Theoretical Methods and Numerical Analysis

The theory of operation of the device can be explained with the help of a linear oscillator consisting of a spring-mass-damper system in conjunction with a bistable potential well created by the force of magnetic repulsion, as explained in 4.2.1. The equivalent mass of the system, linear spring constant and total damping coefficient are denoted by M, k and D respectively. The cantilever onto which the magnets are attached can be modelled to be hinged at one end and the other end is allowed to move up and down in the vertical Z direction. Another repulsively oriented magnet is placed in front of the magnet on cantilever in such a way that the centers of the magnets are a distance ‘d’ apart. On application of an external periodic force F in the vertical (Z) direction, the mass moves periodically in the vertical direction. As the free end is deflected from the initial equilibrium position to a distance ‘z’ in the vertical direction, the cantilever is rotated by an angle ‘θ’ with respect to the equilibrium position. [Figure 4.11]. The displacement of the mass M from its equilibrium position is denoted by z(t).

Figure 4.11: Schematic diagram of the bistable VEH system with repulsively positioned magnets

If the magnetic dipole moments of the repulsively oriented magnets are denoted by 𝑚⃗⃗ ₀ and 𝑚⃗⃗ ₁ respectively, then, 𝑚⃗⃗ ₀ = 𝑚₀cos 𝜃 𝑒 _𝑥+ 𝑚₀sin 𝜃 𝑒 _𝑧 and 𝑚⃗⃗ ₁ = −𝑚₁𝑒 _𝑥 , where 𝑒 _𝑥 and 𝑒 _𝑧 are the unit vectors along the horizontal (X) and vertical (Z) direction respectively [Figure 4.11]. From [1] and [2] the magnetic interaction potential can be expressed as,

𝑈_𝑚= − 𝜇₀

4𝜋𝑟³[3(𝑚⃗⃗ ₀. 𝑒 _𝑟)(𝑚⃗⃗ ₁. 𝑒 _𝑟) − 𝑚⃗⃗ ₀. 𝑚⃗⃗ ₁] 4.10 where 𝑟 = √𝑧²+ 𝑑² is the distance between the centres of the magnetic dipoles and 𝑒 _𝑟 is the unit vector along 𝑟 . Also,

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Equations. (4.10), (4.11), (4.12) and (4.13) can be combined to obtain, 𝑈_𝑚(𝑧) =𝜇₀𝑚₀𝑚₁

The vertical force acting on the cantilever due to magnetic dipole interaction is given by,

𝐹_𝑚(𝑧) = − 𝜕

When z ≥ -w, the dynamical equation of the system can be formulated as, 𝑀𝑧 ̈ + 𝑘𝑧 + 𝐷𝑧̇ + 𝜕

𝜕𝑧𝑈_𝑚(𝑧) = 𝐹 sin(𝜔₀𝑡) ^4.16

In order to model the effect of mechanical impact in the nonlinear bistable system, a rigid wall (glass) at a distance ‘w’ from the mass can be considered. Under low to moderate harmonic excitation (0.2g – 0.5g) the oscillator does not hit the wall. However, with increasing acceleration (0.6g – 1.5g) the amplitude of oscillation increases and the mass starts to collide with the wall as the excitation frequency approaches the peak power

Page | 129 frequency. For higher applied accelerations the oscillator collides with the rigid wall at z = -w when the oscillation amplitude is > w. If the oscillator collides with the wall at the time tc, then [Dankowicz and Jerrelind [8]],

𝑡→𝑡lim_𝑐⁻ 𝑧(𝑡) = −𝑤 _4.17

where CRest is the ‘coefficient of restitution’ for the inelastic impact between mass and the wall. In other words, on each impact with the wall, the motion of the mass is stopped at 𝑧(𝑡) = −𝑤, and the velocity of the mass is changed to −𝐶_{𝑅𝑒𝑠𝑡}∗ 𝑧̇(t). This is imposed as a boundary condition in the numerical model which is simulated using explicit 4th order Runge-Kutta method in MATLAB.

Now, the total potential energy and restoring force can be obtained as,

𝑈(𝑧) =1 repulsively arranged magnets is shown in Figure 4.12. The nature of the potential function can be controlled by manipulating the gap distance between the repulsive magnets.

Specifically, when the gap is large, the potential represents a parabola, which is typical of a linear oscillator. As the gap between the magnets is reduced, two new equilibrium positions appear on either sides of a potential barrier at the centre. The peak of this central

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potential barrier becomes higher if the magnets are brought even closer together. This evolution in potential energy profile is also apparent in the restoring force, which goes from linear to nonlinear with decreasing gap between the repulsive magnets.

Figure 4.12: Variation of potential energy (U (z)) and restoring force (FRest(z)) with d. The wall position is at z = -3.5 × 10^-3 m.

When the system is subject to the external periodic force F due to ambient vibration, the mass can oscillate around the new stable equilibrium positions in each potential well, representing intra-well oscillation. If F is high enough to overcome the potential barrier in between the potential wells, the mass starts to jump from one potential well to the other, producing large amplitude inter-well oscillation. From energy harvesting point of view, the harvester would generate higher energy when it oscillates between the two potential wells, instead of oscillating within a single potential well. On collision with the rigid wall during large amplitude inter-well oscillation, the potential energy becomes discontinuous following impact (Figure 4.12). A similar discontinuity is also observed for the restoring force.

The oscillation of the four-pole magnet assembly (equivalent mass M) across the copper coil changes the magnetic flux linkage across the coil. Thus, current will be induced into the coil, resulting in an electromagnetic damping force on the magnet assembly moving with respect to the fixed coil. The total damping coefficient D in eq. 4.16 can be divided into two components [eq. 4.23], the parasitic damping coefficient, DP and electromagnetic damping coefficient, DEM.

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𝐷 = 𝐷_𝐸𝑀+ 𝐷_{𝑃 4.23}

If ϕ is the magnetic flux across the coil then the flux linkage gradient is represented by ^𝑑𝜙_𝑑𝑧, and the voltage induced across the coil is given by the well-known Faraday’s law,

𝑉_𝐼𝑛𝑑 = −𝜙̇ = −𝑑𝜙

𝑑𝑡 = − (𝑑𝜙 𝑑𝑧) (𝑑𝑧

𝑑𝑡) = − (𝑑𝜙

𝑑𝑧) 𝑧̇ ^4.24

When a load resistance RLoad is connected across the output, then, from Kirchhoff’s voltage law,

𝐿𝐼̇ + 𝑉_𝐼𝑛𝑑+ (𝑅_{𝐶𝑜𝑖𝑙}+ 𝑅_{𝐿𝑜𝑎𝑑})𝐼 = 0 _4.25

Or,

𝐿𝐼̇ − 𝜙̇ + (𝑅_{𝐶𝑜𝑖𝑙}+ 𝑅_{𝐿𝑜𝑎𝑑})𝐼 = 0 _4.26

where L and RCoil are, respectively, the inductance and resistance of the coil. The eq. 4.26 can be rearranged as, constant A can be obtained from eq. 4.28 as,

𝐴 = 1

𝑅_𝑂ln(𝐼₀𝑅_𝑂− 𝜙̇) ^4.29

Substituting A in eq. 4.28 and rearranging yields, 1

Page | 132 Or,

𝐼𝑅_𝑂− 𝜙̇

𝐼₀𝑅_𝑂− 𝜙̇= 𝑒^{−𝑅𝐿 𝑡𝑂 4.32}

Rearranging eq. 4.32 yields I as a function of time,

𝐼(𝑡) = 𝜙̇

𝑅_𝑂+ (𝐼₀ − 𝜙̇

𝑅_𝑂) 𝑒^{− 𝑅𝐿 𝑡𝑂 4.33}

For a fixed rate of change of ϕ, the I(t) relaxes to the value, 𝐼(𝑡) = 𝜙̇ induced in the coil can be calculated from the approximation of a coil moving relatively in a single direction through a magnetic field varying in the direction of movement, which is given by [Saha [9]], quality factor QOC is the lumped representative of different parasitic dissipation mechanisms such as material loss, thermos-elastic loss, clamping loss, surface loss, viscous loss etc. However, detailed analysis and modelling of the various parasitic loss mechanisms are beyond the scope of the present work.

The electrical power generated in the coil is given by [Saha [9]],

𝑃_𝐸𝑀 = 𝐹_𝐸𝑀𝑧̇ = 𝐷_𝐸𝑀(𝑧̇)² _4.37

Page | 133 The average load power across the load resistance RLoad is given as a time integral over an interval from t to t+T, where T is the time period of natural oscillation of the harvester [Saha [9]].

𝑃_{𝐿𝑜𝑎𝑑} = 𝑅_{𝐿𝑜𝑎𝑑} 𝑅_{𝐶𝑜𝑖𝑙}+ 𝑅_{𝐿𝑜𝑎𝑑}

1

𝑇∫ 𝑃_𝐸𝑀𝑑𝑡

𝑡+𝑇

𝑡

4.38

The device model was simulated with the help of FEA tools and numerical techniques.

COMSOL Multiphysics was used to determine the undamped natural frequency, equivalent mass and effective spring constant of the oscillator model consisting of the FR4 folded cantilever and NdFeB magnets. The magnetic flux linkage gradient across the coil was obtained from Ansoft Maxwell. The equation of motion of the system was solved numerically using explicit 4^th order Runge-Kutta method in MATLAB to obtain the voltage and power generated in the system.