical representation of the object as we did in the free body diagram display, and just show the position of the object as a function of time. Run the Initial Conditions display.

Next we will explore the relation between simple harmonic motion (SHM) and uniform circular motion. Consider a circle in the (x,y) plane, centered at the origin, of radius 5 meters with a point on the circle moving around at constant angular speed, w, expressed in radians per second. Next suppose that the initial position of the point on the circle was at angle f from the x-axis. The projection of that point on the x-axis at any time t would be at x = 5*cos(w*t + f) . This expression for x is exactly that we derived for the motion of the SHO. Run the Circular to SHM display to see these relationships.

From this we see that the parameters w and f have a geometric interpretation. The angular frequency w is the angular velocity of the circular motion related to the linear simple harmonic motion. The phase angle f is the starting angle of the related circular motion. We might also note that the amplitude A of the simple harmonic motion is the radius of the related circular motion.

If we had taken the projection of the circulating point onto the Y-axis instead of the x, we would have found the motion to be x = 5*sin(w*t + f) . This is another example of simple harmonic motion that differs only in phase angle from the x motion. If we replace f by f-p/2, cos(w*t + f-p/2) may be replaced with sin(w*t + f) since sin and cosine functions are identical except for a p/2 phase difference.

circular motion to SHM
SHM to circular motion

Now if we turn our thinking inside out, we can synthesize uniform circular motion from the combination of two simple harmonic motions. Two simple harmonic motions along perpendicular lines, of equal amplitude and angular frequency, which differ in phase by p/2 radians combine to produce the circular motion we described. Run the SHM to Circular Motion display for an illustration. In this display we see a green oscillator and a red oscillator making a cyan circle.

In many dynamical systems, motion that appears to be quite complex can be understood as a combination of simple harmonic motions. In the case where the frequencies are the same we have x = Ax*sin(w*t + fx) and y = Ay*sin(w*t + fy) . Suppose the phase angles fx and fy were also equal. Then dividing the y equation by the x equation we get, y = (Ay / Ax) * x , which is the equation of a straight line passing through the origin with slope Ay / Ax.

If the phase angles are different the resulting motion will be elliptical. In the special case where the amplitudes are equal and the phase angles differ by p/2, the motion is circular, as we have seen. Run the Combined SHM display to experiment with various combinations of parameters.

All of this emphasis on simple harmonic motion is appropriate under the heading of vibration. Many of the vibrations encountered in dynamical systems can be very closely approximated by combinations of SHM. Next we will look at a specific vibrating dynamical system. We will study the vibration of atoms in a solid.

In the section of the course on translation when we were developing the notion of center of mass, we talked about the constraint that in a solid the distance between atoms is fixed. It is true that the average distance between atoms is fixed but the atoms do exhibit some vibratory motion about this fixed position. We know from what we have learned here that in vibration, the object stays in the vicinity of a point and that point for an atom is the position where the potential energy is a minimum. Remember that potential energy is the energy it takes to change a system's configuration.

combined SHM
atomic solid

Imagine a solid object made up of atoms. These atoms have parts that have an electrical charge on them. It is the nature of electric charges to effect the space in which they are located so that other charged particles in that space experience a change in potential energy with a change in position. Therefore, work is involved in moving charged particles around in the space. We may think of the space as having a potential energy field in it. If we pick an atom in the solid to examine, we find that the space in which it exists has a potential energy function pe(r) where the potential energy is a function of the atom's position r. All the other atoms in the vicinity create that condition of space.

It turns out that the net effect of all this electric stuff is very similar to having each atom connected to all its immediate neighbors by a little spring. Any displacement of an atom from its equilibrium position would compress some springs and stretch others, building up potential energy. Then as we saw in the discussion on Potential Energy and Fields , the potential energy gets converted to kinetic energy, which goes, back to potential and so on and so forth. To fully model this behavior in three dimensions for a significant number of atoms would bring my computer (and probably yours) to its knees. To give you some sense for how this works, I will put together a nine-atom model in only one dimension. Run the Atomic Solid display to see this in action.

The force between any pair of atoms is k*dx where k is an effective spring constant and dx is the change in distance to its nearest neighbor. We ignore any forces due to displacement relative to more distant neighbors. From Newton's third law we know that this force will be negative on the one atom and positive on the other one. The forces are transmitted from atom to atom through the mutual effect of the disturbance in the potential energy field caused when an atom is out of its equilibrium position.

Actually the restriction that we work in one dimension is not as severe as you might think. In a solid, the arrangement of atoms can be looked at as a series of strings. If we grab an atom and shake it, the motion of that atom may be resolved into components along the directions of all the strings of which it is a member. So the model we show is fairly accurate for one of the components of any atom's motion.

Notice that energy was imparted to the system but the system as a whole did not move except by the amount that the center of mass was affected by the initial displacement of the left atom. If we were looking at the lump of stuff composed of these nine atoms as a particle in a larger system, energy in that system would appear to not be conserved. Basically energy may seem to be lost from systems made up of bulk matter because some of it is transferred into vibratory energy of the matter itself. The amount of energy stored in the vibrations of the atoms making up an object is called heat energy, and the average speed of the vibrating atoms is proportional to the temperature of the object.

You also might note that if you keep your eye on the rightmost atom when you initially click the Action button, you can see that it takes some time for the disturbance to be felt at the far end of piece of matter. The velocity with which the disturbance propagates down the string of atoms is the velocity of sound in the material. It is evidently dependent on the mass of the atoms and the stiffness of the bond between them since those are the only two parameters that have been set in this model. By halting the action with the Cut button, you freeze the positions and velocities at that instant. Then clicking Action again adds an additional shot of energy. That way you can build up a lot of agitation in the atoms. The Reset button restores the atoms to their initial position and zero velocity.

Next we will extend the idea of vibration to include the motion of planets in their orbits about the sun. To do that we need to examine the idea of gravity which we will handle in the next section.

Are there any questions?

linear arrays in crystal
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