Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 3, pp. 717-730, Warsaw 2016 DOI: 10.15632/jtam-pl.54.3.717 ROTATING ORBITS OF PENDULUM IN STOCHASTIC EXCITATION Sze-Hong Teh, Ko-Choong Woo, Hazem Demrdash University of Nottingham Malaysia Campus, Department of Mechanical, Materials and Manufacturing Engineering, Semenyih, Malaysia e-mail: kedx1tsn@nottingham.edu.my; Woo.Ko-Choong@nottingham.edu.my; Hazem.Demrdash@nottingham.edu.my Amethod to extract energy fromanexcitationwhich is stochastic innature ispresented.The experimental rig comprises a pendulum, and a vertical excitation is provided by a solenoid. The control input assumed in the formof a direct currentmotor, and anothermotor, used in reverse, acts as a generator.The stochastic excitation has been achievedby varying the time interval between switching the RLC circuit on and off according to a random distribution. Such non-linear vertical excitations act on an oscillatory system fromwhich a pendulum is pivoted.ThePierson-Moskowitz spectrumhas been chosen as the randomdistributionwhile an inverse transform technique has been used for generation of the randomexcitation signal in LabVIEWenvironment.Moreover, a bang-bang control algorithmhas been implemented to facilitate rotationalmotion of the pendulum. Experimental observations have beenmade for various noise levels of vertical excitations, and their implication on energy generation has been discussed. A positive amount of energy has been extracted for a minimal amount of the control input. Keywords: parametric pendulum, stochastic excitation, energy harvesting 1. Introduction Energy in form of mechanical vibration generally exists everywhere, all the time, and its source can be natural (e.g. sea wave) or artificial (e.g. automobile vibration). While some vibrations may be regarded as useful and necessary, other vibrations can be suppressed or their energy can be converted into electrical energy bymeans of a harvester. For an instance, it can be used to vertically excite a parametric pendulum, which in turns rotates the pendulum and drives a generator to produce electrical energy. That form of vertical excitation can be chaotic or stochastic in the characteristic. Though both chaotic and stochastic excitations appear to be random and irregular, the chaotic excitation is deterministic while the stochastic excitation is characterized by using probability theory. Various investigations have been performed on the dynamics of a parametric pendulum subjected to the stochastic excitation. For example, a stochastic forcing ofGaussian distribution was added to the forced pendulum model (Blackburn et al., 1995) and its effect in stabilizing and destabilizing the periodic motion of the parametric pendulum was studied by Blackburn (2006). Influence of the Wierner phase noise in the chaotic regime of a parametrically excited pendulumwas numerically studied by Litak et al. (2008). A numerical study was conducted to investigate how stable rotational modes of a parametric pendulumwere affected by introducing a stochastic forcing of Gaussian distribution by Horton and Wiercigroch (2008). Rotation of a parametric pendulum system subjected to an excitation with the Pierson-Moskowitz wave spectrumwas studied byNajdecka (2013). Rotationalmotion of theMathieu equation subjected to anarrowband excitationwas investigated byYurchenko et al. (2013). The stochastic response of a rotatingpendulummountedonasingledegree-of-freedom(SDOF)base,whichwas subjected to narrow band excitation, was explored numerically by Yurchenko and Alevras (2013) and 718 S.-H. Teh et al. Alevras and Yurchenko (2014). Stochastic synchronization of rotating pendula mounted on a mutual elastic SDOFbase subjected to narrowband excitation were numerically investigated by Alevras et al. (2014). Moreover, the parametric pendulumhas been a subject of chaotic analysis as well. Influence of a nonlinear oscillator on an attached pendulum in the main parametric resonance region was studied by Warminski and Kecik (2006). The behavior of a parametric pendulum pivoted on a mass-spring-damper system was investigated analytically, numerically and experimentally around the principal resonance region byWarminski andKecik (2009). The dynamics of an autoparametric pendulum-like systemwith a nonlinear suspension subjected to kinematic excitation was numerically and experimentally investigated byKecik andWarminski (2011). The dynamics of a pendulum suspended on a forced Duffing oscillator was numerically explored by Brzeski et al. (2012). As the rotational motion of parametric pendulum is advantageous for energy harvesting, it is desired to maintain that motion when it is subjected to any excitations. A few control strategies have been proposed for the purpose of sustaining rotational motion of the parametric pendulum. A time-delayed feedback method was employed to initiate and stabilize rotational motion of the pendulum by a feedback perturbation proportional to the difference between the present and a delayed state of the pendulum (Yukoi and Hikihara, 2011a). The tolerance of time-delayed feedback method with mistuned delay was experimentally investigated by Yukoi and Hikihara (2011b). The extended time-delayed feedback was proposed to maintain rotating solutions of the parametric pendulum to avoid bifurcations that destabilize the rotating orbit (De Paula et al., 2012). The robustness and sensitivity of the time-delayed feedback method with respect to varying excitation parameters and added noise was investigated by Vaziri et al. (2014). In addition, a velocity control methodwas proposed to effect a control torque about the angular axis if the angular velocity of the pendulumdroped below a threshold (Najdecka, 2013). Alternative methods of control, such as a LQR method (Dadone et al., 2003) were deployed mainly for a pendulum-type systemwithout continuous rotations. The objective of this paper is to show an alternative mean of implementing an experimental stochastic process for facilitating investigations on the parametric pendulum for energy harve- sting. In particular, the observation of rotational state of the pendulumunder such an excitation is of interest in the current study.Theproposed setup could exert a non-linear electromechanical force to thependulumbymeans of anAC solenoid,which is poweredbya solid-state-relay(SSR)- coupledRLCcircuit, and itwas previously reported to exhibit period-1 rotations, period-1 oscil- lations and period-2 oscillations (Teh et al., 2015). The type of the parametric pendulumsystem consideredbelongs a special class of autoparametric systems; it hasamass-spring-damper system as the base and a pendulum is attached to it. Unlike the stochastic excitation of the parametric pendulum as the above-mentioned, the implementation of a stochastic excitation to the pendu- lum system in the current study is achieved by a random distribution of the input signal with respect to the duration of switching on and off the solenoid. In addition, a bang-bang control algorithm is tailored to address the problem of escaping the potential well when the noise is sufficiently destructive for positive energy generation. To implement the control algorithm and monitor potential energy generation, the experimental parametric system is enhanced with a control system by adding a rotational actuator, a generator and some necessary transducers. The main aim of this paper is to show that for stochastic energy sources commonly found in real life it is possible to introduce aminimal amount of the control input, while still harvesting more energy. This is an improvement to the previous work on periodic excitation by Teh et al. (2015). The paper is organized in the following manner. The enhanced version of the experimental setup is first described in Section 2 together with its dynamic equations. The implementation of stochastic excitation for the experimental setup of the current study is described in Section 3. The bang-bang control algorithm that assists the pendulum motion is proposed in Section 4. Rotating orbits of pendulum in stochastic excitation 719 Some results on the pendulum system subjected to different levels of stochasticity are presented in Section 5. Finally, the paper is summarized and concluded in Section 6. 2. Experimental setup The experimental setup of the parametric pendulum system given by Teh et al. (2015) has been extended to further investigation on rotations of the pendulum in the present work. The parametric pendulumsystemassembled for the experimental studyonpendulumrotations under stochastic excitation is depicted in Fig. 1a. This configuration of the parametric pendulum system was previously described by Teh et al. (2015) regarding the functions and connections among the components. In the current study, the pendulum assembly has been modified by adding a L-bracket to mount the motor and the generator on to the pendulum assembly. The motor and generator are connected in parallel with the pendulumpivot bymeans of spur gears. The gear ratio among the pendulum pivot, motor and generator is 2:2:1. The overall assembly is suspended by means of a mechanical hook at the aluminium base plate. Fig. 1. (a) Experimental rig of the present work; (b) schematic of (a) The schematic of the experimental setup is illustrated in Fig. 1b. The interface between the mechanical and electrical part of the system by Teh et al. (2015) has been enhanced in the present work to derivemore information from the parametric pendulum setup while outputting commands simultaneously. Vertical oscillation of the pendulumassembly is achieved by exerting an electromagnetic force on themetal barwithin the solenoid.The solenoid is connected in series with an external power supply and a capacitor to fulfill such forcing requirement, as depicted in Fig. 1b. It is effectively a RLC circuit and it is built in a similar way to Mendrela and Pudlowski (1992). A solid state relay is coupled with the RLC circuit to switch the solenoid on and off according to the input signal generated from the computer system. A non-linear electromechanical excitation of the pendulumassembly is generated by repeatedly switching the solenoid on and off, and a linear tension spring is used to complement the vertical oscillatory motion. 720 S.-H. Teh et al. Four transducers are employed to scavenge useful data from the experimental system. The angulardisplacementof thependulumismeasuredusinganAutonics incremental optical encoder (Model: E6B2-CWZ1X), and the signal is fed tomonitoring systemviaNational Instrument (NI) universal motion interface (Model: NI UMI-7774). Besides, the vertical motion of the pendulum assembly is measured using a Brüel &Kjær piezoelectric accelerometer (Model: 4518-001). Two Allergo hall-effect based linear current sensors (Model: ACS711EX andACS711LC) are used to respectively measure the electric current of the circuit of Crouzet DCmotor (Model: 82800502) and Cytron DC generator (Model: SPG50-100K). The charge amplifier function of a NI data acquisition module (Model: NI USB-4432) is used to amplify the electrical signal generated by the accelerometer before being fed to the monitoring system. On the other hand, the analog voltagemeasurement functionof the samedata acquisitionmodule is used tomeasure thevoltage difference generated by the linear current sensors before being fed to the monitoring system. Themonitoring and logging of the experimental data is performed using NI PXI-embedded computer system (Model: NI PXI-8106). NI PXI-embedded computer system is also capable of simultaneously generating a signal for switching the solid state relay on-and-off andperforminga control action on the pendulumaxis based on themeasured real-time data. LabVIEW is usedt o design a graphical user interface that could perform the above tasks based onNIPXI-embedded computer system. The NI motion controller module (Model: NI PXI-7350) of the computer system is used to generate a digital signal which is software-timed by a graphical user interface under a specific algorithm for switching the solid state relay on-and-off. Besides, the NI motion controller module is used to generate a hardware-timed pulse-width modulation (PWM) signal according to the computed control action, which is in the form of motor input voltage. The PWM signal is received by the Cytron DC motor driver card (Model: MD10C) and it draws electric current from a DC power supply to drive the DC motor according to the PWM signal. In addition to the equations of motion for the mechanical system as well as governing equ- ations describing the electrical circuit of the solenoid (Teh et al., 2015), the variation of current flowing through the DCmotor are expressed as follows LM dIM dt =UM −RMIM −KM θ̇ TM =KMIM (2.1) whereLM is themotor inductance,RM is themotor resistance,KM is themotor torque constant, UM is the voltage applied to the motor armature, and IM is the motor current. Finally, the dynamics of the DC generator is expressed using the following equations LG dIG dt = ηKGθ̇−RGIG TG =KGIG (2.2) where LG is the motor inductance, η is the gear ratio between the pendulum pivot and the generator, RG is the motor resistance, KG is the motor torque constant, and IG is the motor current. 3. Stochastic excitation modeling Implementation of a stochastic excitation in the parametric pendulum system of this work is described in this Section.As aforementioned, the vertical excitation of the pendulumassembly is achievedbyrepeatedly switching the solenoid circuit onandoffviaa solid state relay.Variation of the timing for switching the solenoidonandoffaccording toa randomdistributioncouldgenerate a stochastic excitation on the pendulumsystem.An example of the input signal of the solid state relay with varying on and off timing is shown in Fig. 2. The time interval between each peak of the input signal is different, and these values are generated from a random number generator. Rotating orbits of pendulum in stochastic excitation 721 A frequency range from 1.2Hz to 7.0Hz is selected, in the first instance, as a basis for a random time period generator. The reciprocal of the upper and lower limit of frequency determines the corresponding limits of the time interval between peaks of the input signal. This frequency range has been selected such that these values can be implemented in the experiment such that the pendulumis rotated.This couldmaximize the transmissionof vertical translational energy to the pendulum as it experiences an autoparametric resonance with the electromechanically-excited mass-spring-damper system. Fig. 2. Example of the input signal generated in a stochastic manner The profile of the random distribution can be selected such that the random time period is generated according to either a set of discrete frequencies or a continuous variation of the frequency with respect to the frequency range of interest. Nonetheless, it is intended in this work to attempt emulating stochastic oscillatory motion of the pendulum assembly that could represent the sea wave excitation. The sea wave can be modeled as a stochastic process with a continuous function that can represent the state of the sea in the formof an energy spectrum; the wave spectrum. The Pierson-Moskowitz spectrum (Pierson and Moskowitz, 1964) is considered in the current study as the profile of the randomdistribution for the randomnumber generator. It has been developed under the concept of a fully developed sea, and it is a one-parameter spectrum. The formulation of the Pierson-Moskowitz spectrum is given by S(ω)=A g2 ω5 exp ( − 16π3 ω4T4 z ) Tz =B √ Hs g (3.1) where A = 0.081, B = 11.1, ω is the frequency, and Hs is the significant wave height which is defined as one third of the highest wave observed. From Eqs. (3.1), the profile of the wave spectrum depends solely on the parameter Hs. The same formulation was used by Najdecka (2013) to reconstruct the time history of thewave for stimulating the response of the pendulum system subjected to the sea wave excitation bymeans of an electro-dynamical shaker. Figure 3a depicts the non-dimensionalizedPierson-Moskowitz spectrum.It is computed using Eqs. (3.1), and it features an asymmetric bell-shaped curve with a positive offset. It peaks at a nominal frequency, and that frequency can be expressed explicitly by equating to zero the derivative of S(ω) with respect to ω, which yields the following expression ω= 2 B √ 2g Hs 4 √ π3 5 (3.2) Rearranging Eq. (3.2), one obtain an expression forHs in terms of other parameters Hs = 8πg ω2B2 √ π 5 (3.3) 722 S.-H. Teh et al. Fig. 3. Non-dimensionalized Pierson-Moskowitzwave spectrum; (b) cumulative distribution function of the Pierson-Moskowitz wave spectrum The value of the parameterHs can thus be determined in order tomatch the nominal frequency of the wave spectrum as close as possible to the operating frequency of the pendulum system. Generation of a random time period based on the Pierson-Moskowitz spectrum is performed by using the inverse transform technique in this work. The inverse transform technique, also known as the Smirnov transform, is useful for generating random variates with an arbitrary continuous distribution function (Devroye, 1986). To utilize this technique, the selected random distribution is first integrated and then rescaled to obtain its cumulative distribution functionF, such that itmaps anumber in thedomain to aprobabilitybetween [0,1], as illustrated inFig. 3b. This technique works as follows. A random number u is generated from the standard uniform distribution in the interval of [0,1]. The generated u is then interpolated with the rescaled cumulative distribution function to locate the frequencyω. In other words, the inverse function ofF is sought todetermine the frequencyω that corresponds to the generated randomnumberu, with their relationship being expressed as follows ω=F−1(u) ∆t= 2π ω −T0 Tnew =T0+C1∆t (3.4) In Eq. (3.4)2, the random time period is decomposed into two components, namely the nominal time period T0 and the offset time period introduced by the wave spectrum ∆t. T0 is also the inverse of the nominal frequency of the wave spectrum. Finally, a factor within the range of [0,1]C1 is introduced to control the stochasticity of the random time period inEq. (3.4)3, which yields the new time period for that cycle. To generate a train of square pulses with different timing for switching the solenoid on and off in the experiment continuously, the above procedure can be repeated in a timer loop. For each iteration, the timer will start counting once a new time period is generated, and a square wave and the command signal will be computed based on that time period.The command signal generated by the computer system can be written as a=    1 if t(mod Tnew)< 1 2 Tnew 0 otherwise (3.5) where the units of the timer time t, the new time period Tnew is in seconds. By using the equations above, the generated input signal will be a train of square pulses with a fixed duty cycle of 50%. The timer will be reset automatically once the timer counts until Tnew. The time period for each iteration is then different, and so the on-and-off timing for each square wave in the pulse train, which is illustrated in Fig. 2. LabVIEW environment is used to program the above algorithm in this work. Rotating orbits of pendulum in stochastic excitation 723 Figure 4a depicts the time history of the vertical displacement of the pendulum support generated by the proposed stochastic excitation. It is observed at a root-mean-squared variac voltage of 110Vrms. The nominal frequency of the spectrum is set to be 1.9Hz, such that the corresponding value of Hs is 0.011m. The pendulum is constrained at its downright position during the measurement, and C1 is set to 1 to utilize the full scale of the Pierson-Moskowitz spectrum. From Fig. 4a, the amplitude of oscillation and time interval between two peaks is observed to be varying with time, which suggests that an irregular oscillation of the mass- spring-damper system could be generated by using the proposed stochastic excitation. TheFFT spectrumof the time history (line 1) in Fig. 4b suggests that the vertical oscillation is stochastic in nature. The energy content of the vertical oscillation lies within a range of 1.2Hz to 6Hz, and it also resembles a bell-shaped curve with a positive offset. The spectrum is smoothened (line 2) and it displays a major peak in close vicinity of the nominal frequency of the Pierson- Moskowitz spectrum. By comparing it with the Pierson-Moskowitz spectrum (line 3), it can be seen that the smoothened spectrum of the time history is in good correspondence with the desired power density profile with mild slight discrepancy between the shapes of the spectra, which is representative of many vibratory systems. The above observations show the viability of the excitation mechanism in the current study to attempt emulating a stochastic excitation on a solenoid-powered mass-spring-damper system. Fig. 4. (a) Time history of the vertical support excitation using the proposed stochastic excitation for V =130Vrms and ω0 =1.9Hz; (b) FFT spectrum of (a) compared with the Pierson-Moskowitz wave spectrum generated forH s =0.011m 4. Rotation control of pendulum A bang-bang control is suggested in the current study to promote rotational motion of the pendulumas it is subjected to a random excitation. The objective of control is to assist motion of thependulumdespite randomnessof thevertical excitations.Thenatureof thecontrol strategy is intuitive; while the pendulum is rotating, a control torquewill be exerted along the rotational direction if the control system senses that the pendulum is experiencing a destructive coupling torque arisen from the pendulum support that opposes the current motion of pendulum. The real-time information of the angular displacement and velocity of the pendulumand the vertical acceleration of the pendulum support are assigned as the feedback signal of the controller and some conditions are defined based on those signals to exert control action. The function of each 724 S.-H. Teh et al. signal is detailed as follows. The signal of angular displacement of the pendulum indicates the position of the pendulum from its downright equilibriumposition.On the other hand, the signal of angular velocity of the pendulum indicates the direction and the rate at which the pendulum moves. Lastly, the signal of vertical acceleration of the pendulumsupport indicates the direction of the summation of forces acting on the pendulum support. The rotating plane of the pendulum splits up into two halves, namely left-half plane and right-half plane, and they formcontrol envelopes.The conditions for activating the control action is different for each plane. For the left-half plane (i.e. 2nπ<θ< (2n+1)π, n∈N) UM(t)=          k1+k2 sinθ if z̈ > 0 ∧ 0¬ θ̇ < θ̇min −k3 if z̈ < 0 ∧ −θ̇min < θ̇< 0 0 otherwise (4.1) where UM is the input to the DC motor, k1 and k2 are control parameters applied when the pendulum is climbing from its downright equilibrium, and k3 is the control parameter applied when the pendulum is falling from its upright equilibrium position. Suppose that the pendulum is rotating in the left-half plane. The coupling torquewill be destructive in that plane if the pen- dulum support is suddenly accelerated upwards while the pendulum is rotating in the clockwise direction, and this may cause the pendulum to lose angular velocity and fall to its downright equilibrium position. Hence, a control torque will be exerted along the clockwise direction if the pendulum support is accelerated upwards and the rate at which the pendulum is rotating is lower than the threshold angular velocity θ̇min, as stated in the first condition of Eq.(4.1). A sinusoidal term is added to compensate for the torque due to gravity while the pendulum is climbing from its downright equilibrium position. On the other hand, the coupling torque will also be destructive in the left-half plane if the pendulum support is suddenly accelerated downwards while the pendulum is rotating in the counter-clockwise direction. A control torque of the counter-clockwise direction will then be exerted if the pendulum support is accelerated downwards and the rate at which the pendulum is rotating is greater than the negative thre- shold angular velocity, as stated in the second condition of Eq. (4.1). The control action will otherwise not be exerted in the left-half plane if the above conditions are notmet. Likewise, for the right-half plane (i.e. (2n+1)π<θ< 2(n+1)π, n∈N) UM(t)=          −(k1+k2 sinθ) if z̈ > 0 ∧ − θ̇min < θ̇< 0 k3 if z̈ < 0 ∧ 0¬ θ̇ < θ̇min 0 otherwise (4.2) Due to an inequality condition at θ̇ = 0, the control algorithm is also capable of initiating pendulum motion regardless of the initial conditions besides maintaining the rotation state of the pendulum, which is to be shown in Section 5. 5. Pendulum rotation subjected to the proposed stochastic excitation The experimental rig, as described in Section 2, is used to implement the proposed method of varying the on-and-off timing of the solenoid to induce stochastic excitations on the pendulum system. The pendulum system is tested at different levels of stochasticity by varying the on- and-off timing, and the observation of rotational state of pendulum under such excitation is particularly of interest in view of energy scavenging. The control algorithm, as described in Section 4, is used to drive the rotational actuator in attempting to assist the rotation of the pendulumwhen it could not be sustained. Rotating orbits of pendulum in stochastic excitation 725 An AC supply voltage of 130Vrms and a nominal frequency of 1.9Hz is chosen to generate experimental results in the current study. The selected parameters lie within the resonance area of the pendulum system, which is advantageous for generating rotation of the pendulum. The pendulum system is first excited by switching the RLC circuit of the solenoid on-and-off with a regular time interval. The control algorithm is then activated and the pendulumis actuated from its downright equilibrium position to yield period-1 rotation. A buffer period of approximately 30s is required to let the pendulummotion settle to period-1 rotation. After that, stochasticity is introduced to the excitation and the level of noise is controlled by manipulating C1 in Eq. (3.4)3. The pendulum system is run for a period of 300s, and useful experimental data is logged simultaneously. As the rotational process is stochastic, the measurement for the same noise level is to be repeated for another 9 sets to obtain the average accumulated net energy. The control parameters k1, k2, k3 and θ̇min are experimentally optimized and fixed at 8, 4, 1 and 7.5, respectively, for all noise levels throughout the experiment. Fig. 5. (a) Time response of the pendulum system forC1 =0; (b) FFT spectrum for vertical support excitation generated using experimental results; (c) phase portrait of the pendulum axis Figure 5a demonstrates a sample time history of the pendulum system when C1 = 0 for a period of 21s. In this case, the noise is disabled in the timer loop and the solenoid is switched on andoff in a regular time interval, which is 0.526s.Thus, the pendulumexhibits period-1 rotation as it experiences an autoparametric resonancewith the vertical support excitation, which is also periodic, after it is actuated from its rest position. The quality of the periodic excitation can be assessed via its FFT spectrum, which, see Fig. 5b, shows a considerably discrete spectrumwith amajor peak at the fundamental frequency and subsequentlyminor peaks at its integermultiple with reducing themagnitude. On the other hand, it can be seen from the phase portrait of the pendulum axis in Fig. 5c that the pendulum rotates with fluctuating angular velocity, which is bounded within the range of [7.9,17.2] rad/s, under the regular excitation from the pendulum support. As the source of energy to maintain stable rotation is solely from the vertical support 726 S.-H. Teh et al. excitation, no control action is required from the DC motor. The power of the generator will thus be the only contributor to the accumulated net energy at the end of the measurement, which serves as a benchmark for comparison with the accumulated net energy at higher noise levels. The period-1 rotation of the pendulum system ceases to be stable when C1 is increased up to 0.04. Hence, control assistance is necessarywhenC1 is set to bewithin the range of (0.04,1] in order tomaintain the rotational state of the pendulum. In the next case, the sample of the time history of the pendulum systemwhen it is subjected to stochastic excitation in amoderate scale (C1 =0.3) is illustrated in Fig. 6a. The time interval that the solenoid is switched on-and-off is varied, in this case, within the range of [0.411,0.617] s. Period-1 rotation could still be observed most of the time, as the occurrence of the moments when the vertical support excitation is less destructive. Consequently, less control effort is required in this case with only four instances of the control voltage input observed in this interval of the experimental data. FFT spectrum of the vertical support excitation in Fig. 6b reveals that the spectrum is no longer discrete, and it is stochastic in nature. The amplitude of the vertical excitation at the fundamental frequency becomes one-fourth of its periodic counterpart. The energy is slightly spread in the vicinity of the fundamental frequency as a consequence of slight variation in the on-and-off time interval. The shape of the spectrum, on the other hand, is somewhat similar to that of the Pierson- Moskowitz wave spectrumwith a narrower and sharper peak, as illustrated by the smoothened curve (line 2).Theangular velocity atwhich thependulumrotates, fluctuateswithin the range of [4.8,19.4] rad/s as depicted in Fig. 6c. It is wider than its periodic counterpart. The occasional randomness of the excitation causes the pendulum rotation to lose angular velocity at some instances, which is then rectified by the control action, and hence the lower limit of angular velocity range is extended when stochasticity is introduced to the excitation. Fig. 6. (a) Time response of the pendulum system forC1 =0.3; (b) FFT spectrum for vertical support excitation generated using experimental results; (c) phase portrait of the pendulum axis Rotating orbits of pendulum in stochastic excitation 727 The sample time history of the pendulum system when C1 = 1 is illustratedin Fig. 7a. In this case, the full scale of the Pierson-Moskowitz spectrum is used, and the time interval that the solenoid is switched on-and-off varies within the range of [0.143,0.830] s. It can be observed from the vertical axis data that the vertical support excitation is irregular at higher level of noise, which causes the pendulum trajectory to be intermittently distracted from its rotation attractor. This necessitates frequent control actions to assist the pendulum in regaining angular momentum and synchronizing again with the vertical support excitation. Accordingly, more control efforts are required at a higher level of noise, which is seen in the fifth row of Fig. 7a with 15 instances of the control voltage input observed in this interval of the experimental data. The contents of irregularities can be assessed via the FFT spectrum of the vertical support excitation in Fig. 7b. The smoothened spectrum (line 2) depicts a curve, and the shape of the spectrum is similar to that displayed in Fig. 3a. The spectrum peaks at the fundamental frequency, and the amplitude of the vertical support excitation at the fundamental frequency is lesser than in the previous cases. This is expected as the energy of the vertical excitation has been spread over a wider frequency range due to the maximum difference of 0.687s in varying the time interval. The vertical support excitation does not possess enough energy in the vicinity of the fundamental frequency to sustain the pendulum rotation. This prompts the pendulum to lose its angular velocity and almost stall at some point in time before the controller is alerted to react to sudden disturbances. This is seen in the phase portrait of the pendulum axis, where the fluctuation range of angular velocity is widened to [3.6,19.8] rad/s. Eaccumulated = ηKG t2 ∫ t1 IGθ̇ dt−KM t2 ∫ t1 IM θ̇ dt (5.1) Fig. 7. (a) Time response of the pendulum system forC1 =1; (b) FFT spectrum for vertical support excitation generated using experimental results; (c) phase portrait of the pendulum axis 728 S.-H. Teh et al. Energy generation is measured via current measurement from the generator, and the accu- mulated net energy is computed usingEq. (5.1) in real time during the course of the experiment in all the above cases. In Eq. (5.1), t1 is the time when computation of the accumulated net energy begins and t2 is the timewhen computation of the accumulated net energy ends. Figure 8 illustrates sample time histories of the accumulated net energy for the above three cases compu- ted over a period of t2− t1 =300s. In the case when the noise is disabled (C1 =0), pendulum rotation is sustainablewithout interference fromthe controller. Hence, the power generated from the DC generator is the sole contributor to computation of the accumulated net energy, and it can be seen from Fig. 8 that the accumulated net energy for C1 = 0 increases steadily as the experiment progresses. The average accumulated net energy at the end of 300s, obtained using 10 sets of data, is 24.51J forC1 =0.On the other hand, in the cases when noise is added to the vertical excitation, control effort is required to sustain the pendulumrotation formaximizing the energy output from the generator. Therefore, energy is invested for exerting the control effort by means of actuation of the DC motor and this is taken into consideration while computing the accumulated net energy. For the case when a moderate level of noise is introduced to the excitation (C1 =0.3), occasional control torque is required to maintain the pendulum rotation and some energy is invested to achieve this purpose. The average accumulated net energy at the end of 300s is 14.01J for C1 = 0.3, which is 57.16% of its periodic counterpart. For the case when the full scale of Pierson-Moskowitz spectrum is introduced to the excitation (C1 =1), the accumulated net energy is observed to be the least. Its average accumulated net energy at the end of 300s is calculated as 4.67J, which is 19.05% of the case when C1 = 0. This is expected asmore control input is necessary to prevent the pendulummotion from losing angular velocity when the pendulum is frequently subjected to the destructive vertical excitation. Fig. 8. Comparison of time histories of accumulated net energy forC1 =0,C1 =0.3 andC1 =1 6. Conclusion An experimental pendulum system,which exerts a non-linear vertical excitation on a pendulum via a RLC-circuit powered solenoid is used to conduct investigations on pendulum rotation subjected to a stochastic excitation. The pendulum system, which was previously reported in the open literature, is enhancedwith a control system by adding a rotational actuator and some necessary transducers. In addition, a generator is mounted in parallel to the pendulum axis for extracting energy out of rotating motion of the pendulum. The stochastic excitation of the pendulumsystem is accomplished by a variable time interval for switching the RLC-circuit on and off according to a random distribution. In particular, the Pierson-Moskowitz wave spectrum is selected as the random distribution, and the generation of the randomtime interval is implemented in the experimentusing an inverse transformtechnique. On the other hand, a bang-bang control algorithm is tailored to assist rotational motion of the pendulum as it is subjected to the random excitation. The main inputs of the controller are Rotating orbits of pendulum in stochastic excitation 729 the displacement and velocity of the angular axis and the acceleration of the vertical axis. The output is the command signal to the rotational actuator. Using the proposed method of stochastic excitation and control scheme, the experimental results are illustrated for various levels of noise introduced to the vertical support excitation. 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