Oscillators

Background

Most things around you are in stable mechanical equilibrium: push them a bit and then remove your force and they move back towards their original position. All sorts of systems exhibit this behavior: piano strings, the surface of a lake, air in a cylinder, atoms in a solid or molecule... the list is virtually endless.

In addition to a force that works to restore the system to its equilibrium position, most real systems have damping forces that serve to convert the organized motion of large objects into the random thermal motion of the particles of the object or its surroundings. These dissipative forces brake the motion and bring the object to rest, unless power is continually supplied from an outside source. In this experiment you will investigate the oscillations of mass/spring systems, and explore the influence of damping on their behavior.

Theory

For a force to restore an object to its equilibrium position it must push for one sign of displacement and pull for the other. The simplest smooth function that does this is F = -kx, where k is a constant and x is the displacement from the equilibrium position. This equation for a spring was first enunciated by Robert Hooke (1635 - 1703) and is called Hooke's Law; actual springs follow Hooke's law quite well over a range of elongations and/or compressions.

You can work out the theory of both simple and complex oscillating systems by applying Newton's laws to each mass. This will produce one or more equations similar to

        (1)

To solve this kind of equation, in which the acceleration depends not on time but on position, the easiest way is to guess an oscillatory solution of the form

        (2)

(Cultural aside: the fancy German name for this is kind of guess is Ansatz.) In this equation, A and w are constants needed to make the dimensions work out right, and f_o is the initial phase of the motion. You just guess this kind of answer, stuff it into the equation(s) of motion, and see what values of w work.

w is called the angular frequency of the oscillator, and is usually measured in rad/s or just s^-1. When the system evolves in time for an interval equal to the period T, the product w t changes by 2 p. This brings the trig function back to the same value. Therefore, w = 2 p f = 2 p / T.

When the system includes damping (dissipation of mechanical energy into thermal energy), life gets a bit more complicated. If the damping force is proportional to the velocity,

        (3)

then you're lucky, because you can make a slightly different Ansatz and solve the problem with almost no further effort. The equation of motion for the unforced but damped oscillator is now

        (4)

The Ansatz is, perhaps surprisingly,

        (5)

where i is the square root of -1 and A and w are (complex) constants. What we really mean by this shorthand is that x is the real part of the complex quantity, but it is very convenient to do all the algebra before taking the real part.

To see how this can possibly make sense, let's try to solve the simpler undamped oscillator equation (1) with this Ansatz. You can work out a solution to the damped oscillator equation yourself. Taking two derivatives of x with respect to time. Each derivative brings down a factor of iw and leaves A and the exponential untouched. So

        (6)

        (7)

        (8)

a most magical equation due to Abraham de Moivre (1667 - 1754) or Leonhard Euler (1707 - 1783), which you can verify using Taylor series expansions for the exponential and trigonometric functions, when we take the real part of x we get

        (9)

where A_R is the real part of A, and A_I is the imaginary part of A. This form of solution may look a bit different from Eq. (2), but expanding Eq. (2) using

        (10)

you can show that they are the same. Thus, using the complex exponential Ansatz of Eq. (4), we can obtain the same sort of solution as if we assume the more pedestrian trigonometric form of Eq. (2). The great advantage of the complex exponential form is that it allows us to reduce the second-order differential equation (Eq. (1) or (4), for example) to an algebraic equation for the frequency w.

How would equation (4) be modified if the end of the spring were jiggled sinusoidally with an amplitude x_o?

Apparatus

The equipment you have to work with includes a 2.2-m rail, Pasco carts, weights, springs, Photogates, a vibration driver whose frequency you control by varying the DC voltage, a DC (direct current) power supply and voltmeter, and magnetic dampers.

Vibration Driver

The vibration driver is a DC motor that spins at a rate set by the applied DC voltage. To a very good approximation, the rate at which the motor turns (the frequency of the motor) is a linear function of the applied voltage (f = aV + b), where a and b are constants. You can use the photogate timer to determine the period of revolution of the driver, which is the reciprocal of the frequency f (typically measured in Hz = cycles per second).

Do not exceed the maximum voltage and current ratings of the DC motor! They are listed on the motor's casing. Please read the instructions for operating the DC Power supply very carefully.

Magnetic Dampers

The rail on which the carts ride is made of aluminum, which is not a magnetic material. That is, a magnet will not stick to it. However, when a magnet is placed close to the aluminum rail and moved parallel to the rail, a force is created that opposes the motion. The moving magnetic field causes an electric current to flow in the aluminum rail, with the result that charges in the magnet are pushed in a direction opposite to the motion. The strength of this damping force is proportional to the velocity of the moving magnet and is a very strong function of the separation between the magnet and the rail. The magnets are mounted on the end of a micrometer to allow you to vary the strength of this damping force in a smooth and reproducible way.

Please take care of the carts and springs. Do not allow the carts to smash against the ends. If the amplitude of motion is too great, either use a greater damping force or reduce the amplitude of the drive. A good strategy to avoid having to retake data points is to begin on resonance. If the amplitude of motion is large but not dangerously large, you should be fine.

Suggestions

Simple System

To begin your exploration of mass-spring systems, I suggest you start with a single cart and one spring on either side. Use a meterstick and weights to measure the spring constant of the springs, and measure the mass of the cart with an electronic scale. Warning: magnets in the cart can affect the electronic scale. It may be necessary to place the cart on a block of styrofoam to increase the distance between the magnets and the scale. Connect a spring to the oscillating drive using some thread and connect the second spring to the stop at the end of the track. Use enough thread that the springs are stretched about 40 cm each.

Set up the power supply, voltmeter, and photogate so you can measure the period and frequency of the motor. Set the amplitude of the drive at a moderate value (somewhat less than half the maximum should work well).

Set the voltage on the oscillator at about 3.5 V and observe the motion of the cart until it stabilizes. How would you describe the motion?

• How long do you have to wait after changing the frequency before the system settles into a steady-state motion?
• When you change the drive frequency, what aspect(s) of the cart's motion change (besides its frequency)?
• What is the resonant frequency for your mass-spring system? (what frequency leads to the greatest amplitude of motion?)
• How does the phase of the cart's motion compare to the phase of the drive? That is, each executes an oscillation in the same period, but they don't reach their points of maximum extension at the same time. What fraction of a complete rotation separates their points of maximum extension?
• How does the behavior of the cart change when you place a massive bar in the bed of the cart?
• Measure the amplitude of the cart's motion as a function of the frequency of the driving motor. Start at resonance and plot as you go, so you can tell where you need to take more data points. You may find it useful to set the display mode of the amplitude axis to logarithmic (just double-click it).
• How does magnetic damping affect the motion of the cart? Can it be accounted for using the linear damping expression of Eq. (3)?

Follow-up Investigations

A giant crystal	Once you understand the motion of a single cart, try adding one or two additional carts, with springs between each successive cart. Masses connected by springs is a good simple model for the vibrations of a crystalline solid. The electronic bonds between the atoms of the crystal behave like springs to a very fine approximation. Explore the frequency dependence of the motion very carefully. Can you account for the behavior of the carts and springs using the Ansatz of equation (5)? To do this you will have to draw isolation diagrams of each cart and keep careful track of the amount by which each spring is stretched. Watch out for some surprising and wonderful behavior! Can you explain it?
Eddy-current damping	The maximum amplitude of oscillation on resonance is limited by the damping parameter b. For fixed drive amplitude, the resonant amplitude is approximately inversely proportional to b. How does b depend on the distance between the magnet and the rail.
Phase	The phase of the drive with respect to the mass plays a very important role in the motion. Using a protractor to monitor the angular orientation of the drive when the cart passes its equilibrium position, measure the phase of the drive with respect to the cart as a function of frequency. What happens to the phase at resonance?

Updated 10/25/99 by Peter N. Saeta .