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1. What is it

2. Different types of pendulums -explore

3. Reference to a use

4. Definitions (beat period, frequency, etc.)

5. Reference to similarities inside the course but then explain what is beyond it

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¹ Department of Physics, Federal University Dutse, Jigawa State, Nigeria

² Department of Physics, Sule Lamido University Kafin-Hausa, Jigawa State, Nigeria

³ SRM Research Institute, SRM Institute of Science and Technology, Kattankulathur, Chennai, Tamil Nadu, India

Correspondence to: Hafeez Y. Hafeez, Department of Physics, Federal University Dutse, Jigawa State, Nigeria.

Email:

Copyright © 2019 The Author(s). Published by Scientific & Academic Publishing.

This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/

Abstract

The power flow between two weakly coupled pendula is calculated analytically and applied to explain the behavior of an optical switch. The general equations of motion are derived for different variations of the coupled pendula with identical and non-identical masses of the pendulum bobs. The power flow under white noise random excitation between the pendula in each system is calculated. The effect of the driving voltage in an optical switch is analogous to the changing length ratio of the pendula system. The net power flow corresponds to the system with identical pendula and for non-identical pendula, the phase constants are not equal and hence there will be no net power flow between the waveguides.

Keywords: Coupled oscillators, Power flow, Optical switch

Cite this paper: Chifu E. Ndikilar, Sumayya I. Saleh, Hafeez Y. Hafeez, Lawan S. Taura, Analytical Calculation of Power Flow between Two Weakly Coupled Pendula and Its Application to an Optical Switch, International Journal of Theoretical and Mathematical Physics, Vol. 9 No. 2, 2019, pp. 25-35. doi: 10.5923/j.ijtmp.20190902.01.

1. Introduction

Many objects around us move in a specific pattern like the motion of a swing or a chalking chair, such objects are said to undergo an oscillatory or vibratory motion. A periodic motion is a regular motion that is repeated in equal intervals of time while an oscillatory motion is a periodic motion of a particle about a mean position or point [1]. A body that undergoes oscillatory motion is called an oscillator. If there are no causes of damping like friction, an object will oscillate infinitely. Oscillations are physical phenomena that occur in all scales starting from atoms which are the basic units of matter to very big galaxies [2]. Examples of oscillations include the motion of the earth crust during an earthquake, beating of the human heart [3], the motion of the lungs during respiration, the motion of the hands of clocks, the economic cycle, motion of planets around the sun, the moon around the earth and the motion of stars around the galactic center [4].The knowledge of oscillations is applied in many fields like construction of earthquake proof buildings, construction of musical instruments like violins and guitar, diving boards, metronomes, construction of auditory devices like microphones and speakers and implementation of shock absorbers in vehicles that stabilize and control the vehicles movement and minimize tire wear [5]. Coupled oscillators are two or more oscillators that are connected to one another with the aid of a coupler. The coupler may be a string, a rod or any medium through which the energy is transferred between the oscillators. Coupled oscillators are important in science because their motion can be used to simulate numerous natural phenomena [6]. The study of coupled oscillators has become a very active area of research in the last few decades. Applications of coupled oscillators nowadays are uncountable in many disciplines like physics, engineering, biology, and chemistry. Coupled oscillators are useful paradigms used to study many complex physical, biological and chemical systems. For instance, mathematical biologist use coupled oscillators to mimic synchronization processes in biological systems like: pacemaker cells in the heart [7], while civil engineers, use it to study the interaction between bridges and vehicles passing on them to avoid the unwanted excitations that can lead to structural damage and to understand the periodic behavior of many dynamical systems [8].In the last decades, systems of coupled pendula have been given attention to understand natural phenomena like clustering or chimera states and to model many physical, biological and mechanical systems [9]. Such systems are simple to analyze and easy to construct experimentally. Since such systems are important because of the synchronization modes they demonstrate depending on the initial conditions [10]. Hence knowing the frequencies of vibrations and understanding how the energy is transferred between the pendula are important in all types of models. In this article, we calculate the power flow between two weakly coupled pendula and apply the results to an optical switch.

2. Theoretical Methods

2.1. Description of the Coupled Pendula Systems

2.1.1. Two Coupled Identical Pendula

The system considered consists of two identical pendula of mass m and length l coupled with a massless spring. There is no kinetic energy associated with the motion of the spring with spring constant k (Fig. 1). The system has two degrees of freedom and which are the angles the rods make with the vertical [11].

Fig ure 1. Two coupled identical pendula

The kinetic energy of the system T is or Equation (2.2) can be written in matrix form as:where Also the potential energy of the system V is or Equation (2.6) can be written in matrix form as where
2.1.2. Two Coupled Pendula of Different Masses

The system considered consist of two pendula of the same length l and different masses and . The system has two degrees of freedom and which are the angles the rods make with vertical (Fig. 2).

Fig ure 2. Two coupled pendula of different masses

The kinetic energy of the system T is orAssuming that can be written in terms of such that where n is any positive real number, then from (2.10) and (2.11)Equation (2.12) can be written in matrix form aswhere also the potential energy of the system V is or and in matrix form where
2.1.3. Two Coupled Pendula of Different Lengths

The system consist of two pendula of the same mass m and of different lengths and . The system has two degrees of freedom and which are the angles the rods make with vertical.

Fig ure 3. Two coupled pendula of different lengths

The kinetic energy of the system T is Assuming that can be written in terms of such that where h is any positive real numberthen Equation (2.21) can be written in matrix form aswhere Also the potential energy of the system V is where
2.1.4. Two Coupled Pendula of Different Masses and Different Lengths

The system considered consists of two pendula of masses and lengths coupled with a massless spring. The system has two degrees of freedom which are the angle the pendulum rods make with the vertical. This system is a generalization of the previous three systems and can be reduced to any of them by choosing the suitable values of h and n.

Fig ure 4. Two coupled of different masses and lengths

The kinetic energy of the system T is ororwhere Also the potential energy of the system V iswhere
2.2. Calculation of Power Flow of the System

The potential and kinetic energies are used to obtain the normal frequencies of the system using the secular equation where is the natural frequency. Since all the four systems have two degrees of freedom, two values of are obtained in each case [11].The normal modes of each system are obtained using the equation where is the normal co-ordinates and [11].To convert the equations of motion from normal to generalized coordinates we use the equation Since for all the four systems , we can writewhere and are components of the eigenvector which are computed by setting and using [12] The same procedure applies for and which are components of eigenvector The values of the and are obtained in terms of m and l using where is the Hermitian transpose of a and I is the identity matrix. After obtaining the equations of motion in terms of generalized coordinates for each system, the effects of damping and external forces is taken into consideration and the equations of motion are written in the form [13]:where , and are called the coupling parameters, is the blocked frequency of the i th oscillator and is the damping ratio of the i th oscillator.The parameters from equation (2.39) are used to calculate the power flow. The first order approximation to the power flow between weakly coupled oscillators under white noise random excitations is given as where and White noise is a random signal that is a combination of all audible frequencies (20Hz-20KHz) with constant power spectral density.

3. Results and Discussion

3.1. Identical Pendula System

Substituting equations (2.4) and (2.8) in equation (2.32) the natural frequencies of the system and are obtained as Substituting (3.1) in (2.33) gives Using equation (2.37) it can be shown that and henceSimilarly, andThus and using equation (2.38) we have or and we obtainandand hence equation (3.25) for this system isFrom (3.10) we obtain the relationship between the normal coordinates and generalized coordinates for this system as andIf then and the two pendula oscillate in phase with an angular frequency and this is the first natural mode of the system. Also, if then and the two pendula oscillate in an out of phase mode with an angular frequency and this is the second natural mode of the system [11].The explicit equations of motion for this system are obtained by substituting (3.11) and (3.12) in (3.2) and (3.3) to giveBy respectively adding and subtracting equations (3.13) and (3.14) and re-arranging we obtain Multiplying equations (3.15) and (3.16) by the mass, m and adding the effects of damping and external forces we obtainand comparing with (2.39) we can write (3.17) and (3.18) as where , and are two independent sources of stationary Gaussian white noise random excitation.If , then equation (3.20) can be written as and hence from equation (2.40), the rate of power flow from the first to the second pendulum can be written as The power input is dissipated by the damping of the second pendulum and hence the mean rate of power dissipation is Thus, equating equations (3.22) and (3.23) gives or The blocked kinetic energy of the first pendulum is defined as the energy when the first oscillator is clamped so we can write Substituting equation (3.25) in (3.24) we obtain Equation (3.26) shows that the rate of power flow from the first pendulum to the second (when the latter is under white noise random excitations) is dependent on the blocked frequencies , and the damping coefficients , . Since the blocked frequencies and are equal for this case, the rate of power flow will be maximum [13].

3.2. Two Coupled Pendula of Different Masses

Substituting equations (2.14) and (2.18) in (2.32) we obtain Solving equation (3.27) and retaining only positive frequencies (physically sound) we obtain the natural frequencies of the system and asThe system has two natural frequencies; the first is independent of the mass ratio and identical to that in section 3.1. The second natural frequency is dependent on n and for large values of n, tends to a constant value as .The equations of motion are Since is the same as that of identical pendula, we can conclude thatSimilarly, for we obtain:Thus and using (2.38) givesSolving equation (3.34), the values of and are obtained asand Thus Equation (3.37) givesand Using (3.38) and (3.39) the equations of motion are obtained asAdding equations (3.40) and (3.41) and re-arranging we obtain Assuming that can be written in terms of as , where s is any positive real number and taking , then (3.42) becomeswhere Subtracting equation (3.40) from (3.41) and re-arranging we obtainMultiplying equation (3.43) and (3.44) by the respective masses, adding the damping and the external force terms givesEquations (3.45) and (3.46) can be re-written as where Setting , then the rate of power flow from the first to the second pendulum can be written as where Thus, since the power input is dissipated by the damping of the second pendulum, the mean rate of power dissipation can be equated to (3.49) giving The blocked kinetic energy of the first pendulum is related to (3.50) aswhere is the coupling parameter and the greater the value of C the stronger the coupling. For weak coupling, and . Thus equation (3.60) can be plotted using the calculated parameters in Table 3.1 as in Fig. 3.1.

Table 1. Relationship between ratios of kinetic energy
and frequencies

Fig ure 5. Relationship between ratios of kinetic energy and frequencies

Equation (3.51) shows the rate of power flow from the first to the second pendulum. Fig. 5 is a plot of the kinetic energy of the second pendulum T 2 to the blocked energy of the first T 1B against the ratios of blocked frequency . Fig.3.1 shows that T 2/ T 1Bis maximum when equals one and is minimum when it tends to zero or infinity [13].
3.3. Two Coupled Pendula of Different Lengths

Equation (2.32) for this system givesIf we let then Expanding the term inside the square root sign binomially and neglecting higher terms, givesConsidering only linear terms in l and k gives the normal frequencies of the system as and The system has two natural frequencies both dependent on the length ratio l.The normal equations of motion of a system of two pendula of the same mass and different lengths is obtained asComponents of (2.36) are derived as where andThusThe relations between coordinates are and The equations of motion are obtained after adding the effects of damping and external forces as or where .The rate of power flow from the first to the second pendulum is obtained as It can be showed thatwhere which is the coupling coefficient An illustration of (3.70) is shown in Table 2 and Fig. 6.

Table 2. Relationship between ratios of kinetic energy
and frequencies

Fig ure 6. Relationship between ratios of kinetic energy and frequencies

It is observed from Fig. 6 that the power flow is maximum when the ratio of blocked frequencies is 1.
3.4. Compound Pendula with Different Masses and Lengths

Here, equation (2.32) gives Considering only terms that are linear in l we can write the natural frequencies and of the system asThus, the normal equations of motion are Also,andHenceandAssuming that where q is any real number, we can re-write equations (3.80) and (3.81) as The equations of motion are obtained as:or Where and The rate of power flow from the first to the second pendulum is and the blocked kinetic energy of the first pendulum is related to the power flow as: where Equation (3.90) can be illustrated as in Table 3 and Fig. 7.Equation (3.90) shows the rate of power flow from the first to the second pendulum when the first is under white noise random excitations and Fig. 7, is a plot of T 2/ T 1B against . It can be observed that the power flow is maximum when the ratio of blocked frequencies is 1.

Table 3. Relationship between ratios of kinetic energy
and frequencies

Fig ure 7. Relationship between ratios of kinetic energy and frequencies

3.5. Application of Power Flow in Compound Pendulum to an Optical Switch

Fig. 8 shows an optical switch which is a device that allows optical signals to be switched on and off or switched from one channel to another. An optical switch is made of two parallel waveguides coupled with a directional coupler [14]. A directional coupler is made of two optical waveguides that are brought close to each other in such a way that their respective optical modes can interact and split into even and odd symmetries that allow the transfer of power between them [15]. For an optical switch, the relative phase difference between the supermodes of the coupling system must be π and hence the value of the coupling length L c must be chosen to satisfy this condition.

Fig ure 8. A schematic diagram of an optical switch [16]

Fig. 9 explains the theory of switching in an optical directional coupler, in Fig. 9 (a) the optical signal is transferred through the upper waveguide, at , the two modes interact, the odd mode is shown with red while the even mode is blue. The region where both symmetries interfere (shown in green in (Fig. 9 (b))) is where is optical signal is transmitted to. Similarly at , the phase shift between the even mode and odd mode is π and the interaction between the two mode leads to the switching of the optical signal from the upper to the lower waveguide (Fig. 9 (c) and (d)).

Fig ure 9. Theory of switching [16]

Coupled pendulum system is analogous to coupled transmission lines and understanding the behavior of coupled pendulum system can aid in explaining the behavior of coupled transmission lines and other related devices. In 1996, the coupled mode theory was used to analyze both systems and it was found that a system of two weakly coupled identical pendula and two weakly coupled pendula of different lengths can be used to model the basic behavior of an optical switch. In an optical switch, a deriving voltage is used to change the phase constants of the waveguides, when the phases are equal, the power is transferred between the waveguides and when the voltage is applied, the phase constants are no longer equal and there is no power transfer between the waveguides [17].From Fig 9, it can be deduced that the effect of the driving voltage can be visualized as changing the length ratio of the pendula system and the net power flow corresponds to the case when h=1 i.e. when the system reduces to the identical pendula case. When , the phase constants are not equal and hence there will be no net power flow between the waveguides.
4. Conclusions

The coupled pendula system has been used for decades as a model to simulate natural phenomena and hence knowing the natural and blocked frequencies, general equations of motion and rate of power flow between the components of such systems is crucial. In this research work, four cases of two weakly coupled pendula systems were taken into consideration. Analytical tools were used to derive the rate of power flow under white noise random excitations between the two oscillators. Higher order terms of the frequencies can be included to obtain more accurate results. Future researches may be done to obtain the same parameters for strongly coupled pendula. Computational and experimental components can be added to this research.

References

[6]

Mallada, E. and Tang, A. Synchronization of weakly coupled oscillators: coupling. Journal of Physics,2013, 46(50): 501-510.

[10]

Altshulera E. and Garcia, R. Josephson junctions in a magnetic field: Insights from coupled pendula. American Journal of Physics, 2003, 71(4): 405-408.

[12]

Shapiro, J.A. Classical mechanics.Rutgers university, University press New Jersey, United States of America, 2003, 1-35.

[14]

Vakili, B. Bahadori, S.and Ghayour, R. All-optical switching using a new photonic crystal directional coupler. Advanced Electromagnetics,2015, 4:63-67.

# data[].attributes.attachment_link.title
Analytical Calculation of Power Flow between Two Weakly Coupled Pendula and Its Application to an Optical Switch

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The power flow between two weakly coupled pendula is calculated analytically and applied to explain the behavior of an optical switch. The general equations of motion are derived for different variations of the coupled pendula with identical and non-identical masses of the pendulum bobs.

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Two pendulums, connected by a massless spring, are allowed to vibrate in the same vertical plane. For the limiting case of small oscillations, the equations of motion for the system are given by

[more]

, ,

where is the acceleration of gravity. You set the masses, lengths and initial displacements. The resulting graph shows the angle between the two pendulums as a function of time.

[less]

Contributed by: Stephen Wilkerson (Towson University)
With contribution by Franz Brandhuber

Snapshots

Details

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Coupled Pendulum Oscillations - Wolfram Demonstrations Project

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Two pendulums connected by a massless spring are allowed to vibrate in the same vertical plane For the limiting case of small oscillations the equations of motion for the system are given by where is the acceleration of gravity You set the masses lengths and initial displacements The resulting graph shows the angle between the two pendulums as a function of time

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Coupled Pendulum (Normal Modes and Frequencies) | Lagrangian | EOM - Matrix form

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Coupled Pendulum | UCSC Physics Demonstration Room

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Physics Forums | Science Articles, Homework Help, Discussion

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Now, because this is a classical mechanical system, the instantaneous state of the system is given by telling the coordinates and velocities of the two pendulums. The system has some equilibrium point, which means that the pendulums will stay motionless indefinitely if they are put to rest at the equilibrium positions. Let's choose coordinates as ##x_1## and ##x_2##, which are the horizontal coordinates of pendulum 1 and pendulum 2 relative to their equilibrium points (at equilibrium ##x_1 = x_2 = 0)##.

Now, the equations of motion, which tell the acceleration of both pendulums as a function of their positions, are:

##m_1 \frac{d^{2}x_{1}(t)}{dt^2} = -k x_1 (t) + k_c (x_2 (t) - x_1 (t))##
##m_2 \frac{d^{2}x_{2}(t)}{dt^2} = -k x_2 (t) + k_c (x_1 (t) - x_2 (t))##

Here ##m_1## and ##m_2## are the masses of the pendulums, ##k## is a Hooke's law constant that depends on the strength of gravity and on the length of the strings. The parameter ##k_c## is another Hooke's law constant that tells the strength of the coupling between the pendulums. In these equations it is assumed that the oscillations of the system are small enough for the motion to be harmonic.

Solving the normal modes means that we find out numbers ##a, b, c, d, \omega_1## and ##\omega_2##, for which

##\frac{d^{2}}{dt^2}(ax_1 (t) + bx_2 (t)) = -(\omega_{1})^2 (ax_1 (t) + bx_2 (t))##
##\frac{d^{2}}{dt^2}(cx_1 (t) + dx_2 (t)) = -(\omega_{2})^2 (cx_1 (t) + dx_2 (t))##

Here the ##ax_1 + bx_2## and ##cx_1 + dx_2## are the two normal modes of the system. Mathematically they are weighted averages of the coordinates ##x_1## and ##x_2##, which behave as if they were independent uncoupled oscillators that oscillate with frequencies ##\omega_1## and ##\omega_2##.

With linear algebra that involves matrices and determinants, it can be shown that the two possible frequencies ##\omega## can be solved from a quadratic equation for ##\omega^2##:

##\left(\frac{-k - k_c}{m_1} + \omega^2 \right)\left(\frac{-k - k_c}{m_2} + \omega^2 \right) - \frac{k_c^2}{m_1 m_2}= 0##

You can solve this with Wolfram Alpha if you want to. A more detailed explanation is here: http://users.physics.harvard.edu/~schwartz/15cFiles/Lecture3-Coupled-Oscillators.pdf

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I would like to know how to solve the coupled pendulums problem when the masses of the pendulums are different. BUT the ratio of the masses are known and all other factors are kept constant. Need to find its affect on the beating frequency, but i cannot an equation for this with different...

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This simulation shows two pendula coupled by a spring which has a small spring constant (weak coupling). For such systems it is characteristic that the oscillation's energy permanently moves from one partial system to the other.

The "Reset" button brings the system to its initial position. You can start or stop and continue the simulation with the other button. If you choose the option "Slow motion", the movement will be ten times slower. It is possible to change the initial positions of the two pendula by using the corresponding text fields. A positive (negative) angle means an elongation to the right (left) side.

This browser doesn't support HTML5 canvas!

You can get the harmonics of the system as follows:

If you write two equal numbers into the text fields (e.g. twice 10°), the two pendula will always have same phases.

By choosing two opposite numbers (e.g. 8° and −8°), you can get an oscillation with opposite phases.

If you write 0° into one of the text fields for the initial positions, the entire energy of the oscillation will alternately be transferred to one of the pendula: That pendulum which first was in the mid-position oscillates with an ever-increasing amplitude, whereas the amplitude of the other pendulum decreases. After some time there is a moment in which only the first-mentioned pendulum oscillates; the other one stops in the mid-position at that time. Afterwards this process happens again with reversed roles and so on.

# data[].attributes.attachment_link.description
This simulation shows two pendula coupled by a spring which has a small spring constant (weak coupling). For such systems it is characteristic that the oscillation's energy permanently moves from one partial system to the other. The "Reset" button brings the system to its initial position.

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Coupled Pendulums: two pendulums connected by a spring

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Learn to find the equations of motion of coupled pendulums connected by a spring using the lagrangian method.

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A beautiful demonstration of how energy can be transferred from one oscillator to another is provided by two weakly coupled pendulums.

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Abstract

The topic of coupled oscillations is rich in physical content which is both interesting and complex. In this research work we aimed to study the kinematics behaviour of an important physical system known as coupled pendulum, in which two identical pendulums were coupled together by a spring. The Lagrangian of the system was first constructed, then the equations of motion have been derived. Analytical solutions were obtained for these equations, and in addition we apply a reliable algorithm based on an adaptation of the standard differential transformation method is presented, which is the multi-step differential transformation method to obtain numerical solutions for the considered system for some selected initial conditions. The solutions of the equations of motion were obtained by the multi-step differential transform method. Of course, this method is needed if it predicts the motion of the considered system. For this reason figurative comparisons between the multi-step differential transformation method, the standard differential transformation method and the classical fourth-order Runge-Kutta method are given. The results reveal that the proposed technique is a promising tool to predict the behaviours of the considered system in long time interval. Moreover, the motion of curves represents the initial conditions that sensitive depend and how much oscillatory motion. In conclusion, the proposed technique is a reliable method to solve the considered double pendulum problem in long time interval. We believe that the system considered in this work along with the method used is interesting for physicists, mathematicians and engineers.

Keywords

Differential transformation method

Multi- step differential transformation method

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The kinematics behaviour of coupled pendulum using differential transformation method

# data[].attributes.attachment_link.description
The topic of coupled oscillations is rich in physical content which is both interesting and complex. In this research work we aimed to study the kinematics behaviour of an important physical system known as coupled pendulum, in which two identical pendulums were coupled together by a spring.

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How to Cite Sources in APA Citation Format | Mendeley

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This is a complete guide to APA (American Psychological Association) in-text and reference list citations. This easy-to-use, comprehensive guide makes citing any source easy. Check out our other citation guides on MLA 8 and Harvard referencing.

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Coupled Oscillators (Normal Modes and Frequencies)