Simple, Damped and Forced Harmonic Motion on Torsion Pendulum Bruno Murta, Churchill College, CRSid bm442 Experiment performed Thursday 17 th January 2013 (Practical partner Linh Pham, St John’s College) Coll ege) Abstract The oscillatory motion of a torsion pendulum was investigated under free and forced conditions. The overall aims were to study the effect of damping and forcing on simple harmonic motion, understand the concept of resonance and measure the quality factor of a damped system. The apparatus included a bronze disk subject to a restoring force provided by a coiled spring, two coils of wire, which were responsible for the damping couple, and an electric motor that exerted the driving force. The bronze disk was rotated from the equilibrium position and released from rest. The period of oscillation was measured using a stopwatch and the amplitude of oscillation was measured with a scale ring placed above the oscillating disk. When no current flowed through the coils, some degree of damping was found to exist because the coiled spring was not perfectly elastic; the quality factor was 95.5 ± 1.2. For a current of 0.6 A the quality factor was 7.1 ± 0.4. The effect of greater damping was also studied by varying the current from the initial 0.6 A up to a maximum of 2.0 A; the amplitude decayed more rapidly for greater degrees of damping. The application of a driving force under two different degrees of damping led to the conclusion that the amplitude of response is greater when the damping is lower. In both cases an amplitude peak was observed at frequencies very close to the natural frequency, corresponding to the resonance of the mechanical system. s ystem.
1. Introduction Oscillatory motion is ubiquitous in nature. Indeed, from a swinging pendulum in classical mechanics to atomic bonds in quantum physics, physical systems act like oscillators when displaced from stable equilibrium. In addition to this, the simple harmonic oscillator is one of the few systems in physics whose motion can be solved exactly. For both these reasons, oscillating systems play a critical role in physics. The aims of the experiment are to study the effect of damping and forcing on the simple harmonic motion of a torsion pendulum, understand the physical meaning of resonance in a mechanical system, measure the quality factor of a damped system and determine how the degree of damping affects both the quality factor and the amplitude of response at resonance. The experimental procedure adopted involves using a torsion pendulum consisting of a copper disk. Such disk is connected to a coiled spring, which is responsible for the restoring force, whose magnitude is linearly proportional to the angular displacement. The equipment is housed inside a scale ring, which is used for measuring the displacement of the pendulum with respect to equilibrium. Two coils of wire form an “eddy break”, providing with the damping
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couple in this experiment. The damping couple increases with the current that flows through the “eddy break” . An electric motor, with frequency control, has an arm connected to the pendulum and exerts a driving couple on it. The frequency of the motor is controlled by the voltage across it. A stopwatch is used to measure time intervals in order to determine the frequency of oscillation. The pendulum is displaced from equilibrium and released from rest to undergo oscillatory motion. A more detailed and rigorous explanation of the theory is disclosed in the next section. Section 3 provides with a description of the method used. Section 4 includes the results of the experiment. In section 5 such results are discussed, including potential errors and problems with the method, as well as improvements and suggestions to further work. The overall conclusions are presented in section 6.
2. Theoretical background 2.1
Undamped Harmonic Motion
Invoking Newton’s Second Law, the equation of motion for an undamped oscillator subject to a restitution torque is:
where I is the moment of inertia of the oscillator, is the angular displacement with respect to equilibrium and is the torsional stiffness. A solution to this equation is:
where
is the natural angular frequency of oscillation. The system is
assumed to be displaced from equilibrium by an angle
2.2
and released from rest.
Damped Harmonic Motion
̇
For a damped torsion pendulum an additional damping torque must be included in the equation of motion. Such torque is proportional to the angular velocity and of opposite direction, since it opposes the motion:
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The general solution to this equation of motion considers three distinct cases:
⁄
i)
where
Underdamping:
⁄
ii)
Critical Damping:
iii)
Overdamping:
where
⁄ ⁄ ⁄
⁄
and A and B are constants.
In the last two cases the motion is not oscillatory and the system returns to equilibrium without reversing its direction of motion.
2.3
Forced Harmonic Motion
The equation of motion must also include a sinusoidal driving couple, giving:
The general solution to this equation of motion corresponds to the sum of two parts: a transient response, which corresponds to the solution of the homogeneous equation, and the steady-state response, which is the particular integral that arises from the inhomogeneous equation:
where
Transient Response (Underdamped Case):
⁄
⁄
3
Steady-State Response:
⁄ () ⁄
where
,
The steady-state response has a frequency equal to that of the driving couple, w. Both its amplitude and its phase difference relative to the driving couple are a function of the frequency of oscillation of the motor. The amplitude converges to zero as w
→
and goes to
possible to show that the amplitude has a maximum at
⁄ → as w
. It is also ,
which converges to w0 for small damping [see Appendix A1].
Initially, the pendulum response will be a sum of the transient response, oscillating at w1, and the steady-state response, oscillating at w. This will therefore lead to an irregular oscillation. However, because the amplitude of the transient response decreases exponentially with time, this part of the solution will eventually die away and the pendulum will then oscillate at constant amplitude with frequency w.
2.4
Quality Factor
The quality factor, Q, is a concept that assesses how slow the decay of the amplitude of oscillation is. The quality factor can be defined as π times the number of oscillations, N, required to reduce the amplitude by a factor of e [1]:
where
⁄ ⁄ ⁄
⁄
is the exponential decay factor.
An alternative definition is the ratio of amplitude at resonance to amplitude at zero frequency for the same amplitude of force. Using the expressions derived in Appendix A1 we get:
→ →
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3. Method 3.1
Setting up and checking the apparatus
Figure 1: Torsion Pendulum [2]
Figure 2: Block diagram of circuit.
The block diagram (Figure 2) illustrates the connections between the different components of the electric circuit. The power supply was set up with Channel 2 (CH2) fixed at roughly 24 V. This voltage was not changed throughout the experiment. In addition, before switching the power supply on, the two tracking buttons were set up in the OUT position, the voltage knob on channel 1 (CH1) was turned fully clockwise and the current knob on CH1 was turned fully anticlockwise. The arm joint, which connects the push rod to the drive arm (Figure 3.1), was also positioned at the highest point possible in the slit on the drive arm.
3.2
Transient response
i) Determination of the natural frequency w 0 Before starting the oscillatory motion, the current through the eddy brake (CH1) and the voltage across the driving motor (voltmeter) were both confirmed to be zero. The drive wheel was rotated to align the white pointer on the bronze disk with “0” on the scale ring. The bronze disk was then rotated until the white pointer was aligned with “10” on the ring scale and released from rest. A stopwatch was used to measure the period by taking four measurements of the time interval required for the pendulum to undergo 10 oscillations. From this period, the angular frequency of oscillation, w 1, was determined. Although damping was noticeable, its degree was very low, so w 1 ≅ w0. 5
ii) Analysis of motion for two degrees of damping Before the bronze disk was set to motion, the voltage across the driving motor was again checked to be zero and the pointer in the disk was aligned with “0” in the ring scale.
Firstly, with no current through the “eddy brake”, the pendulum was released from rest at “10” and the maximum displacement was measured for the first 15 oscillations using the ring scale. Then, the experiment was repeated with a current of 0.6 A through the “eddy brake”. In the latter case, the amplitude was measured until it was below 0.1 units of the ring scale. iii) Increased damping To analyse the effect of increased damping, the braking current was increased from the initial 0.6 A to the maximum of 2.0 A. The coils of wire that make up the “eddy brake” can get very hot if a high current flows through them for a long period of time. Therefore, the breaking current should not be raised above 2.0 A or be above 1.8 A for a long period of time.
3.3
Forced response
The multimeter connected across the terminals of the motor was set in the “DC Voltage” mode in the appropriate scale (2-20 V). The damping current was set to 0.3 A. Then, the driving motor was turned on to a low speed (the multimeter read roughly 6 V). Because it takes between 25 to 30 seconds for the pendulum to reach a steady state, during that time interval the period (and hence the frequency) of the driving couple was measured. The amplitude of the steady-state response was measured for different frequencies in the range of voltages between 4 V and 12 V. In particular, a greater amount of measurements was performed in the region close to the frequency of resonance to obtain a well-defined peak of amplitude, since the amplitude changes rapidly with frequency within such range.
The amplitude was measured on both sides of “0” and the average of the two measurements was taken to avoid “zero offsets”. The experiment was repeated with the damping current set to 0.6 A. The amplitude at w = 0 was also measured by slowly rotating the drive wheel by hand and reading off the maximum displacement on either side of “0”.
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7
8
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4. Discussion 4.1
Transient response
Free Motion of the Pendulum
Although there was no current through the “eddy brake”, there was some degree of damping, since there was an evident decrease in the amplitude of the motion, which accounts for loss in energy. This is mainly due to losses in (not perfectly elastic) coiled spring. However, the degree of damping was still small, and therefore the assumption that the frequency of oscillation is roughly the natural frequency is valid. The obtained natural frequency was 3.318 ± 0.007 rad s -1.
Free Oscillation for Two Degrees of Damping
In both cases, the amplitude decreased exponentially with time. However, when the current through the eddy brake was of 0.6 A, the slope of the regression line (Figure 4) was more negative than when the breaking current was 0 A, that is, in the former case the decay was faster. This was expected, since a greater current leads to more damping and a greater dissipation of energy. The quality factor Q confirms this analysis. Indeed, the quality factor for I = 0 A was determined to be 95.5 ± 1.2, whereas that for I = 0.6 A was of 7.1 ± 0.4. Therefore, there is a greater energy loss in the latter case, as expected.
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Increased Damping
As the breaking current was gradually increased from 0.6 A up to the maximum of 2.0 A, the amplitude of oscillation decayed more rapidly, as damping was greater. However, even though for current values close to 2.0 A the damping was heavy, the pendulum would always pass through the equilibrium position before reversing its direction of oscillation. Hence, the pendulum was always underdamped and critical damping was not observed within this interval.
4.2
Forced response
Analysis of Amplitude versus Frequency plots
The two graphs in figure 6 exhibit what was expected. To begin with, they both have a clear peak corresponding to resonance. In addition, both appear to tend to zero as the frequency tends to infinity [Appendix A1]. The resonance frequency for I = 0.3 A appears to be greater than that of the case when I = 0.6 A, which was expected, because γ is greater in the latter case and, as . In both cases, however, the proven in Appendix A1, natural frequency (determined to be 3.318 ± 0.007 rad s -1) is within the error interval of the respective frequencies of resonance. Because the experiment was carried out under small damping conditions, this was expected, since the expression stated above implies when γ is small.
⁄
The amplitude at resonance is greater for I = 0.3 A, as expected, because damping is lower and, from Appendix A1,
Estimate of quality factor
The quality factors obtained for the two degrees of damping were 25 ± 3 for I = 0.3 A and 8.9 ± 1.2 for I = 0.6 A. As expected, the quality factor is greater when the pendulum is subject to a lower damping couple (i.e. lower current). In the case when I = 0.6 A, under unforced conditions the quality factor was previously determined to be 7.1 ± 0.4, which is lower than the quality factor for forced conditions (8.9 ± 0.3). Because the quality factor is defined as , and since both and are equal in the two cases ( is an intrinsic property of the oscillating system and is just a damping coefficient), these two results should be equal. However, their uncertainty intervals don’t overlap.
⁄
Estimation of amplitude of resonance (no eddy breaking)
For free motion, Q = 95.5 ± 1.2. Using the second definition of Q (see section 2.4):
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The amplitude at w = 0 rad s -1 was determined to be 0.8 ± 0.1 for both values of damping current used. Assuming this value is also valid when I = 0 A, we get:
This amplitude could not be experimentally measured using this apparatus, since the amplitude of motion is bounded to 20 units of ring scale by two metal stops on each side of “0”.
4.3
Errors
The quality factor determined for a breaking current of 0.6 A under unforced conditions was not consistent with the quality factor obtained for the same degree of damping under forced conditions. The linear regression in the unforced case (Figure 4) appears to suggest that there is a systematic error; in fact, the traced line has a more negative slope than the line that fits best the first eight data points. This is due to the greater difficulty in accurately measuring the amplitude of motion at values very close to zero, which affected the last two measurements.
⁄
If the last two points were neglected, the absolute value of the slope would be smaller; since (m is the slope) the quality factor would be greater and hence closer to 8.9 ± 0.3, the value of Q under forced conditions. Regarding the determination of the quality factor under forced conditions when I = 0.3 A, the value obtained is not accurate, because the actual amplitude at resonance could not be measured. In fact, since the value used for the amplitude at resonance is simply a lower bound (the fact that the pendulum strikes the metal stops simply means that the amplitude is greater that 20 units) the value of Q is also a lower bound. The uncertainties in the quality factors are in all cases considerably large and the largest contribution is due to the amplitude of oscillation. Therefore, to substantially reduce the uncertainty in the quality factors a more rigorous method to measure the amplitudes would have to be used.
4.4
Improvements to the experiment
The strategy used to determine the frequency of resonance was very inaccurate. It could be improved by making a greater number of measurements to obtain a well-defined resonance peak and use some computational method that could approximate it to a continuous function and give the maximum. The pendulum should be able to have greater displacements from the zero position than 20 units in order to measure the actual value of the quality factor under forced conditions and I = 0.3 A
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5. Conclusions This experiment was performed in order to study the effect of damping and forcing on the simple harmonic motion and understand the concept of resonance in a mechanical system. The quality factor for zero damping current was 95.5 ± 1.2. For a current of 0.6 A the quality factor was 7.1 ± 0.4. Therefore, the quality factor decreases for greater damping. In addition, the decay of the amplitude of oscillation was found to be more rapid for greater degrees of damping. When a driving force was applied for two different degrees of damping the amplitude was found to be greater when the damping was lower. In both cases resonance was observed as the amplitude of oscillation had a peak at a frequency close to the natural frequency, determined as 3.318 ± 0.007 rad s -1.
6. References 1. NST IA Physics Course B, Oscillating Systems (Handout 1), p. 106, G A C Jones, Department of Physics, University of Cambridge 2. NST IA Physics Practicals, Lent Term, 2013, p. 10, Figure E5.4. Laboratory Manual: Department of Physics, University of Cambridge 3. NST IA Physics Practicals, Lent Term, 2013, p. 12, section 5.3.2. Laboratory Manual: Department of Physics, University of Cambridge
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Appendix A1. Steady-State Solution: Limiting Cases
→ → ⁄ ⁄ √ → →
Since for X(w) the numerator is constant, the maximum value of the amplitude will occur for the lowest value of the argument of the root in the denominator:
[ ] ⁄ → → ⁄
NOTE: For small damping At
, so
, the amplitude of the steady-state response is:
In the limit of very low damping X(w) can be simplified to:
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A2. Expressions for Errors
where m is the slope obtained from the linear regression.
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