Validation of a Nonlinear Electromagnetic Vibration Energy Harvester

Proceedings of the ASME 2014 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference

IDETC/CIE 2014August 17-20, 2014, Buffalo, USA

DETC2014-35093

TEST AND VALIDATION OF A NONLINEAR ELECTROMAGNETIC ENERGYHARVESTER

Mohamed Bendame ∗

University of Waterloo

Dept. Systems Design

Waterloo, Ontario, Canada

Email: [email protected]

Eihab Abdel-Rahman

University of Waterloo

Dept. Systems Design

Waterloo, Ontario, Canada


Mostafa Soliman

Electronics Research Institute

Dept. Energy Conversion and Power Electronics

Giza, Egypt


ABSTRACT

Vibration energy harvesting is an alternative power source

for low power electronic devices. Earlier designs of vibration

energy harvesters (VEHs) utilized linear oscillators as the en-

ergy harvesting element to capture ambient vibrations and con-

vert them into electrical power. These harvesters were designed

to harvest energy within a narrow frequency band close to the

natural frequency of the oscillator. Even though these harvesters

are capable of generating mWatts of electrical power, a number

of challenges remain, such as low power density and limited op-

erational bandwidth. To overcome these limitations, researchers

turned their attention to nonlinear oscillators as a harvesting el-

ement. In this paper, we investigate a new type of nonlinear vi-

bration energy harvester that uses a double impact oscillator as

a harvesting element. A prototype of the harvester is analyzed

numerically and experimentally when aligned vertically. Results

show that the new architecture enhanced the output power as well

as the frequency bandwidth significantly. The new harvester is

capable of generating up to 250 mV and has a harvesting band-

width of about 8 Hz. The optimal load for 0.7 g input accelera-

tion is found to be 5.5Ω and the corresponding optimal power is

determined to be 8 mWatts.

∗Address all correspondence to this author.

INTRODUCTION

Energy harvesters extract energy from ambient sources and

convert it into electrical energy. While ambient energy sources

such as solar, thermal and wind have been used extensively in

a variety of applications, another source of energy that has at-

tracted the attention is ambient vibration energy. The attraction

of these new sources of energy is their capability to power wire-

less and micro-systems, thus replacing or complementing the use

of batteries [1]. While the idea of converting vibrations into elec-

trical power has been used before, advances in micro-electronics

have given it an added significance. Electromagnetic vibration

energy harvesters use mechanical oscillators and electromagnetic

transducers to harvest kinetic energy from ambient vibrations

and transform it into electrical energy [2]. A number of vibra-

tion energy harvesters (VEHs) have been proposed and some are

commercially available. Most of these works have focused on the

harvesting with a linear oscillator with harmonic excitation [3].

These Harvesters implement a single degree-of-freedom linear

spring-mass-damper, Figure (1), as the harvesting element. In

this setup, the seismic mass moves under the influence of base

excitation supported by a spring. The oscillator attains maxi-

mum velocity, and thus input kinetic energy, in a frequency band

around its natural frequency:

ω =

√

k

m, (1)

1 Copyright c© 2014 by ASME

where k is the linear stiffness of the spring and m is the effective

mass of the oscillator.

While these types of energy harvesters are capable of gener-

Generator ω (Hz) Accel

(ms−2)

m (g) Power

(µW)

[4] 52 0.589 0.66 45

[5] 99 6.85 2.96 4990

[6] 110 95.5 0.192 830

TABLE 1. Electromagnetic micro-power generators

FIGURE 1. Conventional VEHs

ating electrical energy with output power on the order of few

milli-Watts [4,7], their natural frequency must be tuned to match

the frequency of ambient vibrations. In fact these harvesters are

designed to harvest at a single frequency. However, in the vast

majority of cases ambient vibrations are distributed over a wide

spectrum of frequencies, with significant predominance of low

frequency components, and frequency tuning is not always pos-

sible due to geometrical and dynamic constraints [8]. It is there-

fore impractical to use traditional VEHs that have relatively high

center frequency (≥ 20Hz) to harvest low frequency (< 20Hz)vibrations. Some of the harvesters that have been proposed over

the years are listed in Table (1). We note the proposed harvesters

have high frequencies and low power densities. For example,

the VIBES harvester, which is electromagnetic, has a center fre-

quency of 52Hz and a maximum power of 48µ W . Due to these

limitations, there has been an interest in realizing low center fre-

quency, high power density, and wideband VEHs. To overcome

these limitations many researchers are following a different ap-

proach by exploiting the properties of nonlinear oscillators, and

within the past few years, however, dynamicists have begun to

demonstrate the efficacy of nonlinear systems to overcome mod-

ern challenges in vibratory energy harvesting [9]. For instance,

in [10], the author proposed a wideband MPG that utilized a

nonlinear oscillator by adding a one stopper within the stroke

of a cantilever beam. When the cantilever oscillates it engages

the stopper and hence increases its stiffness, a piecewise func-

tion was used to account for the change in the stiffness in the

system. In [11], the author discussed different ways of deliber-

ately introducing dynamic nonlinearities in energy harvesting de-

vices in order to improve their performance. Another energy har-

vester specifically designed to exhibit a nonlinear response was

described in [3]. In particular, the authors showed how magnetic

levitation could be used to extend device bandwidth through a

hardening frequency response. In this paper, we analyze a new

type of nonlinear VEH that uses a double-impact oscillator as its

harvesting element. Specifically, we study the response of the

vertically aligned configuration of the VEH experimentally and

numerically.

1 Vibration Energy HarvestersThe springless VEH consists of an electromagnetic trans-

ducer and a mechanical oscillator. The mechanical oscillator

used in this architecture is a double-impact oscillator as shown in

Figure 2. The inertial mass comprised of 4 magnets and a steel

cage, and is running freely along a linear guide. The magnets are

arranged with alternating polarities such that a closed magnetic

circuit exists with the magnetic flux passing through the coil. The

carriage moves along the linear guide carrying the assembly with

respect to the stationary concentric coil in response to base ex-

citations. The motion of the carriage induces a voltage V across

the coil terminals and proportional to the time rate of change of

the magnetic flux within the coil in accordance with Faraday’s

law of induction.

1.1 Electromagnetic Transduction

VEHs use electromagnetic transducers to transform kinetic

energy to electrical energy. The induced voltage across the coil

terminals is given;

V =dφ

dx

dx

dt= Bl

dx

dt, (2)

where φ is the total magnetic flux and x is the displacement of

the magnetic field with respect to the coil. From equation (2), we

note that the induced voltage V across the coil terminals depends

on the strength of the magnetic filed B, the length of the coil l,

and the relative velocity between the magnetic field and the coil.


FIGURE 2. Schematic of the horizontally-aligned springless VEH

Dimensions (mm) 25.4 x 12.7 x 1.588

Material Sintered Neodymium

Weight (g) 4

Magnet Strength (T) 0.6

TABLE 2. Magnets Specifications

In this new architecture a concentric coil with an effective

length l = 75 cm, and four magnets arranged in S-N and N-S

configuration as shown in Figure (3). The magnets properties are

shown in Table 2.

1.2 Magnetic Field

One of the main elements of the electromagnetic VEH is

the magnetic flux density, it is the magnetic flux that induces

voltage across the coil during movement of the seismic mass. It

is therefore important to accurately predict the strength of the

magnetic field by analyzing the magnetic circuit.

1.3 FEM Model

The magnetic circuit of the VEH shown in Figure 3, con-

sists of four magnets arranged as shown in the figure, a steel

cage, and an air gap separating the two sets of magnets. The

material for the steel cage is mild steel and the magnets are Sin-

tered Neodymium. The finite element modeling software AN-

SYS was used to determine the magnetic flux density. The FEM

simulation results are compared with measured results for val-

idation purposes. The FEM simulation results of the magnetic

field strength obtained from ANSYS are shown in Figures 4, 5,

while the measured results are shown in Figure. 6 The measured

as well as the simulated results of the magnetic circuit show that

the magnetic flux density is constant in amplitude but with op-

FIGURE 3. Magnetic Circuit

FIGURE 4. Measured magnetic flux density

FIGURE 5. Magnetic Flux Lines

FIGURE 6. Measured magnetic flux density


posite signs on each side of the magnetic circuit. This is due to

the fact that the polarities of the two sets of magnets are opposite

(S-N and N-S). This setup allows the induced voltage across the

coil to add up and hence maximize the harvested power.

1.4 Electrical Damping

The current passing through the coil creates a magnetic field

that opposes the field produced by the magnets. The interac-

tion between the two fields produces a force which opposes the

motions of the inertial mass. This interaction force is the electro-

magnetic damping defined as;

Fem = be x, (3)

The electrical power extracted from the mechanical oscillator is

given by;

Pem = Fem x = be x2, (4)

This power is dissipated in the parasitic coil resistance RC and

the load resistance RL. Equating the power dissipated in the coil

and load to that extracted from the oscillator gives

be x2 =V 2

RL +RC + jωL, (5)

where L is the coil inductance. Substituting equation (2) into

equation (5) we obtain the electromagnetic damping be as

be =1

RL +RC + jωL

(dφ

dx

)2

, (6)

Since the coil inductance is negligible and assuming that the coil

moves in a region of constant magnetic field, the electromagnetic

damping coefficient can be expressed as

be =(N Bl)2

RL +RC

, (7)

where B is the magnetic field intensity, N the number of turns

and l is the effective length of the coil. The electrical damping

for the prototype under test can be calculated using equation (7)

and the parameter values in Table (3).

1.5 Mechanical Damping

The frequency-response curve of the open-loop harvester is

used to determine the mechanical quality factor Qm of the VEH

Parameter Value

Magnetic Field: B (T) 0.67

Effective Coil Length: l (m) 0.75

Load Resistance: RL (Ω) ∞

Coil Resistance: RC (Ω) 2.4

TABLE 3. Electromagnetic Transducer Parameters

from the formula

Qm =f

∆ f, (8)

where f is the center frequency and ∆ f = f2 − f1, with f1 and

f2 the two half-power frequencies. The mechanical damping co-

efficient is then found as

bm =

√mk

Qm

, (9)

We can find the mechanical damping bm using equations (9, 8)

and the values of the systems parameters given in Table (4) as

bm = 0.9kg/s . The center frequency and half power bandwidth

were found from a frequency-sweep of the base acceleration of

the VEH at an amplitude of A = 0.03 g. The total damping of the

VEH is sum of mechanical damping and electromagnetic damp-

ing: b = be +bm.

Parameter Value

Mass: m (Kg) 0.11

Stiffness: k1 (N/m) 980

Center Frequency: f (Hz) 21.25

Damping: bm (kg/s) 0.9

TABLE 4. VEH Parameters

2 Model

In this section the mathematical model of the VEH is derived

and the shooting method for finding periodic orbits of the system

is described.


2.1 Equation of Motion

The vertical implementation of the VEH, is suitable for en-

vironments where motions are predominantly in the vertical di-

rection. The linear guide, aligned vertically, allows the carriage

to move freely along the rail. When the assembly vibrates due

to a base excitation y(t), the seismic mass moves with respect

to the housing producing a relative displacement x(t). The elec-

tromagnetic transducer “drains" some of the “apparent" kinetic

energy converting it to electric energy and acting as an electric

damper be on the relative velocity between the seismic mass and

the frame. The equation of motion of the vertically-aligned har-

vester can be written as:

mx =−(be +bm) x−F (x)− my, (10)

where x and y are the displacements of the seismic mass m and

frame, respectively, and F(x) is the restoring force. The VEH

harvests kinetic energy transmitted to it from the host vibrations

represented by the base acceleration

y = A cos(Ω t), (11)

where A and Ω are the amplitude and frequency of the external

excitation. Two identical springs are used as limiters on either

end of the linear guide. The origin of the coordinate system is

placed at the point where mass m rests on the lower spring. The

seismic mass m is assumed to be a point mass, as shown in Figure

7. The free distance along the rail (not occupied by the cage)

between the upper and lower uncompressed springs is denoted

xL. The uncompressed length of each spring is denoted xs and the

fully compressed length is denoted xc. The restoring force F(x)

FIGURE 7. Simplified schematic of the VEH

varies with the the position of the inertial mass m as it moves

between the two end limiters according to the equation

F(x) =

0 0 ≤ x(t)≤ xt

k1x(t) xc − xs < x(t)≤ 0

k1 (xc − xs)+k2 (−xc + xs + x(t)) −xs ≤ x(t)≤ xc − xs

k1 (x(t)− xt) xt < x(t)<−xc + xs + xt

k1 (xs − xc)+k2 (xc − xs − xt + x(t)) −xc + xs + xt < x(t)< xs + xt

(12)

where xs is the position where the mass touches the free (un-

compressed) spring, xc is the position where the spring is fully

compressed, xL is the length of the linear guide, k1 is the linear

spring stiffness and k2 the linear stiffness of the fully compressed

spring. The force-displacement relationship is shown in Figure

(8).

k1

k2

k1

k2

xs xc

-0.005 0.005 0.010 0.015 0.020xHmL

-10

-5

5

10

FHNL

FIGURE 8. Restoring force-displacement relationship

2.2 Shooting Method

The shooting method is a very powerful tool for finding sta-

ble and unstable periodic orbits of nonlinear dynamical systems.

The method uses a series of initial guesses for the desired period

and trajectory. Because the initial guesses are chosen randomly,

they are corrected after each iteration using Newton-Raphson

scheme until the error is within a predefined tolerance [12].

The equation of motion of the VEH given by equation (12) is

written as a system of first order differential equation as follows:

X ′1(t) = X2(t)

X ′2(t) = −X2(t)ce+cm)

m− F1

m− y(t)−9.81

(13)

In these approaches, the initial-value problem is converted

into a two-point boundary-value problem, and the resulting sys-

tem of equation is given by equation (14)


X ′1(t) = X2(t)

X ′2(t) =−X2(t)(ce+cm)

m− F1

m− y(t)−9.81

X ′3 =

ddt( ∂X1

∂η1) = ∂X2

∂η1

X ′4 =

ddt( ∂X1

∂η2) = ∂X2

∂η2

X ′5 =

ddt( ∂X2

∂η1) =−( ce+cm

m) ∂X2

∂η1− 1

m∂F1∂η1

X ′6 =

ddt( ∂X2

∂η2) =−( ce+cm

m) ∂X2

∂η2− 1

m∂F1∂η2

(14)

Initial guess values are needed to solve the system given by

equation (14), and one way to choose such guesses is to solve

the initial problem by long time integration and then pick the

guesses from the obtained orbit. In section (3), experimental and

numerical results obtained by solving the shooting model given

by equation (13) are presented. The results show the VEH’s the

frequency response for different amplitudes of base excitation.

3 Results

A prototype of the VEH is mounted on an electromagnetic

shaker as shown in Figure (9) and a base acceleration is applied

as input excitation with amplitude Ao and frequency Ω. The volt-

age across the coil terminals is measured using an oscilloscope.

Experimental results show that, based on the amplitude of the in-

put acceleration, the vertically-aligned harvester possesses three

different regions of operation. For amplitudes (Ao ≤ 0.05g), the

VEH response is linear, since the seismic mass remain attached

to the lower spring during motion, this region will be known as

regime 1. For amplitudes between (0.1g < Ao < 0.5g), the mass

will detach from the lower spring without impacting the upper

one, in this region the response is that of a single-impact oscil-

lator, and will be called regime 2. For amplitudes (Ao > 0.5g),the mass will impact both springs and the response is that of a

double-impact oscillator, this will be called regime 3. The exper-

imental and numerical results for the three different regimes are

presented next.

3.1 Regime 1

In this setup, the mass remains in contact with the lower

spring, and test results show that the response of the VEH

is linear. Figure (10) shows the measured and the computed

frequency-response curve of the voltage RMS for a base accel-

eration amplitudes of Ao = 0.03g and a frequency sweep from

12Hz to 30Hz. Since the response of the VEH system is linear,

the piecewise restoring force reduces to a linear relationship be-

tween stiffness and displacement, and the equation of motion is

reduced to a simple spring-mass-damper model given by equa-

tion (15):

FIGURE 9. Experimental Setup of the VEH

x =− (be +bm)

mx− k1

mx− y−g, (15)

The steady-state response x of linear model under a base accel-

eration

y = Ω2 Y0 cosΩ t = A0 cosΩ t (16)

is given by:

x(t) = a cos(Ω t +Φ) (17)

where a and Φ are the amplitude and phase of the system re-

sponse. The open load RMS voltage is given by:

VL = Bl x (18)

In this regime, the harvester’s response obtained by integrat-

ing the model given by equation (15) match the results obtained

experimentally as shown in Figure (10). The stiffness and the

seismic mass of the harvester were measured and shown in Ta-

ble. 4. Since the response of the system is linear in this regime,

the damping of the system was easily obtained from the fre-

quency response curve and it is bm = 0.9Kg/s. The maximum

output voltage in this regime is 18mV and it occurs at a frequency

ω = 21Hz, and the bandwidth of system in is about 3Hz.

3.2 Regime 2

The experimental frequency response of the VEH for mod-

erate excitations is shown in Figure (11). The figure shows the


12 16 20 24 28

Frequency (Hz)

0

0.004

0.008

0.012

0.016

0.02

Vo

ltag

e (

V)

Math Model

Experiment

Regime 1: Vertical MPG Frequency Response for 0.03 g Input Axcitation

FIGURE 10. The seismic mass displacement for an excitation of am-

plitude A = 0.4g

up and down frequency sweeps starting at f = 5Hz and finish-

ing at f = 35Hz. We note the existence of hysteresis between

the up and down frequency sweeps and a jump in the frequency

response curves at certain frequencies .The jump phenomenon is

a nonlinear phenomenon that is usually associated with soften-

ing and hardening nonlinearities. As the frequency is decreased

the amplitude jumps to a lower amplitude for soft springs and

a higher amplitude for hard springs [12]. From Figure (11) we

notice a jump to lower amplitude as the frequency is decreased

indicating a softening nonlinearity in the VEH due to the springs

when they impact the seismic mass.

For regime 2, the system’s response for an input accelera-

tion A = 0.4g was obtained numerically and the results match

those obtained experimentally as shown in Figure (12). For this

regime the stiffness as well as the damping were reduced, this

was expected since in this regime the mass looses contact with

the spring and spend a great deal of time in the air and thus re-

ducing the effective stiffness and damping of the system. The

stiffness in this case reduced to 880/,N/m and the damping was

reduced to 0.6Kg/s. We note from the model as well as the

experimental frequency response curves the jump phenomenon

as well as the shift in the center frequency that are associated

with softening nonlinearities. The frequency shifts from 21Hz

to 12Hz which indicates that nonlinearities in the system do al-

low the harvesting of low frequencies compared to their linear

counterparts. For the aforementioned input amplitude accelera-

tion, the maximum output voltage was 110V and it occurred at a

frequency ω = 12Hz.

FIGURE 11. Frequency-response of the VEH under moderate excita-

tions

10 20 30

Frequency (Hz)

0

0.04

0.08

0.12

Vo

ltag

e (

V)

Model

Experiment

Regime 2: MPG Model Matching

FIGURE 12. The open-loop voltage in the coil for an excitation of

amplitude A0 = 0.4g


3.3 Regime 3

By increasing the amplitude of the base excitation the seis-

mic mass m flies away from the lower limiter and impacts the

upper limiter. For large excitations, the base excitation ampli-

tudes were set to; 0.6g, 0.8g, 0.9g, and 1g. As in the case of

moderate excitations, we also note the jump phenomenon that is

due to nonlinearity caused by the impact with the two springs.

FIGURE 13. Frequency-response of the VEH under large excitations

In Figure (13), various up and down frequency sweep curves

of the VEH are shown. These curves show the effect of chang-

ing the base excitation amplitude on the frequency response of

the VEH. The base excitation amplitude was maintained fixed

at 0.6g, 0.8g, 0.9g or 1g while the frequency was varied from

5Hz to 35Hz. From test results, we note that as the excitation

amplitude was increased, the frequency response curves of the

VEH started to bend to the left. In the case of 0.2g, the VEH re-

sponse was maximal at a frequency f ≈ 16.5Hz, and as the am-

plitude was increased to 0.3gand0.4g, we noticed the response

was maximal at f ≈ 14.5Hz and 12Hz respectively. We note

that as the base amplitude is increased the frequency correspond-

ing to the peak amplitude of the VEH decreased, indicating a

softening type nonlinearity.

4 Conclusions

In this paper, we modelled and analyzed the response of a

nonlinear vertically-aligned vibration energy harvester. In this

study, a double-impact oscillator was used as the harvesting el-

ement. In regime 1, the seismic mass remains in contact with

the lower spring while subjected to low base excitations and the

response of the harvester is linear. The linearity of the system

was observed experimentally as well as numerically. The maxi-

mum open-loop output voltage obtained in this regime is 18mV

and the center frequency is about 21.5Hz. In regime 2, the seis-

mic mass leaves the lower spring and the harvester behaves as

a single-impact oscillator. Numerical as well as experimental

results show a consistent bending to the right of the frequency-

response curves of the coil RMS voltage indicating an effective

hardening-type nonlinearity. In this regime the maximum out-

put voltage was 150 mV for an input acceleration A = 0.5g. In

Regime 3, the dynamics of the harvester are reminiscent of those

of a double-impact oscillator. The results show that within a cer-

tain range of frequency, and depending on the amplitude of the

input excitation, the output voltage varies linearly with the fre-

quency. This is in agreement with the theory since the voltage

is the derivative of the displacement of the seismic mass. In this

regime a maximum output of over 200mV was recorded and a

frequency bandwidth of about 10Hz.

REFERENCES

[1] Zhu, D., Tudor, M. J., and Beeby, S. P., 2009. “Strategies

for increasing the operating frequency range of vibration

energy harvesters: a review”. Journal of Measurement Sci-

ence and Technolgy, 21, Dec.

[2] Williams, C. B., and Yates, R. B., 1995. “Analysis of a

micro-electric generator for microsystems”. Vol. 1, IEE-

EXplore, pp. 369–372.

[3] Mann, B. P., and Sims, N. D., 2009. “Energy harvest-

ing from the nonlinear oscillations of magnetic levitation”.

Journal of Sound and Vibration, 319, pp. 515–530.

[4] Priya, S., and Inman, T. D., 2009. Energy Harvesting Tech-

nologies. Springer.

[5] Glynne-Jones, P., 2001. “Vibration powered generators

for self-powered microsystems”. Phd thesis, University of

Southampton, Southampton, England.

[6] Ching, N., Wong, H., Li, W. J., Leong, P. H. W., and Wen,

Z., 2002. “A laser-micromachined multi-modal resonating

power transducer for wireless sensing systems”. Sensors

and Actuators A: Physical, 97-98, Apr, pp. 685–690.

[7] Hadas, Z., Kluge, M., Singule, V., and Ondrusek, C., 2007.

“Electromagnetic vibration power generator”. In Diagnos-

tics for Electric Machines Power Electron, Vol. 1, IEEEX,

pp. 451–455.

[8]

[9] Stanton, S. C., McGehee, C. C., and Mann, B. P., 2010.


“Nonlinear dynamics for broadband energy harvesting: In-

vestigation of a bistable piezoelectric inertial generator”.

Physica D: Nonlinear Phenomena, 239(10), May, pp. 640–

653.

[10] Soliman, M., Abdel-Rahman, E., El-Saadany, E. F., and

Mansour, R. R., 2009. “A Design Procedure for Wideband

Micropower Generators”. Journal of Microelectromechan-

ical Systems, 18(6), Dec, pp. 1288–1299.

[11] Green, L. P., 2012. “Nonlinear energy harvesting”. Phd

thesis, University of Sheffield, Sheffield, England, October.

[12] Nayfeh, A., and Mook, D., 1997. Nonlinear Oscillations.

John Wiley & Sons.


Documents

Validation of a Nonlinear Electromagnetic Vibration Energy Harvester