Driven simple harmonic oscillator with damping
(Mercer University Physics)

Objectives

In the several parts of this experiment, you will perform both qualitative and quantitative exercises involving a mechanical oscillator.

Equipment List

(i) Multi-purpose undergraduate laboratory (MUL) apparatus, with a (ii) symmetric differential capacitive (SDC) electronics control unit, a (iii) frequency and amplitude adjustable electronic oscillator, along with a (iv) linear amplifier to drive (i), and (v) a Pasco interface to monitor motion of the MUL on the computer.

Description of the MUL:

The mechanical oscillator (MUL) used in this experiment, which is conventionally referred to as a `balance' (and which could be
called a physical pendulum) is illustrated in Fig. 1
. sho_app.jpg
An SDC sensor, S, is used to sense balance position. Offset of friction which otherwise dissipates the motion is provided by passing an electric current through a wire, part of which is positioned in a field that is provided by the horse-shoe drive magnet. The vertical position of this wire, relative to the magnet pole faces, is determined by the horizontal placement of the mass, W.

Controlled damping of the balance is provided by eddy currents that are induced as the small rare earth magnet, M, moves in the presence of the metal ring, R. Without the ring in place under the magnet, damping is small, as shown in Fig. 2.

freed_0.gif
freed_al.gif
In Fig. 3, the much greater damping as compared to Fig. 2, derived from the use of an aluminum ring.

Theory

Equation of Motion:

The equation of motion of the damped, driven simple harmonic oscillator is:
d2x/dt2 + 2bdx/dt + w02 x  =  ad(t)
 		 (1)
where w0  =  2p/T0, with T0 being the period if there were no damping. With damping, the angular frequency of the balance is given by
w  =  2p/T  =  (w02 - b2)1/2
 		 (2)
For the present experiment, x may be considered as a point on some part of the wire which is positioned between the poles of the horseshoe magnet. Although the motion of such a point is really circular, for small amplitudes it is approximately linear. The constant, b, is determined by the amount of damping. It is governed by Faraday's and Lenz's Law, which explain the eddy currents that flow because of relative motion between M and R of Fig. 1.

The right hand side of Eq. (1) is the amount of acceleration provided by the external driving force. It is proportional to the current supplied to the wire. When the system is monitored in free decay (zero forcing), this term is zero. Otherwise, this right hand side is established in accord with the external force. In the present experiment, this derives from the Lorenz force which is present when a current flows in a wire placed in a magnetic field.

Free-decay Solution:

For free decay, the solution to Eq. (1) in the ideal case is
x(t)  =  A e-btcos(wt + f)
 		 (3)
where the phase factor, f, depends on the zero (start) of the plot of position versus time, and the initial amplitude, A is determined by the `vigor' with which the oscillation was initiated.

For a viscous damped system, the period of the balance (reciprocal of the frequency) depends on the magnitude of the dissipation as seen from Eq. (2). The magnitude of the damping constant (which determines the amount of dissipation) is quantified by the logarithmic decrement (hereafter referred to as the log-decrement).

Procedure, General

Free decay:

In free decay measurements, the balance is initially disturbed by an external, short-lived force; and then monitored as the motion decays. In practice, the balance usually does not execute motion around zero, but rather around an equilibrium point, xe; which is a negative or positive number that must be added to x in Eq. (3) above.

In the present experiment, we will not bother to calibrate the sensor so as to work with x directly. Rather, we will work with the output voltage from the sensor. This is possible because the SDC sensor is essentially linear; i.e., its output voltage, V is proportional to x. (Note: In assuming the replacement, x ¨ V, it is important that no gain changes be made in the SDC electronics during the course of the experiment.)

The decay constant, b, in Eq. (3) is the time required for the amplitude to decay from the starting value A to A/e  =  0.3679 A. It can be estimated from the free decay record as follows. Let xn, n  =  1,  2, ... be the turning points of the motion (x1  =  A + xe for f  =  0). It can be shown that the log-decrement of the motion, (bT), is expressible in terms of any three adjacent turning points as follows:

bT  =  - 2 ln[(xn+1-xn+2)/(xn+1-xn)]
 		 (4)

Driven system:

When an external periodic force is applied to the balance, its response is determined by (i) the initial state of the balance, (ii) the amplitude of the external drive, and (iii) the frequency of the drive. The steady state response results once the balance has entrained with the drive; i.e., after transients have decayed. During the time of transients, beating occurs if the frequency of the drive is different from the natural frequency of the balance. Once entrained, the frequency of the balance becomes the same as that of the drive.

Resonance Response:

You will measure the steady state response as a function of drive frequency of a sinusoidal source, while holding the amplitude of the source constant. The well-known curve which results is known by the name, Lorentzian; for which a data set collected with
the MUL is shown in Fig. 4.
res.jpg

In general, the Lorentzian is of the form

A  =  A0 [(w2-w02)2+4b2w2]-1/2
 		 (5)
In Fig. 4, the values for A0, f0 = w0/(2p), and b were set at 0.70, 1.13 Hz, and 0.25 respectively. To estimate b, one measures four quantities: T0, xn, xn+1, and xn+2 (actually V¢s rather than x¢s), working with a conveniently small value of n, typically n  =  1. (Note: one must be careful to include the sign on any negative values of xn.)

The period with damping, T is most readily obtained from T0, since it is difficult to accurately measure T directly when b is large. From Eq. (2), the period with damping is found to be

T  =  T0 [1 + (bT)2/(4p2)]1/2,
 		 (6)
from which b may be found, using Eq. (4). For example, in Fig. 2, the first three turning points are 0.35, -.41, and 0.33 respectively; and the period is 0.96 s. Thus bT  =  0.053 and b  =  0.055 s-1. In like manner, the decay constant for the case with the aluminum ring (Fig. 2) is 0.93 s-1.

When determining T0, one should let the balance execute at least ten cycles of the motion (in turn dividing the total time interval by 10), to reduce the influence of random errors in the measurement.

Procedure, Specific

Software setup

  1. Select `other options' in the software menu.
  2. Select channels `A' on, `B' and `C' off.
  3. Select `change sample time' and set at `5'
  4. Select `main menu'
  5. Select `oscilloscope mode'
  6. With the balance configured so that the wire through the horse-shoe magnet is approximately centered in the pole gap (by moving W and the horse-shoe magnet to the proper positions)-(i) set the SDC electronics control unit gain at maximum, and (ii) zero the output (trace at center of monitor) by rotating the coarse and/or fine adjust knobs.
  7. Select `real time graph' mode, with options: (i) line connecting points `on', all else off, (ii) min. V = -0.5, max V = 0.5, (iii) store in memory `yes', and (iv) duration 0 - 10 s.

Recording Free Decay records

To generate a free decay record, begin by laying a piece of paper on the balance in the vicinity of the horse-shoe magnet, so that the wire through the magnet `bottoms out' in contact with the magnet.
  1. First do a decay without a dissipation ring (minimum log-decrement).
  2. For the `graph real time' mode that has been configured as above, hit `enter'; and as the trace begins to form on the monitor screen, lift the paper from the balance, to initiate oscillation.
  3. When the trace has finished (audible alert indicating the end),
  4. Select `return to main menu'.
  5. Select `Plot graph', with `line on', all else off, auto-scaling.
  6. Select `examine data'.
  7. Using the right/left arrow keys, move the cursor from a turning point (maximum or minimum) to the next adjacent one . Repeat for the third turning point. At each of the three turning points, record the voltage that is displayed at the bottom of the monitor. To obtain the period, record also the times of these turning points. Collect three separate sets of data in this manner and average your results, noting the variance between cases as an estimate of errors. Additionally, with the right arrow key you can accurately note the time required to complete 10 cycles of the motion; from which the period, T is conveniently obtained.
  8. Using Eq. (4), determine the log-decrement for the record that was saved.
  9. Repeat the above process with the aluminum dissipation ring in place.

QUESTIONS:

  1. How do your log-decrement mean values compare with the example cases (Figs. 2 and 3)?
  2. What is the range of variation in each of your estimates (express as a percentage)?
  3. What is your opinion concerning the cause of the variance (systematic error, random error, or combination)?
  4. How could you reduce the variance of your log-decrement estimates, assuming the errors are random?

Resonance response

With the aluminum ring placed on the board beneath the magnet, M, generate a resonance curve. You will do this by selecting each of the following drive frequencies (all in Hz): 0.7, 0.8, 0.9, 0.95, 1.0, 1.05, 1.1, 1.2, and 1.3. In all cases, monitor the motion using `oscilloscope mode' of the software.

First set the amplitude of the drive oscillator, so that at resonance (near 1.1 Hz) the wire between the pole pieces is oscillating so as to nearly impact the magnet. Once this drive amplitude has been set, it must remain henceforth at that setting.

At each frequency setting, wait until the transient has decayed, following a lift of the paper that has been placed on the balance while the drive oscillator frequency was adjusted. Pay particular attention to the `beat' that exists while the balance is becoming entrained to the drive. The beat is most readily noted when the drive is far from the natural frequency of the balance.

Once steady state is achieved at a given frequency, measure (from the screen of the monitor) the peak-to-peak motion of the balance at that frequency. In the vicinity of resonance, fluctuations may be observed. If this should be the case, roughly estimate the mean amplitude of the motion; i.e., note the minimum and maximum values and select a representative intermediate value.

On graph paper that is linear in both axes, plot the nine amplitudes recorded above as a function of drive frequency.

QUESTIONS:

  1. How does your curve compare with the example provided in Fig. 4?
  2. What would happen to the shape of your resonance curve if the log-decrement were smaller?
  3. In which case would fluctuations near resonance be greater-for small damping or large damping?
  4. Do air currents in the room disturb the balance and contribute to the fluctuations? (You may want to test your hypothesis by blowing on the balance.)
File translated from TEX by TTH, version 1.95.
On 23 Dec 1998, 09:20.