In this paper energy harvesting from vibrating surfaces through electromagnetic induction is addressed. A double pendulum subject to base excitations generates electrical energy through energy harvesting coils mounted on the pendulum masses. Optimum double pendulum generates energy at a rate of 9 mW for excitation parameters characteristic of bridge vibration, frequency of 2 Hz and amplitude of 1 mm

Chunawala, Taizoon

Ghandchi Tehrani, Maryam

Yan, Jize

Southampton (e-Prints Soton)

An optimum design of a double pendulum in autoparametric resonance for energy harvesting applications  Taizoon Chunawala1, Maryam Ghandchi-Tehrani2, Jize Yan2 1Birla Institute of Technology and Science-Pilani, Pilani, India 2University of Southampton, Southampton, UK  E-mail: 1taizoon.chunawala@gmail.com, 2 m.ghandchi-tehrani@soton.ac.uk, 2j.yan@soton.ac.uk    Abstract. In this paper energy harvesting from vibrating surfaces through electromagnetic induction is addressed. A double pendulum subject to base excitations generates electrical energy through energy harvesting coils mounted on the pendulum masses. Optimum double pendulum generates energy at a rate of 9mW for excitation parameters characteristic of bridge vibration, frequency of 2 Hz and amplitude of 1mm.   Keywords: energy harvesting, double pendulum, autoparametric resonance, in-phase, out-of-phase  1. Introduction  Energy harvesting is a process of transforming ambient vibrational kinetic energy into useful electrical energy to power devices and sensors with modest energy requirements. If implemented successfully, it can save time and costs required to maintain the batteries currently being used to power sensors. Systems in which the force of excitation depends the upon a system parameter are called parametrically excited systems. Rand in [1], discusses the most basic parametrically excited system, Mathieu equation. It is shown that the amplitude of oscillation of pendulum grows exponentially when the forcing frequency is twice the natural frequency of the pendulum. Jial and Seshia in [4] show unlike the linear resonance, oscillatory amplitude growth in parametric resonance is not limited by linear damping. Zaghari, et al in [2] showed that a small excitation can produce a large response when the system is parametrically excited with a frequency near to twice of its fundamental frequency. Krzysztof Kecik [3] proposed conception of simultaneous elimination of vibration and energy induction by a pendulum motion. Michal Marszal et al in [6] derived the double pendulum natural frequency for the out-of-phase and in-phase modes. A.C. Skeldon explored parametrically excited double pendulum in [5] found a range of complicated dynamical phenomena in a region where the excitation frequency was close to twice the natural frequency. The work presented aims at harvesting energy by harnessing the advantages offered by autoparametric resonance.   2. Double pendulum governing equation  The double pendulum is the simplest mechanical apparatus that exhibits a range of dynamic responses from periodic oscillations to chaotic rotations. This utility can be harnessed to maximize harvest energy from a double pendulum subject to base excitation.  Figure 1 Schematic of an energy harvesting double pendulum subject to base excitation facing a magnet wall. Consider the double pendulum shown in Figure 1. Let the two pendulum lengths be L1 and L2. The pendulum masses are cylindrical coils of radius r and masses M1 and M2 and pendulum link masses m1 and m2. The generalized co-ordinates for the double pendulum system are the angles made by the two pendulum links with the vertical,  1 and  2. The double pendulum is excited by surface vibrations with amplitude A and frequency Ω. The governing equation of motion for a dissipative system can be derived using the Euler-Lagrangian equation given by, 𝐸𝑂𝑀𝑖 =𝑑𝑑𝑡𝑑𝐿𝑑𝑞?̇?−𝑑𝐿𝑑𝑞𝑖+𝑑𝑅𝑑𝑞?̇?= 0 (1) where, Lagrangian function, L is the difference between the system kinetic energy and potential energy, L=T-U, R is the Rayleigh dissipation function and qi is the system generalized coordinate. The system kinetic energy, T, is obtained by differentiating mass co-ordinates with time,  =   𝑀 𝐿    ̇(𝑡) + 𝑀 𝐿    ̇(𝑡) +  𝑀 𝐿   ̇(𝑡)𝐿   ̇(𝑡)  𝑠(  (𝑡) −   (𝑡)) + 𝑀 𝐿    ̇(𝑡)   +       𝐿    ̇(𝑡) +  𝐿 (𝐿  𝐿   ̇(𝑡) +𝐿     ̇(𝑡) + 𝐿 𝐿    ̇(𝑡)  ̇(𝑡)  𝑠(  (𝑡) −   (𝑡)))  (2) Potential energy of the system assigning pivot point zero potential energy reference is given by:  = −𝑀  𝐿   𝑠(  (𝑡)) −   𝐿    𝑠(  (𝑡)) − 𝑀  (𝐿   𝑠(  (𝑡)) + 𝐿   𝑠(  (𝑡)))−    ( 𝐿   𝑠(  (𝑡)) + 𝐿   𝑠(  (𝑡))) (3) Frictional torque at the bearing as given in reference [8] is given by,  =   𝑑    where,   is the coefficient of friction, d is the bearing bore diameter and F is the dynamic radial load on the bearing. The Rayleigh dissipation function for frictional damping, 𝑅  =     −  ((𝑀 ( +   ̇(𝑡) 𝐿 ) + 𝑀  ( +   ̇(𝑡) 𝐿  ))  ̇(𝑡)+ (𝑀 ( +   ̇(𝑡) 𝐿  )(  ̇(𝑡) −   ̇(𝑡))) (4) Viscous air drag on the coil is approximated as the viscous force acting on a sphere of the same radius. The air drag force acting on a sphere of radius r, moving with a velocity v, is given by,  =    𝑟 , where,   is dynamic coefficient of viscosity, 2e-5 kg/ms for air. Substituting  =   𝑟 =       −  , the Rayleigh dissipation function to incorporate viscous losses, 𝑅  =  (  𝐿    ̇(𝑡) +   (𝐿    ̇(𝑡) + 𝐿    ̇(𝑡) +  𝐿 𝐿   ̇(𝑡)  ̇(𝑡))) (5) Power harvested by induced electromagnetic voltage is given by the equation, 𝑃 =−(𝑁     ( (𝑡))  ̇(𝑡))  𝑅, where, N is the number of loops in the coil, B is the magnetic field strength, A is the area of enclosed by each coil,   is the angle between the normal to the coil area vector and magnetic field lines, R is the coil resistance. Taking 𝑟 = (𝑁  )  𝑅, Rayleigh dissipation function for electromagnetic damping, 𝑅 =  (𝑟 𝑠  (  (𝑡))   ̇(𝑡) +𝑟 𝑀 𝑀 𝑠  (  (𝑡))   ̇(𝑡) ) (6) Total Rayleigh function, 𝑅 = 𝑅  + 𝑅  + 𝑅 . Following constants are substituted for the corresponding ratios to non-dimensionalize expressions and simplify notation,  =𝐿 𝐿      =𝑀 𝑀       =  𝑀       =  𝑀  Non-dimensional equations governing the two degrees of freedom  1(t) and  2(t) are given by,   ̇(𝑡) 𝑠  (  (𝑡) −   (𝑡))  +     ̈(𝑡)  𝑠(  (𝑡) −   (𝑡)) +   ̈(𝑡)( +  )+ ( +  )𝐿 𝑠  (  (𝑡)) + (𝑃 − 𝑃 )𝑝𝑀 𝐿  +𝑟 𝑠  (  (𝑡))   ̇(𝑡)𝑀 𝐿  +     ̇(𝑡)𝑀 +   ̇(𝑡)  𝑠(  (𝑡) −   (𝑡))  𝑀 +   ̈(𝑡)(   +   ) +      ̈(𝑡)  𝑠(  (𝑡)−   (𝑡)) +      ̇(𝑡) 𝑠  (  (𝑡) −   (𝑡)) − (    𝐿 +   𝐿 𝑠  (  (𝑡))− Ω 𝑠  (𝑡)𝑠  (  (𝑡))𝐿           =           0 (7)  −  ̇(𝑡) 𝑠  (  (𝑡) −   (𝑡))  +     ̈(𝑡)  𝑠(  (𝑡) −   (𝑡)) +      ̈(𝑡) +   𝐿 𝑠  (  (𝑡))+ 𝑃 𝑝𝑀 𝐿  +𝑟  𝑠  (  (𝑡)) 𝑀 𝐿    ̇(𝑡) +   ̇(𝑡)  𝑠(  (𝑡) −   (𝑡))  𝑀  +   ̇(𝑡)   𝑀 +       ̈(𝑡) +      ̈(𝑡)  𝑠(  (𝑡) −   (𝑡))−      ̇(𝑡) 𝑠  (  (𝑡) −   (𝑡)) −     𝐿 𝑠  (  (𝑡))−   Ω 𝐿 𝑠  (𝑡)𝑠  (  (𝑡))             =          0 (8)  3. Numerical simulations  A standard copper coil with dimensions and wire properties listed in Table 1 is taken as pendulum one mass M1. Working out the electromagnetic damping coefficient and mass from the dimensions gives us, rh=0.00056kg/s and M1=1.6kg. Since the coil is massive, the ratio link mass to coil mass can be approximated to zero,   =   = 0   Table 1 Standard coil dimensions Coil ID Coil OD Wire diameter(d) Turns(N) Resistivity(  ) Wire density (  ) Magnetic field(B) 23mm 40mm 1.4mm 1100 1.7e-9Ω/m2 8900kg/m3 0.01T/m2  Natural frequency of the double pendulum for small oscillation amplitudes is obtained by implementing method of harmonic balance as employed by Roy et. al in  [7]. Neglecting damping and forcing in equations 7 and 8, the equations reduce to,     ̇(𝑡) 𝑠  (  (𝑡) −   (𝑡)) +     ̈(𝑡)  𝑠(  (𝑡) −   (𝑡)) +   ̈(𝑡)( +  )+ ( +  )𝑠  (  (𝑡))𝐿             =         0 (9)  −    ̇(𝑡) 𝑠  (  (𝑡) −   (𝑡)) +     ̈(𝑡)  𝑠(  (𝑡) −   (𝑡)) +      ̈(𝑡)+   𝑠  (  (𝑡))𝐿                     =          0 (10) Trigonometric ratios sine and cosine are expanded to first two terms. Substituting   =   𝑠(  𝑡) and   =    𝑠(  𝑡) where    is the natural frequency of double pendulum, higher powers of trigonometric ratios are expanded as linear combinations of higher harmonics. Coefficient of   𝑠(  𝑡) in the resulting equations is equated to zero to remove secular terms to obtain,   −     𝐿      −    𝐿      −     𝐿      −       −   𝐿     −      −   𝐿    +    ( +  ) +        𝐿      +    𝐿      +     𝐿      +     𝐿      −   𝐿        =  0 (11)       𝐿      +      𝐿      +     𝐿      +     𝐿      −   𝐿       +   −   𝐿      −      𝐿      −     𝐿      −   𝐿       −        +       = 0 (12) Equations 11 and 12 are treated as two linear conditions on the amplitude of oscillation C and D, and for consistency demand that, −     𝐿      −    𝐿      −    𝐿      −       −   𝐿     −     −   𝐿    +    ( +  )       𝐿      +    𝐿      +    𝐿      +     𝐿      −   𝐿           𝐿      +      𝐿      +    𝐿      +     𝐿      −   𝐿        −   𝐿      −      𝐿      −    𝐿      −   𝐿       −        +                               =                            0  (13) When solved for frequency,    , two solutions obtained correspond to in-phase oscillation   (0) = C,   (0) =D and out-of-phase oscillation   (0)=C,   (0)=-D. In-line with the small angles of oscillation assumption, substituting C=D=0.3 in Equation 13, natural frequencies of the two modes of oscillation,      = (   (     +      +     +       √(       +      ( +  ) +    (  +   +  ) − 0     −    ( −  ))) ((      +       +       +      +     )𝐿 )      (14) In Figure 3, energy harvested is plotted against the frequency ratio, Ω/   for in-phase and Ω/   for out-of-phase oscillating double pendulum. Pendulum length, L1 is changed to satisfy the forcing frequency to natural frequency ratio, n, according to, 𝐿 = (   (     +      +     +        √(       +      ( +  ) +    (  +   +  ) − 0     −    ( −  ))) ((      +       +       +      +     )Ω )   (15)   Figure 2 Energy harvested in 100 seconds,1<n<5 for two modes of oscillation, a=1, b=1, A=1mm, Ω=10πrad/s Distinct energy peak is seen at frequency ratio 2 in both Figures 2a and 2b for the two modes of oscillation. The analysis below, attempts to identify optimum a and b to harvest maximum energy from a source vibrating with fixed amplitude and frequency. The autoparametrically resonant pendulum link length, L1, is modified as a and b vary according to Equation 15 for n=2. In the Figures 3a and 3b energy harvested in 100 seconds for steady state oscillations is plotted along the Z-axis for in-phase and out-of-phase oscillations respectively. a and b vary over the domain,0.1 < a, b < 2.5.              Figure 3 Energy harvested in 100 seconds by autoparametrically resonant double pendulum over the a-b domain, A=1mm, Ω=10πrad/s The ideal energy harvester, autoparametric in-phase oscillating double pendulum cannot be realized for practical implementation as seen readily by the dimensions given in Table 3. The resonant out-of-phase oscillating double pendulum is a more pragmatic alternative.  Table 2 Dimension of optimum double pendulum for the two modes of oscillation   L1 M1 L2 M2 Power Ω =     1.25cm 1.6kg 3.75cm 2.24kg 30mW Ω =     31cm 1.6kg 10cm 3.5kg 9mW   (a)Ω=2𝜔  (b) Ω=2𝜔  (a)In-phase oscillation (b)Out-of-phase oscillation  Figure 4 Energy harvested in 100 seconds by double pendulum with optimal a and b for the two modes of oscillation over realisable L1.  Amplitude of in-phase oscillation of the two angles,    and   , for practically realizable link lengths decays harvesting negligible steady state energy as seen by the blue plot in Fig. 4. In-case of out-of-phase initial conditions (red plot in Fig. 4) Eh plot rises exactly at autoparametrically resonant L1=31cm, generating 9mW in steady state validating importance of autoparametric resonance. The Eh peak at L1=15cm corresponds to out-of-phase 1:4 resonance. Maximizing energy harvested requires a comprehensive study of such variety of responses, especially so for high frequency base excitation where attaining resonance entails an even shorter pendulum link length.  4. Conclusions  The in-phase oscillating pendulum harvests energy at a rate of 30mW, more than five times that of an equivalent autoparametrically resonant simple pendulum, for modest excitation parameters, A=1mm, Ω= 10πrad/s. Pendulum one link length obtained for in-phase resonance is too short for practical implementation. While, the optimum out-of-phase oscillating double pendulum, harvests energy at 9mW, which might just be sufficient to power maintenance sensors, the dimensions obtained for double pendulum are realizable, making out-of-phase oscillation a more desirable mode of oscillation for energy harvesting. Application specific design may help design by adding constrains on more equation parameters, which can help us in optimizing the remaining parameters more comprehensively giving better results.  References  [1] Richard Rand, http://www.math.cornell.edu/rand/randdocs/nlvibe52.pdf, May 2016. [2] B. Zaghari, M. Ghandchi Tehrani, E. Rustighi, Mechanical modeling of a vibration energy harvester with time-varying stiffness, 9th International Conference on Structural Dynamics, EURODYN 2014, 2079-2085. [3] K. Kecik, M. Borowiec, An autoparametric energy harvester, European Physical Journal Special Topics 222, 2013, 1597-1605. [4] Y. Jia, A.A. Seshia, A parametrically excited vibration energy harvester, Journal of Intelligent Material Systems and Structures, 2014, Vol 25(3), 278-289. [5] Y. Jia, A.A. Seshia, Directly and parametrically excited bi-stable vibration energy harvester for bistable operation, The 17th International Conference on Solid-State Sensors, Actuators and Microsystems (Transducers and Eurosensors XXVII), June 2013, 454-457. [6] Michal Marszal, Krzysztof Jankowski, Przemyslaw Perlikowski, Tomasz Kapitaniak, Bifurcations of oscillatory and rotational solutions of double pendulum with parametric vertical excitation,Mathematical Problems in Engineering Volume 2014 (2014), Article ID 892793, 9 pages [7] Jyotirmoy Roy, Asok K. Mallik, Jayanta K. Bhattacharjee, Role of initial conditions in the dynamics of a double pendulum at low energies, Nonlinear Dyn (2013) Volume 73, 993-1004. [8] www.roymech.co.uk/UsefulTables/Tribology/Bearing.html, May 2016.  

An optimum design of a double pendulum in autoparametric resonance for energy harvesting applications

Taizoon Chunawala

Maryam Ghandchi-Tehrani

Jize Yan

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 163 
An optimum design of a double pendulum in 
autoparametric resonance for energy harvesting 
applications 
Taizoon Chunawala1, Maryam Ghandchi-Tehrani2, Jize Yan2 
1Birla Institute of Technology and Science-Pilani, Pilani, India 
2, 3University of Southampton, Southampton, UK 
1Corresponding author 
E-mail: 1taizoon.chunawala@gmail.com, 2m.ghandchi-tehrani@soton.ac.uk, 3j.yan@soton.ac.uk 
(Received 7 September 2016; accepted 14 September 2016) 
Abstract. In this paper energy harvesting from vibrating surfaces through electromagnetic 
induction is addressed. A double pendulum subject to base excitations generates electrical energy 
through energy harvesting coils mounted on the pendulum masses. Optimum double pendulum 
generates energy at a rate of 9 mW for excitation parameters characteristic of bridge vibration, 
frequency of 2 Hz and amplitude of 1 mm. 
Keywords: energy harvesting, double pendulum, autoparametric resonance, in-phase, 
out-of-phase. 
1. Introduction 
Energy harvesting is a process of transforming ambient vibrational kinetic energy into useful 
electrical energy to power devices and sensors with modest energy requirements. If implemented 
successfully, it can save time and costs required to maintain the batteries currently being used to 
power sensors. 
Systems in which the force of excitation depends the upon a system parameter are called 
parametrically excited systems. Rand in [1], discusses the most basic parametrically excited 
system, Mathieu equation. It is shown that the amplitude of oscillation of pendulum grows 
exponentially when the forcing frequency is twice the natural frequency of the pendulum. Jial and 
Seshia in [4] show unlike the linear resonance, oscillatory amplitude growth in parametric 
resonance is not limited by linear damping. Zaghari, et al. in [2] showed that a small excitation 
can produce a large response when the system is parametrically excited with a frequency near to 
twice of its fundamental frequency. Krzysztof Kecik [3] proposed conception of simultaneous 
elimination of vibration and energy induction by a pendulum motion. Michal Marszal et al in [6] 
derived the double pendulum natural frequency for the out-of-phase and in-phase modes. 
A. C. Skeldon explored parametrically excited double pendulum in [5] found a range of 
complicated dynamical phenomena in a region where the excitation frequency was close to twice 
the natural frequency. The work presented aims at harvesting energy by harnessing the advantages 
offered by autoparametric resonance.  
2. Double pendulum governing equation 
The double pendulum is the simplest mechanical apparatus that exhibits a range of dynamic 
responses from periodic oscillations to chaotic rotations. This utility can be harnessed to maximize 
harvest energy from a double pendulum subject to base excitation. 
Consider the double pendulum shown in Fig. 1. Let the two pendulum lengths be ܮଵ and ܮଶ. 
The pendulum masses are cylindrical coils of radius ݎ and masses ܯଵ and ܯଶ and pendulum link 
masses ݉ଵ and ݉ଶ. The generalized co-ordinates for the double pendulum system are the angles 
made by the two pendulum links with the vertical, ߠଵ and ߠଶ. The double pendulum is excited by 
surface vibrations with amplitude ܣ and frequency Ω. The governing equation of motion for a 
dissipative system can be derived using the Euler-Lagrangian equation given by: 
AN OPTIMUM DESIGN OF A DOUBLE PENDULUM IN AUTOPARAMETRIC RESONANCE FOR ENERGY HARVESTING APPLICATIONS.  
TAIZOON CHUNAWALA, MARYAM GHANDCHI-TEHRANI, JIZE YAN 
164 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
ܧܱܯ௜ =
݀
݀ݐ
݀ܮ
݀ݍపሶ −
݀ܮ
݀ݍ௜ +
ܴ݀
݀ݍపሶ = 0, (1)
where, Lagrangian function, ܮ is the difference between the system kinetic energy and potential 
energy, ܮ = ܶ − ܷ , ܴ  is the Rayleigh dissipation function and ݍ௜  is the system generalized 
coordinate. The system kinetic energy, ܶ, is obtained by differentiating mass co-ordinates with 
time: 
ܶ = 12 [ܯଵܮଵ
ଶߠଵሶ (ݐ)ଶ + ܯଶܮଵଶߠଵሶ (ݐ)ଶ + 2ܯଶܮଵߠଵሶ (ݐ)ܮଶߠଶሶ (ݐ)cos൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ + ܯଶܮଶଶ ߠଶሶ (ݐ)ଶ] 
 + 12 ൥
1
3 ݉ଵܮଵ
ଶߠଵሶ (ݐ)ଶ +
݉ଶ
ܮଶ ൭ܮଵ
ଶܮଶߠଵሶ (ݐ)ଶ +
ܮଶଷ
3 ߠଶሶ (ݐ)
ଶ + ܮଵܮଶଶ ߠଵሶ (ݐ)ߠଶሶ (ݐ)ܿ݋ݏ൫ߠଵ(ݐ) − ߠଶ(ݐ)൯൱൩. 
(2)
 
Fig. 1. Schematic of an energy harvesting double pendulum subject  
to base excitation facing a magnet wall 
Potential energy of the system assigning pivot point zero potential energy reference is given 
by: 
ܷ = −ܯଵ݃ܮଵcos(ߠଵ(ݐ)) −
݉ଵ݃ܮଵ
2 cos(ߠଵ(ݐ)) − ܯଶ݃(ܮଵcos(ߠଵ(ݐ)) + ܮଶcos(ߠଶ(ݐ))) 
      − ݉ଶ݃2 ቀ2ܮଵcos൫ߠଵ(ݐ)൯ + ܮଶcos൫ߠଶ(ݐ)൯ቁ.
(3)
Frictional torque at the bearing as given in reference [8] is given by, ܶ = ߤܨ݀/2, where, ߤ is 
the coefficient of friction, ݀ is the bearing bore diameter and ܨ is the dynamic radial load on the 
bearing. The Rayleigh dissipation function for frictional damping: 
ܴௗଵ = 1.5݁ − 5((ܯଵ(݃ + ߠଵሶ (ݐ)ଶܮଵ) + ܯଶ (݃ + ߠଶሶ (ݐ)ଶܮଵܽ))ߠଵሶ (ݐ)
      + ൬ܯଶ൫݃ + ߠଶሶ (ݐ)ଶܮଵܽ൯ ቀߠଶሶ (ݐ) − ߠଵሶ (ݐ)ቁ൰. (4)
Viscous air drag on the coil is approximated as the viscous force acting on a sphere of the same 
radius. The air drag force acting on a sphere of radius ݎ, moving with a velocity ݒ, is given by, 
ܨ = 6ߨߟݎݒ , where, ߟ  is dynamic coefficient of viscosity, 2e-5 kg/ms for air. Substituting  
ܿ = ߨߟݎ = 12.56e-7, the Rayleigh dissipation function to incorporate viscous losses: 
ܴௗଶ =
1
2 ൬6ܿܮଵ
ଶߠଵሶ (ݐ)ଶ + 6ܿ ቀܮଵଶߠଵሶ (ݐ)ଶ + ܮଶଶ ߠଶሶ (ݐ)ଶ + 2ܮଵܮଶߠଵሶ (ݐ)ߠଶሶ (ݐ)ቁ൰. (5)
Power harvested by induced electromagnetic voltage is given by the equation,  
௛ܲ = − ቀܰܤܣ sin൫ߠ(ݐ)൯ ߠሶ(ݐ)ቁ
ଶ /ܴ, where, ܰ is the number of loops in the coil, ܤ is the magnetic 
AN OPTIMUM DESIGN OF A DOUBLE PENDULUM IN AUTOPARAMETRIC RESONANCE FOR ENERGY HARVESTING APPLICATIONS.  
TAIZOON CHUNAWALA, MARYAM GHANDCHI-TEHRANI, JIZE YAN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 165 
field strength, ܣ is the area of enclosed by each coil, ߠ is the angle between the normal to the coil 
area vector and magnetic field lines, ܴ is the coil resistance. Taking ݎ௛ = (ܰܤܣ)ଶ/ܴ, Rayleigh 
dissipation function for electromagnetic damping: 
ܴ௛ =
1
2 ൬ݎ௛sin൫ߠଵ(ݐ)൯
ଶߠଵሶ (ݐ)ଶ +
ݎ௛ܯଶ
ܯଵ sin൫ߠଶ(ݐ)൯
ଶߠଶሶ (ݐ)ଶ൰. (6)
Total Rayleigh function, ܴ = ܴௗଵ + ܴௗଶ + ܴ௛ . Following constants are substituted for the 
corresponding ratios to non-dimensionalize expressions and simplify notation: 
ܽ = ܮଶܮଵ ,    ܾ =
ܯଶ
ܯଵ ,   ߟଵ =
݉ଵ
ܯଵ , ߟଶ =
݉ଶ
ܯଵ.
Non-dimensional equations governing the two degrees of freedom ߠଵ(ݐ) and ߠଶ(ݐ) are given 
by: 
ߠଵሶ (ݐ)ଶsin൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ܾܽ + ܾܽߠଶሷ (ݐ)cos൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ + ߠଵሷ (ݐ)(1 + ܾ)
      + ݃(1 + ܾ)ܮଵ sin൫ߠଵ(ݐ)൯ +
ߤ( ଵܲ − ଶܲ)݌
ܯଵܮଵଶ +
ݎ௛sin൫ߠଵ(ݐ)൯ଶߠଵሶ (ݐ)
ܯଵܮଵଶ +
12ܿߠଵሶ (ݐ)
ܯଵ  
      + 6ߠଵ
ሶ (ݐ)cos൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ܿܽ
ܯଵ + ߠଵ
ሷ (ݐ) ቀߟଵ3 + ߟଶቁ +
ߟଶܽ
2 ߠଶሷ (ݐ)cos(ߠଵ(ݐ) − ߠଶ(ݐ)) 
      + ߟଶܽ2 ߠଵሶ (ݐ)
ଶsin(ߠଵ(ݐ) − ߠଶ(ݐ)) − ቆ
ߟଵ݃
2ܮଵ +
ߟଶ݃
ܮଵ sinߠଵ(ݐ)ቇ −
ܣΩଶsin(ݐ)sin൫ߠଵ(ݐ)൯
ܮଵ = 0, 
(7)
−ߠଵሶ (ݐ)ଶsin൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ܾܽ + ܾܽߠଵሷ (ݐ)cos൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ + ܽଶܾߠଶሷ (ݐ)
      + ܾܽ݃ܮଵ sin൫ߠଶ(ݐ)൯ +
ߤ ଶܲ݌
ܯଵܮଵଶ +
ݎ௛ܾsin൫ߠଶ(ݐ)൯ଶ
ܯଵܮଵଶ ߠଵ
ሶ (ݐ) + 6ߠଵ
ሶ (ݐ)cos൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ܿܽ
ܯଵ  
      + 6ߠଵ
ሶ (ݐ)ܿܽଶ
ܯଵ +
ߟଶܽଶ
3 ߠଶሷ (ݐ) +
ߟଶܽ
2 ߠଵሷ (ݐ)cos൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ 
      − ߟଶܽ2 ߠଵሶ (ݐ)
ଶsin൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ −
ߟଶ݃ܽ
2ܮଵ sin൫ߠଶ(ݐ)൯ −
ܣܾܽΩଶ
ܮଵ sin(ݐ)sin൫ߠଶ(ݐ)൯ = 0. 
(8)
3. Numerical simulations 
A standard copper coil with dimensions and wire properties listed in Table 1 is taken as 
pendulum one mass ܯଵ. Working out the electromagnetic damping coefficient and mass from the 
dimensions gives us, ݎ௛ = 0.00056 kg/s and ܯଵ = 1.6 kg. Since the coil is massive, the ratio link 
mass to coil mass can be approximated to zero, ߟଵ = ߟଶ = 0. 
Natural frequency of the double pendulum for small oscillation amplitudes is obtained by 
implementing method of harmonic balance as employed by Roy et. al in [7]. Neglecting damping 
and forcing in Eqs. (7-8), the equations reduce to: 
ܾܽߠଶሶ (ݐ)ଶsin൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ + ܾܽߠଶሷ (ݐ)cos൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ + ߠଵሷ (ݐ)(1 + ܾ)
      + ݃(1 + ܾ)sin൫ߠଵ(ݐ)൯ܮଵ = 0,
(9)
−ܾܽߠଶሶ (ݐ)ଶsin൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ + ܾܽߠଵሷ (ݐ)cos൫ߠଵ(ݐ) − ߠଶ(ݐ)൯ + ܽଶܾߠଶሷ (ݐ)
      + ܾ݃ܽsin൫ߠଶ(ݐ)൯ܮଵ = 0.
(10)
Trigonometric ratios sine and cosine are expanded to first two terms. Substituting  
AN OPTIMUM DESIGN OF A DOUBLE PENDULUM IN AUTOPARAMETRIC RESONANCE FOR ENERGY HARVESTING APPLICATIONS.  
TAIZOON CHUNAWALA, MARYAM GHANDCHI-TEHRANI, JIZE YAN 
166 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
ߠଵ = ܥcos(߱௡ݐ) and ߠଶ = ܦcos(߱௡ݐ) where ߱௡  is the natural frequency of double pendulum, 
higher powers of trigonometric ratios are expanded as linear combinations of higher harmonics. 
Coefficient of cos(߱௡ݐ) in the resulting equations is equated to zero to remove secular terms to 
obtain: 
ܥ[−2ܥଶܦଶܮଵܾܽ߱௡ଶ − 6ܦସܮଵܾܽ߱௡ଶ − 48ܦଶܮଵܾܽ߱௡ଶ − 12ܥଶܾ݃ − 96ܮଵܾ߱௡ଶ − 12ܥଶ݃ 
      −96ܮଵ߱௡ଶ + 96݃(ܾ + 1)] + ܤ[6ܥଶܦଶܮଵܾܽ߱௡ଶ + 2ܦସܮଵܾܽ߱௡ଶ + 36ܥଶܮଵܾܽ߱௡ଶ
      +12ܦଶܮଵܾܽ߱௡ଶ − 96ܮଵܾܽ߱௡ଶ] = 0,
(11)
ܥ[2ܥସܮଵܾܽ߱௡ଶ + 6ܥଶܦଶܮଵܾܽ߱௡ଶ + 12ܥଶܮଵܾܽ߱௡ଶ + 36ܦଶܮଵܾܽ߱௡ଶ − 96ܮଵܾܽ߱௡ଶ]
      +ܦ[−6ܥସܮଵܾܽ߱௡ଶ − 2ܥଶܦଶܮଵܾܽ߱௡ଶ − 48ܥଶܮଵܾܽ߱௡ଶ − 96ܮଵܽଶܾ߱௡ଶ
      −12ܦଶܾܽ݃ + 96ܾܽ݃] = 0.
(12)
Table 1. Standard coil dimensions 
Coil 
ID 
Coil 
OD 
Wire diameter 
(݀) 
Turns 
(ܰ) 
Resistivity 
(ߩଵ) 
Wire density 
(ߩଶ) 
Magnetic field 
(ܤ) 
23 mm 40 mm 1.4 mm 1100 1.7e-9 Ω/m2 8900 kg/m3 0.01 T/m2 
Eqs. (11-12) are treated as two linear conditions on the amplitude of oscillation ܥ and ܦ, and 
for consistency demand that: 
−2ܥଶܦଶܮଵܾܽ߱௡ଶ − 6ܦସܮଵܾܽ߱௡ଶ − 48ܦଶܮଵܾܽ߱௡ଶ − 12ܥଶܾ݃ − 96ܮଵܾ߱௡ଶ − 12ܥଶ݃
      −96ܮଵ߱௡ଶ + 96݃(ܾ + 1) = 0,
6ܥଶܦଶܮଵܾܽ߱௡ଶ + 2ܦସܮଵܾܽ߱௡ଶ + 36ܥଶܮଵܾܽ߱௡ଶ + 12ܦଶܮଵܾܽ߱௡ଶ − 96ܮଵܾܽ߱௡ଶ = 0, 
2ܥସܮଵܾܽ߱௡ଶ + 6ܥଶܦଶܮଵܾܽ߱௡ଶ + 12ܥଶܮଵܾܽ߱௡ଶ + 36ܦଶܮଵܾܽ߱௡ଶ − 96ܮଵܾܽ߱௡ଶ = 0,
−6ܥସܮଵܾܽ߱௡ଶ − 2ܥଶܦଶܮଵܾܽ߱௡ଶ − 48ܥଶܮଵܾܽ߱௡ଶ − 96ܮଵܽଶܾ߱௡ଶ − 12ܦଶܾܽ݃ + 96ܾܽ݃ = 0. 
(13)
When solved for frequency, ߱௡ , two solutions obtained correspond to in-phase oscillation 
ߠଵ(0) = ܥ , ߠଶ(0) = ܦ  and out-of-phase oscillation ߠଵ(0) = ܥ , ߠଶ(0) = −ܦ . In-line with the 
small angles of oscillation assumption, substituting ܥ = ܦ = 0.3 in Eq. (13), natural frequencies 
of the two modes of oscillation: 
߱ଶ,ଵଶ = (57݃(1.2ܾܽ + 1.15ܽ + 1.2ܾ + 1.2 
      ± ඥ(1.2ܽଶܾଶ + 2.5ܾܽ(ܽ + ܾ) + 1.2(ܽଶ + ܾଶ + 1) − 0.11ܾܽ − 2.5(ܽ − ܾ))ቁ 
      /((5.8ܽଶܾ + 11.8ܾܽ + 132.7ܽ + 5.83ܾ + 5.83)ܮଵ).
(14)
In Fig. 3, energy harvested is plotted against the frequency ratio, Ω/߱ଵ for in-phase and Ω/߱ଶ 
for out-of-phase oscillating double pendulum. Pendulum length, ܮଵ  is changed to satisfy the 
forcing frequency to natural frequency ratio, ݊, according to: 
ܮଵ = (57݃(1.2ܾܽ + 1.15ܽ + 1.2ܾ + 1.2
      ±ඥ(1.2ܽଶܾଶ + 2.5ܾܽ(ܽ + ܾ) + 1.2(ܽଶ + ܾଶ + 1) − 0.11ܾܽ − 2.5(ܽ − ܾ))ቁ
      /൫(5.8ܽଶܾ + 11.8ܾܽ + 132.7ܽ + 5.83ܾ + 5.83)Ωଶ൯݊ଶ.
(15)
Distinct energy peak is seen at frequency ratio 2 in both Figs. 2(a) and 2(b) for the two modes 
of oscillation. The analysis below, attempts to identify optimum ܽ and ܾ to harvest maximum 
energy from a source vibrating with fixed amplitude and frequency. The autoparametrically 
resonant pendulum link length, ܮଵ, is modified as ܽ and ܾ vary according to Eq. (15) for ݊ = 2. 
In the Fig. 3(a) and 3(b) energy harvested in 100 seconds for steady state oscillations is plotted 
along the ܼ-axis for in-phase and out-of-phase oscillations respectively. ܽ and ܾ vary over the 
domain, 0.1< ܽ, ܾ < 2.5. 
AN OPTIMUM DESIGN OF A DOUBLE PENDULUM IN AUTOPARAMETRIC RESONANCE FOR ENERGY HARVESTING APPLICATIONS.  
TAIZOON CHUNAWALA, MARYAM GHANDCHI-TEHRANI, JIZE YAN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 167 
 
a) In-phase oscillation 
 
b) Out-of-phase oscillation 
Fig. 2. Energy harvested in 100 seconds, 1 < ݊ < 5 for two modes of oscillation,  
ܽ = 1, ܾ = 1, ܣ = 1 mm, Ω = 10ߨ rad/s 
 
a) Ω = 2߱ଵ 
 
b) Ω = 2߱ଶ 
Fig. 3. Energy harvested in 100 seconds by autoparametrically resonant double pendulum  
over the ܽ-ܾ domain, ܣ = 1 mm, Ω = 10ߨ rad/s 
The ideal energy harvester, optimum autoparametric in-phase oscillating double pendulum 
cannot be realized for practical implementation as seen readily by the dimensions given in Table 2. 
The resonant out-of-phase oscillating double pendulum is a more pragmatic alternative. 
Table 2. Dimension of optimum double pendulum for the two modes of oscillation  
 ܮଵ ܯଵ ܮଶ ܯଶ Power 
Ω = 2߱ଵ 1.25 cm 1.6 kg 3.75 cm 2.24 kg 30 mW 
Ω = 2߱ଶ 31 cm 1.6 kg 10 cm 3.5 kg 9 mW 
 
Fig. 4. Energy harvested in 100 seconds by double pendulum with optimal  
ܽ and ܾ for the two modes of oscillation over realisable ܮଵ 
AN OPTIMUM DESIGN OF A DOUBLE PENDULUM IN AUTOPARAMETRIC RESONANCE FOR ENERGY HARVESTING APPLICATIONS.  
TAIZOON CHUNAWALA, MARYAM GHANDCHI-TEHRANI, JIZE YAN 
168 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
Amplitude of in-phase oscillation of the two angles, ߠଵ and ߠଶ, for practically realizable link 
lengths decays harvesting negligible steady state energy as seen by the blue plot in Fig. 4. In-case 
of out-of-phase initial conditions (red plot in Fig. 4) ܧ௛ plot rises exactly at autoparametrically 
resonant ܮଵ = 31 cm, generating 9 mW in steady state validating importance of autoparametric 
resonance. The ܧ௛ peak at ܮଵ = 15 cm corresponds to out-of-phase 1:4 resonance. Maximizing 
energy harvested requires a comprehensive study of such variety of responses, especially so for 
high frequency base excitation where attaining resonance entails an even shorter pendulum link 
length. 
4. Conclusions 
The in-phase oscillating pendulum harvests energy at a rate of 30 mW, more than five times 
that of an equivalent autoparametrically resonant simple pendulum, for modest excitation 
parameters, ܣ = 1 mm, Ω = 10ߨ rad/s. Pendulum one link length obtained for in-phase resonance 
is too short for practical implementation. While, the optimum out-of-phase oscillating double 
pendulum, harvests energy at 9mW, which might just be sufficient to power maintenance sensors, 
the dimensions obtained for double pendulum are realizable, making out-of-phase oscillation a 
more desirable mode of oscillation for energy harvesting. Application specific design may help 
design by adding constrains on more equation parameters, which can help us in optimizing the 
remaining parameters more comprehensively giving better results. 
References 
[1] Rand Richard http://www.math.cornell.edu/rand/randdocs/nlvibe52.pdf, 2016. 
[2] Zaghari B., Tehrani M. Ghandchi, Rustighi E. Mechanical modeling of a vibration energy harvester 
with time-varying stiffness. 9th International Conference on Structural Dynamics, 2014, p. 2079-2085. 
[3] Kecik K., Borowiec M. An autoparametric energy harvester. European Physical Journal Special 
Topics, Vol. 222, 2013, p. 1597-1605. 
[4] Jia Y., Seshia A. A. A parametrically excited vibration energy harvester. Journal of Intelligent 
Material Systems and Structures, Vol. 25, Issue 3, 2014, p. 278-289. 
[5] Jia Y., Seshia A. A. Directly and parametrically excited bi-stable vibration energy harvester for 
bistable operation. 17th International Conference on Solid-State Sensors, Actuators and Microsystems 
(Transducers and Eurosensors), 2013, p. 454-457. 
[6] Marszal Michal, Jankowski Krzysztof, Perlikowski Przemyslaw, Kapitaniak Tomasz 
Bifurcations of oscillatory and rotational solutions of double pendulum with parametric vertical 
excitation. Mathematical Problems in Engineering, Vol. 2014, 2014, p. 892793. 
[7] Roy Jyotirmoy, Mallik Asok K., Bhattacharjee Jayanta K. Role of initial conditions in the 
dynamics of a double pendulum at low energies. Nonlinear Dynamics, Vol. 73, 2013, p. 993-1004. 
[8] www.roymech.co.uk/UsefulTables/Tribology/Bearing.html, 2016. 


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https://www.jvejournals.com/article/17715/pdf

An optimum design of a double pendulum in autoparametric resonance for energy harvesting applications

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