An oscillation discovery of the forced vibrating system predicted by the multi-time differential equation

In this article, an experiment of the forced oscillating pendulum system was set up. Here, a pendulum is driven sinusoidally by the loudspeaker. From this experiment, the spectrum of output signal of a Halleffect sensor (UGN3503) which is used to transform the oscillatory motion of the pendulum into an electrical signal, exhibits four sharp frequency peaks, and it is also found that two of these frequency peaks cannot be described by the solution of ordinary differential equation from conventional ordinary differential equation text books. However, it can be solved by multi-time variable technique, a mathematical tool. The latter solution consists of the sum of four terms: Natural response, forced response and the two new terms being the result of multiplying between natural and forced responses. This analytical solution reveals the frequency components and their behaviors more precisely and corresponds well to the spectrum of the experimental result.


INTRODUCTION
The dynamics of a simple pendulum is one of the most popular topics analyzed in textbooks and undergraduate physics courses (Keller et al., 1993).The simple pendulum is also one of the examples presented when nonlinear oscillations are studied (the oscillation amplitude is not small) (Marion, 1970;Mickens, 1996).For small values of the oscillation amplitude, it is possible to linearize the equation of motion of the pendulum (Zill, 2005) and, in this regime, the oscillatory motion is a simple harmonic motion, that is, the restoring force is proportional to the angular displacement.
Pendulum is also the common example of a mechanical oscillator.When perturbed, it begins to oscillate with a natural frequency.Yet, this free oscillation decays exponentially with time due to friction forces, that is, these forces cause the mechanical energy of the oscillator to decrease.Any system that behaves in this way is known as a damped harmonic oscillator.The oscillation of the damped harmonic oscillator *Corresponding author.E-mail: thongchaim@nu.ac.th.
(pendulum) acted on by an external driving force is called forced oscillation or driven oscillation.The oscillations occur mainly in machinery and in electric circuits.Using the conventional method (Taylor, 2005;Arya, 1997;James et al., 1989) for finding the solution of a linear differential equation describing the oscillation of such system, the solution is given by the sum of two parts.The natural response, the solution for a damped harmonic oscillator discussed earlier, delays out eventually.Another one is the forced response, the solution due to the external driving force, persists after the natural response has died away.The natural response depends on the initial conditions at time 0 t = and on the initial values of the forced response also at time 0 t = (Dimarogonas, 1996).
Recently, the analysis of a second-order oscillator based on a multi-time variable technique was proposed (Maneechukate et al., 2008).The result demonstrates that the amplitude of the natural response ( ) x t of the system depends on initial value 0 x and the forced response ( ) f x τ at any arbitrary time τ according to the following equation: x t τ is the complete solution of the separated time scales.This solution is somehow different from the one achieved by the conventional method: It is remarked that amplitude of the natural response ( ) x t depends on the initial condition 0 x and the forced response ( ) In this paper, we apply the multi-time variable technique to solve a mathematical model in the form of second order differential equation describing the damped harmonic oscillation of the pendulum acted on by the sinusoidal external force.The obtained analytical result consists of the sum of four components: natural response, forced response and the new two terms coming from the product of natural and forced responses.These two new terms do not appear when we use conventional method.
To verify that such analytical result is valid, an experiment of the forced-oscillating system is set up.Here, a mechanical pendulum acted by a sinusoidal external force is used as an example to illustrate behavior of such solution.A large loudspeaker is used to generate the sinusoidal external force to the pendulum.The oscillatory motion of the pendulum is transformed as an electrical signal by the popular UGN3503 Hall Effect Sensor, which is placed near a magnet fixed next to pendulum's rotating point.From such experiment, the spectrum of output signal of the sensor exhibits four sharp frequency peaks, which agree well with the solution of such mathematical model.Moreover, the solution can clearly describe the behavior of each frequency peak in the spectrum.

MATERIALS AND METHODS
In this research, the forced vibration of the pendulum is described by a linear second order differential equation system, with multitime variable technique.First of all, we would therefore like to Prompak et al. 3293 describe the multi-time variable technique briefly as follows.

Definition (I)
The concept of multi-time variable technique is that the time of the system can be separated into two different variables, t and τ , which the natural response is a function of t , and the forced response is a function of τ .As a result, the complete response of system can be written as: where τ is time considered after the beginning of the time of the system, t , by the time t ∆ .Alternatively, we can say that t is shifted τ by t ∆ , namely:

Definition (II)
The linear second order differential equation that describes the oscillation of system acted on by an external input can be written as: These definitions are applied to solve the linear system as shown in Figure 1.Dividing Equation 3 by 2 c gives the equation of motion: and the constants a and b of the natural response depend on the initial conditions and on the forced response ( ) f y τ , which will be discussed later.In addition, Equation 6 can usefully be rewritten as:   4as: Using the technique of Fourier transform, we define the input of system ( ) , where 0 X and f ω are the amplitude and angular frequency of the input, respectively, then we obtain the forced response output: where ( ) is the phase shift of the system at frequency f ω (Appendix B) .Finally, substituting Equations 8 and 10 in Equation 1, the complete response in the viewpoint of multi-time variable is ( ) where ( ) f y τ was defined in Equation 10.
Upon finding a from the initial position 0 (0, ) y y τ = , we set 0 t = in Equation 11, getting: Thus, Equation 11 can be written as: ( ) In Appendix C, the solution of ordinary second order differential equation solved by conventional method is described.It is convenient to write such solution here: ( ) Comparing Equation 13with Equation 14, it is easy to see that the natural response term in Equation 13depends on the values of the forced response for all time 0 t ≥ , whereas the natural response term in Equation 14 depends on the initial values of the forced response at time 0 t = only.This is the obvious difference between the multi-time variable technique and the conventional method.
Next, substituting Equation 10 in Equation 13gives: From a trigonometric identity, , and the assumption of τ in Equation 2, Equation 15 can be rewritten as:  decreases continuously and approaches zero asymptotically.In effect, the magnitudes of the second and third terms, including the first term, will decrease continuously.In addition, the magnitude of forced response becomes large when f ω gets close to 0 2 / c c (by observing the value of ( ) f H jω as mentioned earlier).As a result, the magnitudes of the second and third terms also increase, that is, half the magnitude of forced response at 0 t = .

The forced vibration of a pendulum described by the multi-time second order differential equation
To explain the forced vibration of a pendulum, we consider the pendulum acted on by the external force as shown in Figure 2. Without external forcing, when the bob of the pendulum is pulled from its equilibrium position, the restoring force magnitude acting on the bob is given by: where m is the mass of the bob and g is the acceleration due to gravity.Suppose the pendulum is set in motion, if the angle θ is small then sinθ θ ≈ ; the motion of the pendulum is the damped harmonic oscillation, and the arc y can be approximately considered to be a straight line.From the relation y Lθ = , Equation 17 can be rewritten as: The motion of the pendulum will eventually be damped out, because there is a damping force proportional to the velocity of the bob acting in the direction opposite the motion: When the support of the pendulum is acted on horizontally by a driving force ( ) x τ , using the multi-time variable technique, the equation describing the forced motion of the pendulum is in the form: When we want to find the natural response, Equation 23 can be rewritten as: where ( )  ( ) ( ) ( ) Similar to Equation 16in the multi-time variable for separating time scale of system, Equation 32 can be rewritten as: .As a result, the magnitudes of the second and third terms are also large, that is, half the magnitude of forced response at time 0 t = .
In order to confirm that the aforementioned analytical result is valid, an experiment that has a vibrating pendulum acted on by an external force is designed as shown in Figure 3.In this experiment, a vibration system, the pendulum consists of a light rod of length 0.10 m (pendulum's arm) and a mass 0.10 kg attached at one end.The other end of this rod is suspended to a support fixed on an experimental cart, here, the natural frequency n f of the pendulum's oscillation is about 1.50 Hz (the damping constant β can be roughly calculated from the equation, ).The body of the cart is connected horizontally to the center of a large loudspeaker.Hence, such that the vibrating system is forced to undergo periodic oscillation of the loudspeaker in the direction of the pendulum motion; the loudspeaker is connected with a power amplifier circuit that receives the sine signal from the function generator, so that we may easily adjust the frequency and amplitude of the external force To measure the oscillatory motion of the pendulum, the Hall Effect Sensor, UGN3503, is placed close to a circular flat magnet fixed on pendulum arm close to the rotating point of it as shown To confirm linearity of the designed measurement system in the aforementioned experiment, the pendulum is put first into motion while the external force, applied by loudspeaker, is switched off.The output signal spectrum of sensor should have only one frequency, namely, the frequency of pendulum's oscillation.Second, let the external force operate while the pendulum does not oscillate (the pendulum is fixed at rest on the experimental cart).Spectrum of output sensor signal in this case should also have only one frequency, the frequency of loudspeaker oscillation.
In the next process of experimentation, as the external force is switched on, initially, we slightly pull the mass to one side of its equilibrium position, which the angle of the pendulum with respect to the vertical is small (less than 5°) and then relea se it.This is equivalent to giving energy to the pendulum, that is, the oscillation of the pendulum is maintained, and it is being driven by the periodic external force.In this case, the obtained spectrum of sensor's output signal should correspond to Equation 33.The frequencies of the force used in the experiment are as follows: 1.10, 1.20 and 1.90 Hz.The experimental results of these experiments are illustrated subsequently.

RESULTS
The experimental result which the pendulum oscillates, while the external force is switched off, is shown in Figure 5.The upper trace signal is the output signal of sensor, Prompak et al. 3297  UGN3503, and the lower trace signal is its spectrum.
Figure 6 shows the experimental result that the external force oscillates, while the pendulum does not oscillate; here, the frequency of the external force is 1.10 Hz.Next, the experimental result in case the pendulum is forced by the driving force 1.10 Hz is illustrated in Figure 7.
Similarly, the experimental results of the pendulum forced by the driving forces 1.20 and 1.90 Hz are as shown in Figures 9 and 10, respectively.

DISCUSSION
The experimental results in Figures 5 and 6 confirm that the designed measurement system is linear, since in each figure the spectrum has only one frequency, that is, a sharp frequency peak in Figure 5 showing the frequency of oscillation of the pendulum, namely 1.50 Hz.
The sharp frequency peak in Figure 6 shows the frequency of the external force equal to 1.10 Hz.Next, when the oscillating pendulum is acted on by the external force 1.10 Hz, the obtained spectrum, as shown in Figure 7, matches well with all the terms in Equation 33, where the frequency components 1.50, 2.60, 0.40, and 1.10 Hz of the spectrum are equivalent to the first, second, third, and fourth terms of Equation 33, respectively.Consider Figure 7 in detail, a moment after the mass is pulled away from its equilibrium position and released, the magnitudes of the frequency components 2.60 and 0.40 Hz are about half the magnitude of the frequency component 1.10 Hz, corresponding to the amplitudes of the second and third terms in Equation 33, or of new terms, which are equal to half the amplitude of the fourth term at time 0 t = .After a short time, the magnitude of the frequency component 1.50 Hz slightly decreases, and the frequency components 2.60 and 0.40 Hz cannot be observed, as shown in Figure 8

Conclusions
From the experimental and analytical results corresponding to each other, we may conclude that the two occurring new frequency components are an outcome of an oscillating system driven by external force.The analytical result using the multi-time variable technique can predict four frequency components in the spectrum of the experimental result more precisely.


Here, we consider the solution for the damped harmonic oscillator, that is, ; for this case, the roots will be complex: and we can write the solution, natural response, as: as the natural angular frequency of the damped oscillator and 0 0 2 c c ω = as that of the undamped oscillator, then Equation A1 can be written as: We know, of course, that ( ) n y t is real, whereas the two exponentials in Equation A3 are complex.Therefore, the coefficient 2 D must be the complex conjugate of 1 D .Let us assume that From Equation 9, we can find the frequency response of the system via Fourier transform pair as follows: ( ) τ ω = Here, we have defined ( ) x τ as input of the system.
Substituting these Fourier transform pair into Equation 9, we get: Then, the ratio between ( ) f Y jω and ( ) X jω , the frequency response of system, is: where the system's magnitude response is ( ) and phase response is Since the frequency of ( ) x τ is equal to f ω , the system amplitude gain at frequency f ω , ( ) f H jω , is: ( ) and the phase shift of the system at frequency f ω is: Here, the input of the system is ( ) 0 ( ) cos f x X τ ω τ = , the sinusoidal input, where 0 X is its amplitude.The steadystate response is indicated by: Therefore, it is clear that the sinusoidal steady-state response, or the forced response, has the same frequency as the input, whereas its amplitude and phase angle are determined by the system's magnitude response ( )

Figure 1 .
Figure 1.Block diagram of this system.
is the phase difference between the forced response and the external force.From Equation16, the frequencies of the first, second, third, and fourth terms are n ω , of two new terms, as discussed previously.At 0 t = , the magnitudes of the second and third term, that is, of two new terms, are equal to half the magnitude of the forced response (the fourth term).As t increases,
damped and undamped natural angular frequency of the system, respectively.The constants a and b of the natural response depend on the initial conditions, namely, the initial position 0 y and the initial velocity 0 v , and on the forced response ( )

Figure 3 .
Figure 3.The setup of the forced pendulum experiment.
multi-time variable for separating time scale of system, the forced response of Equation 28 is condition, namely the initial position or the displacement at 0 t = , the parameter a is given by: the magnitudes of both the second and third terms, that is, of new terms, are equal to half the magnitude of the forced response (the fourth term); as t increases, approaches zero.From this effect, the magnitudes of the second and third terms, including the first term, decrease continuously.In addition, the magnitude of forced response becomes large when the oscillating frequency of the driving force, f

Figure 4 .
Figure 4. Hall Effect Sensor placed nearby the flat circular magnet.

Figure 5 .
Figure 5.The experimental result due to pendulum oscillation in the absence of external force.The upper trace is the output signal of the UGN3503 sensor, and the lower trace is its spectrum.

Figure 6 .
Figure 6.The experimental result in case pendulum does not oscillate, while the external force operates.The upper trace is the output signal of the UGN3503 sensor, and the lower trace is its spectrum.

Figure 7 .
Figure 7.The experimental result of the forced pendulum, in case the frequency of the external force is equal to 1.10 Hz.The upper trace is the complete output response of the pendulum in time domain.The lower trace shows the spectrum of the upper trace signal.

Figure 8 .
Figure 8.After a short time, the frequency component 1.50 Hz slightly decreases, and the frequency components 0.40 and 2.60 Hz cannot be observed (continued from Figure 7).

Figure 9 .
Figure 9.The experimental result of the forced pendulum with the frequency of the external force 1.20 Hz.The upper trace is the complete output response of the motion of the pendulum in time domain.The lower trace shows the spectrum of the upper trace signal.
the magnitude of the frequency component 1.10 Hz has steady or fixed amplitude.

Figure 9
is similar to Figure7.It shows experimental result due to the external force frequency 1.20 Hz.In this figure, the frequency components 1.50, 2.70, 0.30, and 1.20 Hz of the spectrum correspond to the first, second, third, and fourth terms of Equation33, respectively.Analogous to Figure7, Figure10shows experimental result due to the external forcing frequency equal to 1.90 Hz.In this figure, the frequency components 1.50, 3.40, 0.40, and 1.90 Hz of the spectrum correspond to the first, second, third, and fourth terms of Equation33, respectively.

Figure 10 .
Figure 10.The experimental result of the forced pendulum with the frequency of the external force equal to 1.90 Hz.The upper trace is the complete output response of the motion of the pendulum in time domain.The lower trace shows the spectrum of the upper trace signal.