Scaling and characterisation of a 2-DoF velocity amplified electromagnetic vibration energy harvester

The VAEGs discussed herein utilise impacts in free-moving masses to achieve velocity amplification, as described in section 2. Each mass can, therefore, be forced upon by a spring or can be free of any spring force, resulting in piecewise-linearity. PWL systems have received significant investigation in the field of nonlinear dynamics [27–29]. Piecewise linearity arising from impacts has also been investigated to improve the performance of VEHs in systems referred to as impact oscillators [18, 30–32].

The 2-DoF VAEGs presented here are modelled as PWL inertial oscillators. A schematic of a 2-DoF PWL system is shown in figure 2. The springs do not transfer load in tension, meaning that the masses can become detached from the springs.

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The transducer is coupled between m₂ and the base. This involves incorporating a magnet in m₂, with a coil that is attached to the device housing arranged concentrically to the mass. Therefore, the electrical damping force, ${F}_{e}={c}_{e}({\dot{x}}_{2}-\dot{y})$, is applied in the equation of motion of m₂ only. As shown by (7), the electrical damping force can also be written as Ki.

As this is a 2-DoF system, there are two equations of motion, which are determined by applying a force balance to each of the masses. The masses are considered to be point masses in the model; therefore, their height is zero. The relative displacements of the masses and the base are ${z}_{1}={x}_{1}-y-{h}_{s}$ and ${z}_{2}={x}_{2}-y-2{h}_{s}$, while the relative velocities are then ${\dot{z}}_{1}={\dot{x}}_{1}-\dot{y}$ and ${\dot{z}}_{2}={\dot{x}}_{2}-\dot{y}$. Considering the 2-DoF system as a PWL oscillator with detachable masses, the stiffness and damping forces are discontinuous. Boundary conditions determine the forces acting on the masses at any instant:

• the base mass m₁ is attached to the spring k₁ under the condition z₁ < 0;
• the two masses are attached to each other through k₂ if z₂ < z₁; and
• m₂ is in contact with the stopper spring k₃ under the condition z₂ > g₀.

The set of coupled equations describing the dynamics of a 2-DoF PWL system are then:

$$\begin{eqnarray}&&{m}_{1}{\ddot{z}}_{1}=-{m}_{1}\ddot{y}-{m}_{1}g-{c}_{{m}_{{f}_{1}}}{\dot{z}}_{1}\,+\\ \quad \left\{\begin{array}{ll}-{k}_{1}{z}_{1}-{c}_{{m}_{{s}_{1}}}{\dot{z}}_{1}, & \mathrm{if}\ {z}_{1}\lt 0\\ 0, & \mathrm{if}\ 0\leqslant {z}_{1}\leqslant {z}_{2}\\ {k}_{2}({z}_{2}-{z}_{1})+{c}_{{m}_{{s}_{2}}}({\dot{z}}_{2}-{\dot{z}}_{1}), & \mathrm{if}\ {z}_{2}\lt {z}_{1}.\end{array}\right.\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{m}_{2}{\ddot{z}}_{2}=-{m}_{2}\ddot{y}-{m}_{2}g-{c}_{{m}_{{f}_{2}}}{\dot{z}}_{2}-{Ki}\,+\\ \quad \left\{\begin{array}{ll}-{k}_{2}({z}_{2}-{z}_{1})-{c}_{{m}_{{s}_{2}}}({\dot{z}}_{2}-{\dot{z}}_{1}), & \,\mathrm{if}\ {z}_{2}\lt {z}_{1}\\ 0, & \,\mathrm{if}\ {z}_{1}\leqslant {z}_{2}\leqslant {g}_{0}\\ -{k}_{3}({z}_{2}-{g}_{0})-{c}_{{m}_{{s}_{3}}}{\dot{z}}_{2}, & \,\mathrm{if}\ {z}_{2}\gt {g}_{0}.\end{array}\right.\end{eqnarray} \tag{ 3 }$$

When a conductor experiences a change in magnetic field, a voltage, V_ind, is induced, which is equal to the temporal rate of change of magnetic flux, ϕ (Faraday's law). This can be rewritten as the product of the magnetic flux gradient and the relative velocity between the coil and magnetic field:

$$\begin{eqnarray}&&{V}_{{\rm{ind}}}=-\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}{t}}=-\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}{z}}\dot{z}.\end{eqnarray} \tag{ 4 }$$

The flux gradient, often termed as the coupling coefficient, K, is the link between the mechanical and electrical domains:

$$\begin{eqnarray}&&K=\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}{z}}={B}_{r}{l}_{w},\end{eqnarray} \tag{ 5 }$$

where B_r is the average radial magnetic flux density and l_w is the total coil length. Substituting (5) into (4) gives a new expression for the induced voltage:

$$\begin{eqnarray}&&{V}_{{\rm{ind}}}=-K\dot{z}.\end{eqnarray} \tag{ 6 }$$

The movement of the coil through the magnetic field also results in an electrical damping force known as the Lorentz force:

$$\begin{eqnarray}&&{F}_{e}={Ki},\end{eqnarray} \tag{ 7 }$$

where i is the current.

The electrical circuit of the VAEG implemented here consists of a generator connected in series with a coil resistance R_C, load resistance R_L, and coil inductance L_C (figure 3(a)). Applying Kirchhoff's voltage law to the circuit:

$$\begin{eqnarray}&&{V}_{{\rm{ind}}}=K\dot{z}={L}_{C}\displaystyle \frac{{\rm{d}}{i}}{{\rm{d}}{t}}+i({R}_{C}+{R}_{L}),\end{eqnarray} \tag{ 8 }$$

where i is current. The coil resistance is given by:

$$\begin{eqnarray}&&{R}_{C}=\displaystyle \frac{{\rho }_{c}l}{{A}_{w}},\end{eqnarray} \tag{ 9 }$$

where ρ_c is the resistivity of copper, and A_w is the coil wire cross sectional area. The coil inductance (in μH) is calculated using the Wheeler approximation [33]:

$$\begin{eqnarray}&&{L}_{C}=\displaystyle \frac{0.8({r}_{m}^{2}{N}^{2})}{0.025\,4(6{r}_{m}+9{{ct}}_{{\rm{ax}}}+10{{ct}}_{{\rm{rad}}})},\end{eqnarray} \tag{ 10 }$$

where the mean coil radius ${r}_{m}=({r}_{o}+{r}_{i})/2$, N is the number of coil turns, ct_ax is the coil thickness in the axial direction, and ct_rad is the coil thickness in the radial direction (figure 3(b)). Rearranging (8) and substituting i = V_L/R_L (Ohm's Law) results in an equation for the load voltage:

$$\begin{eqnarray}&&\dot{{V}_{L}}=K\dot{z}\displaystyle \frac{{R}_{L}}{{L}_{C}}-\displaystyle \frac{{V}_{L}}{{L}_{C}}({R}_{C}+{R}_{L}).\end{eqnarray} \tag{ 11 }$$

This electrical circuit equation (11) is coupled to the equations of motion, (2) and (3), to model both the mechanical and electrical systems of the VAEGs. This system of equations is solved numerically by the MATLAB numerical ODE solver ode15s. It is necessary to use this solver (rather than a nonstiff solver such as ode45), as the set of coupled equations is stiff, i.e. the equations contain terms which rapidly change during the solution, potentially making the nonstiff numerical method of solving the equations unstable.

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For simplicity, the coupling coefficient, K, is often assumed to be constant in the analysis of electromagnetic energy harvesters. This assumption is valid if the displacement of the inertial mass is small. VAEGs, however, exhibit relatively large displacements. Therefore, to accurately model a VAEG, a coupling coefficient that varies with axial displacement, K(z), is necessary.

The transducer employed in the VAEG is illustrated in figure 4(a). The magnet stack consists of two oppositely polarised magnets with a soft magnetic spacer between them. The spacer acts as a flux concentrator to increase the radial flux at the centre of the magnet stack. The three individual coils are located at the regions of high radial flux density and are connected such that voltage cancellation is avoided. The process used to determine K(z) for the transducer employed here is outlined below.

• The static magnetic field of the permanent magnet stack was modelled using COMSOL multi-physics, the finite element analysis software, through the AC/DC module with the magnetic fields, no currents physics. The geometry of the magnet stack was defined in two dimensions about a central axis of symmetry with a surrounding air domain. The air domain was made sufficiently large (10× the magnet stack height and 10× the magnet stack diameter) such that the boundary condition on the exterior boundaries had a negligible effect on the field in the region surrounding the magnet. The magnet geometry and a portion of the surrounding air domain is illustrated in figure 5. The zero magnetic scalar potential condition was applied to the external boundaries in the model (the top, right and bottom sides of figure 5), while the magnetic insulation condition was applied to the axis of symmetry (the left side of figure 5). The model required the residual flux density, B_res, and magnet orientation as input parameters, with the model output being the radial magnetic flux density, B_r—this is the radial component of the normal magnetic flux density shown in figure 5. To get an accurate calculation of the magnetic field, the model mesh (free tetrahedral) was refined until the magnetic flux density no longer changed with reduced mesh size.
• The average radial magnetic flux density, ${\bar{B}}_{r}$, in the axial direction—averaged across the coil width in the radial direction—is shown in figure 4(b).
• K(z) is then the product of ${\bar{B}}_{r}$ over the total coil length, l_w (5). This was calculated for a range of coil positions relative to the magnet. A function was fitted to the result such that K(z) could be determined for any relative coil-magnet position (figure 4(c)).

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The length scale, s, of a harvester is defined as the cubed root of the device volume (s = Volume^1/3). It was shown by [24] that the maximum average power dissipated by a linear energy harvester per cycle at resonance is ${P}_{\max }\,=(1/2)m{\omega }_{n}^{3}{{YZ}}_{{\rm{lim}}}$. The displacement limit of the mass Z_lim scales with the length scale (Z_lim ∝ s), while the mass scales with the volume (m ∝ s³). The excitation amplitude Y and resonant frequency ω_n are independent of scale; therefore, it is stated that the maximum power scales as P_max ∝ s⁴. However, the mechanical damping, c_m, is not accounted for in the analysis of [24].

It was demonstrated by [22] that the scaling rate of the actual displacement amplitude, Z_max, is affected by c_m, which also affects the electrical damping, c_e, required to maximise the power delivered to the load (electrical domain analogue matching (EDAM) condition, [47]). Consequently, although Z_lim—the maximum possible displacement of the mass within a device volume—scales with s, the assumption that Z_max ∝ s may not be valid. An expression of the maximum power delivered to the load at the optimal load resistance was derived by [47]:

$$\begin{eqnarray}&&{P}_{L}=\displaystyle \frac{{m}^{2}{{\rm{Acc}}}^{2}}{8{c}_{m}(\tfrac{{c}_{m}{R}_{C}}{{K}^{2}}+1)}.\end{eqnarray} \tag{ 12 }$$

Assuming that all of the harvester components are scaled uniformly with s, the scaling relationships of the parameters in (12) and other relevant parameters are as listed in table 5—a parameter which is ∝s⁰ is independent of scale. Clearly, in order to determine the load power scaling rate from (12), the scaling relationship of the mechanical damping must be known. This is discussed in the following subsection.

Table 5.  Scaling relationships of the parameters in a linear electromagnetic harvester.

Parameter	Symbol	Relationship
mass	m	∝s3
Base acceleration	Acc	∝s0
Coil volume	Vcoil	∝s3
Coil wire diameter	dw	∝s
Coil wire area	${A}_{w}=\pi {d}_{w}^{2}/4$	∝s2
Coil wire length	${l}_{w}={V}_{{\rm{coil}}}{f}_{f}/{A}_{w}$	∝s
Fill factor	ff	∝s0
Coil resistance	${R}_{C}={\rho }_{{\rm{cop}}}{l}_{w}/{A}_{w}$	∝s−1
Resistivity of copper	ρcop	∝s0
Coupling coefficient	$K={B}_{r}{l}_{w}$	∝s
Radial flux density	Br	∝s0 [23, 48]

In the following analysis, the influence of the mechanical damping coefficient, c_m, on scaling is considered. The scaling of c_m is more difficult to define than the previously discussed parameters as it comprises a number of damping mechanisms including thermoelastic damping, friction, squeeze-film damping, and air resistance. The dominant damping mechanism is dependent on the device design and fabrication. It is stated by [22] that viscid damping is usually dominant, and scales as c_m ∝ s. This scaling relationship is also taken by [25].

In this analysis, two mechanical damping scaling rates are considered—${c}_{m}\propto s$ and c_m ∝ s³. The former, c_m ∝ s, is the scaling relationship which is typically employed. The latter mechanical damping scaling condition, c_m ∝ s³, is discussed here as it is employed in the investigation into energy conversion in the transducer of a VAEG in section 6.2.1, and it is of interest to investigate this effect on linear systems in order to allow direct comparison to the VAEG systems.

${{c}}_{{m}}\propto {s}$: Inserting ${c}_{m}\propto s$ [22] into (12) yields the scaling relationship for linear oscillatory energy harvesters for the given mechanical damping scaling relationship:

$$\begin{eqnarray}&&{P}_{L}\propto \displaystyle \frac{{s}^{7}}{{s}^{2}+1}.\end{eqnarray} \tag{ 13 }$$

As the denominator of (13) contains a sum, two asymptotic scaling relationships can be determined:

$$\begin{eqnarray}&&{P}_{L}\propto {s}^{5}\,\,\,\,\,{\rm{for}}\,s\to \infty ,\end{eqnarray} \tag{ 14a }$$

$$\begin{eqnarray}&&{P}_{L}\propto {s}^{7}\,\,\,\,\,{\rm{for}}\,s\to 0.\end{eqnarray} \tag{ 14b }$$

These scaling relationships were described by [22] based on analysis of the equations derived by [47].

From (13), where the electrical damping dominates (${K}^{2}/{R}_{C}\gg {c}_{m}$) the scaling relationship approaches P_L ∝ s⁵. If, however, the mechanical damping dominates (${c}_{m}\gg {K}^{2}/{R}_{C}$) the scaling relationship approaches P_L ∝ s⁷. Examination of the scaling relationships of these terms—${K}^{2}/{R}_{C}\propto {s}^{3}$ and c_m ∝s—suggests that at larger scales electrical damping will dominate, while at smaller scales mechanical damping will dominate.

The amplitude of the mass displacement scales at a minimum rate of Z_max ∝ s², i.e. the displacement of the mass decreases at a greater rate than the displacement limit, Z_lim. This is determined by inputting the scaling relationships described above into the equation for maximum displacement amplitude, ${Z}_{\max }=Y/2{\zeta }_{T}$. Clearly, assuming c_m ∝ s, scaling has a highly negative effect on the power density of a linear VEH, with a minimum load power scaling rate of P_L ∝ s⁵.

${{c}}_{{m}}\propto {{s}}^{{\rm{3}}}$: This is the ideal case where the mechanical damping scales directly with the volume, and so, the damping is independent of scale. This scaling relationship is unlikely to occur in reality, as the difficulty of fabricating devices with moving components increases as scale is reduced, while aspects of the device such as surface finish become more significant. However, it is informative to understand the influence of this damping scaling relationship, as it allows the scaling behaviour of the system and its parameters to be investigated independent of damping:

• In this instance, the mechanical damping and, consequently, the electrical and total damping scales with the volume (${c}_{m,e,T}\propto {s}^{3}$)—EDAM load matching condition [47].
• The total damping ratio ζ_T is, therefore, independent of scale (${\zeta }_{T}={c}_{T}/2m{\omega }_{n}\propto {s}^{0}$, where m ∝ s³ and ω_n ∝ s⁰),
• Z_max remains constant (${Z}_{\max }=Y/2{\zeta }_{T}\propto {s}^{0}$) and
• P_L ∝ s³ (12).

Z_max does not scale under this mechanical damping scaling condition as the total damping force acting on the mass scales in proportion to the volume, resulting in the motion of the mass (its velocity–frequency response) remaining constant with scale (this is shown explicitly in figure 11(a) for a 2-DoF VAEG). This is only possible if the displacement limits do not scale with the other harvester dimensions. If the spring deflection limit, Z_lim, is less than Z_max, obviously, the mass would not be able to oscillate to its maximum displacement.

Implementation: In the analysis of the influence of scaling on energy conversion in the transducer of a VAEG, presented in this subsection, a number of deviations from the linear scaling theory of [22] are employed. The purpose of these deviations is to achieve, as closely as possible, conditions where the mechanical damping scales directly with the volume (c_m ∝ s³); hence, maintaining the velocity–frequency response across scales. This allows the effect of scaling on energy conversion in the transducer to be considered separately from the mechanical system, i.e. any difference in energy converted in the transducer is a result of the electrical, rather than the mechanical system. The deviations employed in the analysis of the influence of scaling on the electrical system of a 2-DoF VAEG are discussed below.

• The mechanical damping coefficient was assumed to scale with volume (c_m ∝ s³) in the simulations, rather than the dimension (c_m ∝ s), as in [22].
• As described in section 6.1.2, under the damping scaling rate ${c}_{m}\propto {s}^{3}$, the displacement amplitude does not scale. To accommodate this, the spring height was not reduced with scale (h_spring ∝ s⁰) in both the experimental and simulated scaling analysis. This avoided the possibility of the displacement exceeding the maximum spring compression. Additional height contributed by the springs was not considered in the volume and corresponding length scale. The volume used to define the length scale (s = Volume^1/3) was, therefore, the product of the area within the outermost coil, and the summed height of the copper mass, magnet stack and gap. A schematic showing the influence of scaling on the geometry of the 2-DoF VAEG in the current experimental and simulation scaling analysis is illustrated in figure 8, with the area contributing to volume highlighted. Evidently, the copper mass and transducer dimensions scale with s, while the spring height does not scale. Initial gap, g₀, does not scale either—the reason for this is described in the following point.
• The geometric feature of a VAEG which differs most significantly to a linear energy harvester is the presence of an initial gap, g₀, between the masses and springs. As g₀ is a component of the displacement range, it does not scale; this means that the change in dynamics with g₀ is independent of scale and g₀ must remain constant to maintain the velocity–frequency response with scale. This is a consequence of the effective stiffness of the harvester being dependent on the portion of the cycle the masses spend attached/detached from the springs during operation. For the effective stiffness to be maintained and, consequently, the velocity–frequency response to be the same across scales, the portion of each cycle the masses spend in contact with the springs must remain constant. Consequently, in the analysis presented, g₀ remains constant with scale. As a result of this, the rate of change of volume with magnet coil dimension decreases as g₀ increases, i.e. the larger g₀, the smaller the influence of scaling on its volume. The scaling relationship is described for a range of g₀ values.
• Uniform scaling of the dimensions of a uni-axially loaded spring—which can be of cantilever, helical, or planar form—results in the spring stiffness scaling as k ∝ s [49]. Consequently, the natural frequency is inversely proportional to length scale (w_n ∝ s⁻¹). In this analysis, the spring stiffness, k, was scaled as k ∝ s³, such that the natural frequency of an equivalent linear system would remain constant with scale (${\omega }_{n}{(s)=(k/m)}^{1/2}\propto {s}^{0}$).

As a result of the scaling conditions described above, the system dynamics—amplitude and frequency of velocity and displacement responses—were not affected by scale in the scaling analysis of VAEGs presented. The electrical system was, thus, partly isolated from the mechanical system. Any deviations from linear scaling theory were then a result of the effect of scaling on energy conversion in the transducer. The difference in the effect of scaling on a linear system and a VAEG can, therefore, be determined solely in terms of the transduction properties. This is the defining feature of the scaling analysis presented in this section, allowing the mechanical and electrical systems to be considered separately across scales.

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As previously stated, if the transducer is scaled uniformly, the transducer properties scale as follows: d_w ∝ s, l_w ∝ s, A_w ∝ s², R_C ∝ s⁻¹ and K ∝ s. In the scaling analysis presented in this section, the coil wire diameter was kept constant (d_w ∝ s⁰) so that the coil fill factor would be the same for each scale; whereas, if the coil wire diameter was scaled, the fill factor would likely have changed and this would have affected the uniform scaling of the VAEGs. Applying the same analysis as above to the systems with constant coil wire diameter gives: l_w ∝ s³, A_w ∝ s⁰, R_C ∝ s³ and K ∝ s³. Inserting these scaling relationships into (12) yields the same scaling relationships described for d_w ∝ s, i.e. the load power scales at the same rate for both d_w ∝ s and d_w ∝ s⁰ (P_L ∝ s³). This is because the scaling rate of the R_C/K² term in (12) remains the same for both cases, i.e. ${R}_{C}/{K}^{2}\propto {s}^{-3}$. A constant coil wire diameter of d_w = 100 μm was used in this analysis.

The scaling relationships employed in the scaling analysis of a 2-DoF VAEG are summarised in table 6. Parameters described as uniform scaling are those which conform to homothetic scaling, or the scaling relationships employed by [22]. The non-uniform scaling parameters are those which do not conform to homothetic scaling, while the resultant parameters are those whose scaling relationships are a consequence of the non-uniform scaling.

Examining (12), with c_m ∝ s³, the only parameter which could cause a deviation from P_L ∝ s³—the linear scaling relationship—is K, as the remaining parameters are rigidly defined. A variable K becomes K_eff, the effective electromagnetic coupling coefficient, which is the RMS of K over a period of excitation:

$$\begin{eqnarray}&&{K}_{{\rm{eff}}}=\sqrt{\displaystyle \frac{\omega }{2\pi }{\int }_{0}^{\tfrac{2\pi }{\omega }}{K}^{2}{\rm{d}}{t}}.\end{eqnarray} \tag{ 15 }$$

Although the peak K is proportional to the length scale cubed, K_eff can scale at a greater rate. The peak K occurs when the coil and magnet stack are located at the optimal relative position—which is determined by the geometry of the transducer. For small displacement ranges, K_eff remains close to this peak value. In VAEGs, the displacement range varies with g₀; the larger g₀, the greater the displacement range (provided the excitation conditions result in large amplitude oscillations). Large displacements result in values of K_eff which are considerably lower than the peak value. The relative position of the magnet and coil which results in maximum average power generation, for a set of excitation conditions and a given g₀, differs from the optimal relative magnet coil position under very low displacements. In this case, the optimal magnet coil position is the one that results in the highest RMS load voltage, which is a function of K and relative velocity.

In the simulations, ${c}_{m}\propto {s}^{3}$ and the velocity–frequency response remained constant with scale. In the experimental data (figure 10), this was not the case; c_m scaled at a lower rate, resulting in the mechanical damping effect increasing as scale was reduced. Consequently, the experimental velocity response did not remain constant with reduced scale; its amplitude reduced. To allow the experimental and simulation results to be compared across scales, rather than using the load power, the effective coupling coefficient was employed. This was appropriate as K_eff is normalised by velocity, which mitigates the issue of the experimental velocity–frequency response amplitude changing with scale.

The voltage induced in the electrical circuit was derived using Kirchhoff's voltage law in (8). As the harvester is designed to operate at low frequencies (<30 Hz), the effect of the coil inductance was small [50] and, hence, was neglected here—the inductor impedance was Z_L = 56 Ω at 30 Hz, compared to a coil resistance of R_C = 938.9 Ω. Rearranging (8) allows an expression for the load voltage to be found:

$$\begin{eqnarray}&&{V}_{L}={Kv}\left(\displaystyle \frac{{R}_{L}}{{R}_{L}+{R}_{C}}\right),\end{eqnarray} \tag{ 16 }$$

where v is the relative velocity between the magnet and coil. The effective coupling coefficient K_eff is calculated from both simulation and experimental data by rearranging (16):

$$\begin{eqnarray}&&{K}_{{\rm{eff}}}=\displaystyle \frac{{V}_{L}}{v}\left(\displaystyle \frac{{R}_{L}+{R}_{C}}{{R}_{L}}\right).\end{eqnarray} \tag{ 17 }$$

In this case V_L is the RMS load voltage, while v is the RMS velocity.

The experimental K_eff is calculated from the peak RMS load voltage and the corresponding RMS velocity in the experimental frequency sweeps. Although the RMS relative and absolute velocities vary significantly at frequency ranges above and below the peak frequency, the deviation between them is small at the peak frequency. This is illustrated in figure 9 for simulation frequency sweeps at a range of g₀ values, and the findings are valid for all scales—under the mechanical damping conditions employed. As the LDV measures the absolute velocity of the mass, and due to the small difference in absolute and relative RMS velocities at the peak response, the RMS absolute velocity is used in the calculation of the experimental K_eff.

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Results: The simulation and experimental K_eff are plotted as functions of the length scale in figure 10 for acceleration levels of Acc = 10 and 6 m s⁻², for gap lengths ranging from g₀ = −0.2 to 10.5 mm. The experimental data is plotted for the three magnet diameters, d_mag = 6, 8, and 12 mm, while the simulation data is presented for magnet diameters in the range d_mag = 4.3–15.6 mm (range selected to extend beyond experimental range to higher and lower scales). The simulation data was taken at a single frequency which varied for each g₀ and acceleration. The selected frequency remained constant with scale, however. This single frequency was selected based on the response showing a high amplitude over the entire range of scales at this frequency.

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The results of the scaling analysis of a 2-DoF VAEG are discussed below.

• The linear scaling relationship, K ∝ s³, is plotted as two dashed lines from the maximum and minimum K_eff values, respectively (figure 10). These lines are included to compare how the scaling behaviour of the 2-DoF VAEG deviates from the linear scaling relationship.
  • Evidently, K_eff scales at a greater rate than the linear relationship. This is due to the aforementioned reduction in K_eff with increased displacement range, which is a consequence of the relative motion of the magnet stack and coils being in sub-optimal locations for portions of each cycle.
  • At higher values of g₀, the scaling rate of K_eff is increased further. This is a result of g₀ remaining constant with length scale, which causes the volume to decrease at a slower rate than s³. The increased scaling rate of K_eff is seen in the shift in experimental data points to higher length scales with increasing g₀, and in the divergence of the simulation results from each other and the linear scaling relationships as length scale is decreased.
  • Furthermore, as the magnet stack and coil dimensions are reduced, K_eff is effectual over a smaller range, which becomes an ever smaller portion of the displacement range.
• Power law relationships are fitted to the simulation results for the highest and lowest gap lengths.
  • In the simulation results, at the lowest acceleration level (Acc = 6 m s⁻²) and the lowest gap (g₀ = −0.2 mm), the scaling rate is effectively the same as for a linear system (K ∝ s³). This is because, under these excitation and geometric conditions, the harvester effectively behaves as a linear system, i.e. the masses are always in contact with a spring.
  • As g₀ and the acceleration level are increased, the displacement range rises and the system no longer behaves linearly. The scaling rate increases as a result of this larger displacement range. At the highest acceleration level (Acc = 10 m s⁻²) and the highest gap size (${g}_{0}=10.5$ mm), the highest scaling rate of K ∝ s^3.45 is seen.

The scaling analysis described here relates to VAEGs; however, the results are also relevant for other electromagnetic energy harvesters which operate at low frequencies featuring large displacement amplitudes [13, 51]. In these systems, K_eff will be lower than the peak K, and will scale at a greater rate than a system featuring lower displacement amplitudes. This indicates that inertial generators operating at low frequencies will scale poorly, compared to devices operating at higher frequencies.

The influence of the mechanical damping scaling rate on the velocity–frequency response of a 2-DoF VAEG is investigated in this subsection. This analysis is of interest as it explores the limits of damping on the dynamic response of a VAEG. The focus of the scaling analysis presented thus far has been on the influence of geometry and acceleration level on the scaling of the electrical system of a VAEG, relative to that of a linear system. Simulation only is employed in this analysis; the experimental mechanical damping scaling rate is not considered, as the mechanical damping parameters are highly dependent on the fabrication quality. A sample of three fabricated devices is, consequently, not significant enough to draw conclusions of significance from. This section compares the velocity–frequency response of 2-DoF VAEGs at four length scales for three mechanical damping scaling rates, through simulation.

The dominant component of mechanical damping in a VEH is highly dependent on the device design. In the VAEG demonstrated herein, the most significant components of the mechanical damping were thermoelastic damping in the springs, and friction between the masses and Teflon tube.

Simulation velocity–frequency responses for a range of mechanical damping scaling rates and length scales are presented in figure 11. This test series was completed to show the influence of the mechanical damping scaling rate on the frequency response of a 2-DoF VAEG. Scaling rates of c_m ∝ s³, c_m ∝ s², and c_m ∝ s are presented for four length scales. The reasons these scaling rates were selected are as follows: c_m ∝ s³—the ideal scenario, with the velocity–frequency response remaining constant as scale is reduced; c_m ∝ s—the scaling rate stated by [22]; and c_m ∝ s²—shows the transition between the two previously described scaling rates. The length scales presented are lower than those in figure 10, as there is no coil present—the dynamics only are under investigation—which decreases the device volume.

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For ${c}_{m}\propto {s}^{3}$, the system velocity–frequency response does not change with scale (figure 11(a))—as discussed in section 6.2.1. Lowering the scaling rate results in the RMS velocity significantly decreasing with scale. The mechanical damping dominates, such that the masses no longer achieve high velocity impacts that result in velocity amplification. This is particularly evident in figure 11(c), where the peak demonstrated at higher length scales is no longer present at s = 7.2 mm. Comparing the highest and lowest length scales presented for c_m ∝ s, the peak RMS velocity of the magnet stack goes from 4.36 to 1.21 m s⁻¹, which is a reduction of 72.4%. This has a highly negative effect on the potential energy which can be harvested from a VAEG, particularly approaching MEMS scale.

Considering, finally, the scaling of the load power in a 2-DoF VAEG, assuming c_m ∝ s. Incorporating a transducer in the model used in figure 11 allowed the power generated at each scale to be simulated. Going from a length scale of s = 18.7 to 8.4 mm (the increase in volume with the addition of the coil is neglected to keep the length scales consistent with figure 11), the simulated peak power generated drops from P_L = 12.48 to 0.152 mW. This corresponds to a scaling rate of P_L ∝ s^5.51, which is in the range specified by [22] (14a). Decreasing the scale further, this power scaling relationship continues to increase, as the influence of the mechanical damping increases relative to the electrical damping, becoming the dominant damping mechanism. This is further compounded by the volume decreasing at a lower rate than the transducer dimensions, due to g₀ remaining constant with scale.