SPECIAL ISSUE PAPER 27
Energy harvesting from motion using rotating and gyroscopic proof masses E M Yeatman Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK. email:
[email protected] The manuscript was received on 13 April 2007 and was accepted after revision for publication on 13 August 2007. DOI: 10.1243/09544062JMES701
Abstract: Energy harvesting – the extraction of energy from the local environment for conversion to electrical power – is of particular interest for low power wireless devices such as body or machine mounted sensors. Motion and vibration are a potential energy source, and can be exploited by inertial devices, which derive electrical power by the damping of the relative movement of a proof mass mounted in a frame attached to the moving host. Inertial devices using linear motion of the proof mass, which have been extensively studied and developed, have a maximum power output limited by the internal travel range of the proof mass. In the current paper, the potential power of devices using rotating proof masses, powered by linear or rotational host motion, is analysed. Two new operation modes are introduced: rotationally resonant devices, and devices driven by continuous rotation. In each case the maximum achievable power densities are estimated, and these are compared with equivalent expressions for devices with linear proof mass motion where appropriate. The possibility of using actively driven, gyroscopic structures is then introduced, and the potential power of such devices is considered. By avoiding the linear displacement limit and the limited mass of conventional devices, it is shown that increases in obtainable power are possible if parasitic damping is minimized, particularly for cases of low linear source amplitude. Finally, issues of implementation are discussed, with an emphasis on microengineered devices. Keywords: energy scavenging, microelectromechanical systems, wireless sensors, gyroscopes
1
INTRODUCTION
A supply of electrical power is naturally a critical requirement for wireless sensor nodes, and because batteries have limited energy capacity, these frequently dominate the size and weight of such nodes. They also impose a maintenance burden of recharging or replacement if a long sensor lifetime is required. For this reason, devices that extract energy from their surroundings in some way (so called energy harvesting or energy scavenging devices) have attracted attention from many researchers [1, 2]. Ambient light, and temperature differences, are useful potential power sources in some applications, but there will be many instances where neither is sufficiently available. Mechanical motion is another energy source that has attracted considerable interest. Such motion sources generally fall into two JMES701 # IMechE 2008
clear classes – low frequency, high-amplitude motion such as human body movement, and high frequency, low-amplitude motion such as machine vibration. In the first category, the motion amplitudes are typically in the order of or greater than the desired device dimensions, whereas in the latter the reverse is generally true. The extraction of energy from motion may be by direct application of force to the mechanism, such as foot strike in a shoe mounted device [3]. More common, and more universally applicable, are the inertial mechanisms. In these, the generator need only be attached to the moving host at a single point, as illustrated in Fig. 1. A proof mass is suspended within the generator such that internal motion is induced when the device frame moves along with the host; electrical power is then generated by a transduction mechanism, which acts to Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
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have the significant advantage of operating effectively over a wide range of source frequency without requiring active tuning. If energy is extracted in both directions of travel, then the maximum power is simply twice Umax divided by the period 2p/v, giving Pmax ¼ mv3 Y0 Zl =p Fig. 1
Schematic of operating principle of linear inertial energy scavenger
damp this internal motion. Most devices described in the literature consider the case of linear internal motion driven by linear source motion, and this case has been extensively analysed in reference [4]. However, devices with rotating masses do exist, particularly for the generation of power in wristwatches. It is the purpose of the current paper to identify the possible classes of rotating mass generators, and to analyse the limits on the ultimately achievable power levels for such devices. Low output power levels are the principal factor limiting the widespread adoption of motion energy harvesting technologies developed to date, and so it is worth examining whether alternative architectures using rotating masses might offer at least the possibility of increased power, and to identify how the power levels should scale with key device and operational parameters, as has been done previously in considerable detail for linear motion devices. The power levels theoretically achievable from linear inertial scavengers are limited by four parameters [5]: the proof mass and internal travel range of the device, m and Zl, and the amplitude and frequency of the source motion, Y0 and v (assuming harmonic source motion). The maximum frame acceleration for harmonic motion is simply v2Y0; unless the damping force per unit mass is below this level, the proof mass will move together with the frame, so there will be no relative motion and thus no work done against the damper. This places an upper limit on the force of mv2Y0, and thus on the energy per transit of Umax ¼ mv2 Y0 Zl
ð2Þ
Consequently, since mass is proportional to volume and maximum displacement to linear dimension, maximum power scales as linear dimension to the fourth power, or as volume4/3. Thus, power density reduces as device size decreases, obviously an undesirable feature for miniaturization. Furthermore, the very strong dependence on frequency means that for the low-frequency group of applications, such as body-mounted sensors, the power density is poor. For example, for a device of dimensions s s 2s, having a cubic proof mass of volume s 3 and a displacement range of s, and using frequency f ¼ v/2p and acceleration a ¼ v2Y0, the power density limit is Pmax =Vol ¼ rsaf
ð3Þ
with r the proof mass density. This is illustrated in Fig. 2, for a source acceleration of 1 g (10 m/s2), and r ¼ 9 g/cc (nickel), for several linear dimensions. As can be seen, in the frequency range 1– 10 Hz and for devices below 1 cm3, as might be the case for biomedical sensors, the power density is only a few mW/cm3 (or equivalently, mW/mm3). Although useful electronic functions can certainly be carried out with mW of power, it would clearly be beneficial
ð1Þ
It is worth noting that this maximum requires use of the full internal travel range, whereas for high frequency sources this may be significantly greater than the excitation amplitude of the source. In these cases, resonant oscillation of the proof mass on its suspension is typically used to obtain the required internal amplitude. For low frequency, high-amplitude sources, on the other hand, nonresonant devices can be exploited [6], and these Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
Fig. 2
Maximum power density of linear inertial generators versus frequency, for various linear dimensions, assuming source acceleration amplitude 10 m/s2 and proof mass density 9 g/cc JMES701 # IMechE 2008
Energy harvesting from motion using rotating and gyroscopic proof masses
to find ways to overcome this limit. In the current paper, two approaches are considered which may allow the limitation given by equation (2) to be surpassed. The first is to eliminate the range constraint on internal motion by using rotating proof mass motion. The second is to replace the inertia of a simple translating mass by the angular rotational inertia of a gyroscopic body. In the gyroscopic case, the magnitude of the inertia depends not only on the size and density of the proof mass, but also on its rotational velocity, and this latter parameter is not directly constrained by the device dimensions. Before discussing these alternative generator types, a further comment on the linear motion power limits described above is justified. Although equation (2) is presented as the level of maximum power for harmonic excitation, it is derived using the assumption that the damping force per unit mass can approach the maximum external acceleration throughout the motion cycle. This, however, also implies that the mass makes each internal transit in negligible time, since the peak external acceleration is by definition only present instantaneously. Although the assumption is therefore physically unrealistic, it does result in quite accurate power limit approximations. This can be illustrated by comparison with a realistic device in which the internal motion as well as the source motion is harmonic. Analysis of this case shows the maximum power to be reduced only by a factor of p/4 [4], that is Pmax ¼ mv3 Y0 Zl =4
ð4Þ
(Note also that in some references Zl is defined as the maximum internal motion amplitude, rather than the range, in which case the factor ‘/4’ in equation (4) is replaced by ‘/2’.) If harmonic internal motion is not required, the proof mass can be allowed to make each internal transit in less than half a cycle (resting at either end between transits). This allows a larger average force closer to the maximum peak value mv2Y0 to be employed, and brings the achievable power even closer to that of equation (2). The possible improvement is greater for cases where Y0 Zl. 2
29
energy extraction mechanism, and any parasitic damping mechanisms. Although little described in the literature, such devices have been commercially available for some years as electronic self-powering watches, for example those sold by Seiko Corp. under the Kinetic brand name. These are a development of the purely mechanical self-winding watch known since the 18th century [7]. In both cases, the proof mass is roughly semicircular in form, taking up half the area of the watch body and pivoting around a central axis, as illustrated in Fig. 3(a). Although this geometry provides less mass than would be the case for a circular proof mass, which could rotate on its own axis and thus take up most of the device volume, it has the major advantage of allowing rotating motion to be obtained from linear, as well as rotational, excitation. In the case of the electronic watch, the rotating mass is coupled to a gear train to increase the speed, and through this drives a highly miniaturized electromagnetic generator. The watch application is one in which the linear excitations are generally of larger amplitude than the device dimensions, and for this reason resonant operation is not essential. Furthermore, rotating structures are more suited to sliding (as opposed to flexural) coupling between moving parts and frame, as performance is generally better for rotating bearings than for their linear counterparts. And clearly there is no displacement constraint on the proof mass internal motion, since it can rotate in either direction without mechanical limits. A key question is whether this lack of displacement limit offers the possibility of enhanced power density compared with devices having linear internal motion. The non-resonant rotating device can be analysed with reference to the notation of Fig. 3(b). The rotational positions of the frame and proof mass are denoted by V and u, respectively, and the linear position of the frame and proof mass (centre of mass) are denoted by y and x, respectively. R and Rc indicate, respectively, the proof mass radius, and the distance from the axis to the centre of mass. In the case depicted of a half-circular mass, Rc ¼ 4R/3p. The mechanism of the energy extracting transducer is
NON-RESONANT OSCILLATING ROTATIONAL DEVICES
Although most demonstrated linear motion generators are resonant, having a spring suspension coupling the proof mass to the frame, the only reported rotating mass generators are non-resonant, having a proof mass rotating freely on a bearing. Coupling between the rotational motion of the proof mass and the frame movement is caused only by the JMES701 # IMechE 2008
Fig. 3
Schematics of non-resonant rotating generator proof mass: basic structure (a) and notation for analysis (b)
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E M Yeatman
denoted by D, a torque applying coupling between frame and mass rotation. Both linear and rotational motion of the frame can provide a source for energy extraction, and rotation is first considered. In order to obtain equations of motion, it is necessary to choose a mathematical form for the damping. This will depend on the transduction mechanism chosen, as is the case for linear motion devices. The only existing rotating devices use conventional electromagnetic conversion, which can be well approximated by a torque proportional to relative rotational velocity, with D the damping coefficient, that is † † TD ¼ D V u
ð5Þ
††
†
†
ð6Þ
using dot notation for the time derivatives. For the half disc as shown, I ¼ mR 2/4. Now a harmonic drive V(t) ¼ V0sin(vt) is introduced, and since the system is linear, a harmonic response u(t) ¼ u0 sin(vt þ f) can also be assumed. Inserting these into equation (6) and applying a conventional Laplace analysis, the response amplitude and phase are obtained as functions of the damping strength
u0 D ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V0 D þ v2 I 2
ð7Þ
Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
ð8Þ
For a weakly damped case, the absolute proof mass motion is small, and so the relative motion approaches that of the source (in anti-phase). At the strongly damped limit, internal motion approaches zero, and so u(t) approaches V(t). At both extremes the extracted power approaches zero. Instantaneous extracted power is given by the damping torque times the relative velocity, that is ††
ð u Þ2 P ¼ DðV uÞ ¼ I D †
Besides representing an important class of physical implementation, this also has the advantage of leading to equations of motion for which analytic solutions can be straightforwardly obtained, and therefore this form of damper will be adopted for further analysis. However, it must be noted that other mechanisms are possible, particularly electrostatic damping. The electrostatic mechanism is best modelled as a so-called Coulomb force, which is a constant force (or torque in the rotating case) with direction opposing the relative motion, and does not lead to closed form solutions of the equations of motion. This analysis, and the comparison between transduction mechanisms more generally, has been extensively described in reference [4] for the linear motion case. Maximum power levels were found to be the same for operation at resonance for electrostatic and electromagnetic operation, and to vary somewhat off resonance. Similar conclusions are likely to apply for rotating devices. Using equation (5), and taking I as the moment of inertia of the proof mass about the device axis, the differential equation of motion can be obtained I u ¼ DðV uÞ
u0 f ¼ cos1 ð Þ V0
†
2
2
ð9Þ
so for the harmonic case, using equation (7) and †† u ¼ 2 v2u, the average power can be derived as kPl ¼
I 2 v4 u20 I v3 V20 D/vI ¼ 2D 2 1 þ ðD/vIÞ2
ð10Þ
for which the condition for maximum power is easily obtained as D ¼ vI, for which (taking I ¼ mR 2/4) kPlmax ¼
mv3 V20 R2 16
ð11Þ
This has a similar form to the maximum power of the linear motion devices as given by equation (2), and so provides a straightforward basis for comparing the potential of these types of generator. At the optimum damping level
f¼
p ; 4
V0 u0 ¼ pffiffiffi ; 2
V0 C0 ¼ pffiffiffi 2
ð12Þ
with C0 the amplitude of the relative motion, C(t) ¼ u(t)2V(t). Now practical conclusions can be drawn for this system. First, there is the theoretical possibility of having large internal displacements (i.e. rotational amplitude of multiple cycles). However, according to equation (12), this requires a source excitation having an even larger amplitude. Practical sources of rotational oscillation on the other hand, such as limb motion or machine rocking, are likely to have small amplitudes, probably below p/4 rad. In that case, comparing the potential power to that of the linear case in equation (2), the V20 term in equation (11) will be significantly less than 1, and the R 2 term (and thus the power) only compares favourably with Y0Zl in the linear case, if the latter has a source amplitude Y0 significantly less than the device dimensions. In such a case a linear system is likely to be require resonant operation to extract power JMES701 # IMechE 2008
Energy harvesting from motion using rotating and gyroscopic proof masses
effectively, so this comparison is better made with the resonant rotating case (discussed below). Now excitation by linear harmonic source motion y(t) ¼ Y0 sin(vt) is examined. In this case, it is assumed that the frame does not rotate, so that u now represents the internal (relative) as well as the absolute angular orientation of the mass. The linear and angular position of the proof mass are simply linked by xðtÞ ¼ yðtÞ þ Rc sinðuðtÞÞ
ð13Þ
Motion of the frame will result in a linear force Fy on the proof mass axle. This will generate a torque on the proof mass of magnitude FyRc cos(u)around its centre of mass. Summing torques ††
†
I u þ Fy Rc cosðuÞ þ Du ¼ 0
ð14Þ
Taking the second derivative of equation (13) x€ ¼ y€ þ Rc cosðuÞu€ Rc sinðuÞu_
2
ð15Þ
The axial force Fy is the only linear force acting on the proof mass, therefore Fy ¼ mx¨, so using equation (15) Fy can be substituted in equation (14) to obtain ††
†
2 I u þ mR2c cos2 ðuÞu€ mR2c sinðuÞ cosðuÞu_ þ Du ¼ my€ Rc cosðuÞ
ð16Þ
This is the equation of motion linking the driving function y(t) to the angular response u(t). It is nonlinear and, as would be expected, does not yield harmonic response solutions for harmonic excitation. In the absence of explicit solutions for u(t), explicit expressions cannot be obtained for the extracted power. However, the power limit can be estimated by another approach. Rather than analysing the damping power directly, the power applied by the axial force Fy to the system is considered. Integrating the work done by this force over a cycle will give the energy extracted by the energy harvesting transducer, since there are no other dissipative mechanisms present in the formulation. Taking dU as the increment of work done, we have dU ¼ Fy dy ¼ mx¨ dy. Now introduce a quantity for the relative linear displacement of the proof mass, z(t) ¼ x(t) 2 y(t). Then x¨ ¼ y¨ þ z¨, so dU ¼ y€ dy þ z€ dy m
ð17Þ
The first term on the right integrates to zero over a cycle, so the extracted power can be obtained by JMES701 # IMechE 2008
31
integrating the second term. Although z(t) is not harmonic, it is assumed that it is periodic for a periodic excitation, in steady state, P and so can be written as a series expansion z(t) ¼ zn sin(nvt þ fn), and its second derivative as a corresponding series. Using dy ¼ vY0 cos(vt)dt, and neglecting the y¨ dy term since it will integrate to zero, equation (17) can be rewritten as X dU ¼ v 3 Y 0 n2 zn sinðnvt þ fn Þ cosðvtÞ dt m
ð18Þ
It can be seen from equation (18) that all terms for n . 1 will integrate to zero over one excitation cycle, and therefore these terms do not contribute to extracted power. These terms, with multiple oscillations of z(t) per excitation cycle, correspond to proof mass rotation by more than +p/2 rad, and so equation (18) indicates that enhanced output power cannot be obtained by the use of large internal rotation amplitudes. The average power can be obtained by integrating the first term in dU from equation (18) over a period and dividing by the period, to give 1 kPl ¼ mv3 Y0 zl cosðf1 Þ 2
ð19Þ
Although the range of z(t) is limited to +Rc, the amplitude z1 of its fundamental component can be greater. However, Rc can be taken as a reasonable approximation to the maximum z1, and since cos(f1) 4 1 1 kPlmax ffi mv3 Y0 Rc 2
ð20Þ
Since the internal motion range is 2Rc, equation (20) is exactly equivalent to equation (4). Therefore, within the limits of assumptions and approximations, it can be concluded that for a given proof mass size and linear travel range, the extractable power from an inertial energy harvester is the same whether the internal motion is linear or rotating. Which of these is the preferred choice for a particular application is therefore likely to depend on practicalities of implementation. In cases where the size and cost of conventional rotating bearings can be borne, as in watches, the rotating option is likely to be preferred. For a monolithic silicon microengineered device (MEMS), since practical rotating bearings are less developed, linear proof mass motion using a flexural suspension is preferred. Rotating MEMS flexures exist, but tend to have quite limited angular range and relatively excessive size. However, it should be noted that hybrid integration of rotating MEMS generators (powered in this case by air flow) with miniature conventional bearings has been demonstrated Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
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[8], and such a hybrid approach may prove to be well suited to the energy harvesting case. Furthermore, MEMS gas bearings for high speed rotating devices have also been reported [9]. The non-resonant rotating generator shares with its linear motion counterpart the considerable benefit that effective operation is possible over a wide input frequency range, and indeed if the damping coefficient D is dynamically variable, optimum energy extraction over a wide source frequency and amplitude range may also be achievable. This suits well the body-powered application. Furthermore, with an eccentric proof mass, the same structure may draw power from both linear and rotational excitation. Although only electromagnetic transduction has been achieved in rotating harvesters, electrostatic systems should be possible. However, the lack of a defined travel range means that self-synchronous operation, as achieved in reference [6] using physical contact switches, will not be possible, so electronic synchronized switching will be required. Piezoelectric transduction requires direct mechanical coupling between the proof mass and frame, and so would appear to be incompatible with devices having unconstrained proof mass rotation. The choice of transduction mechanism does not affect the maximum power attainable from inertial energy harvesting, but does influence how near the theoretical maximum a practical device can reach. The same can be said of the circuit coupling the transducer to the load. For example, where the source amplitude is large, devices require a strong damping force to operate optimally, and this can be difficult for any transducer type. Electrostatic damping forces are limited both by the magnitude of capacitance that can be achieved in a mechanically variable form for a given volume, and by the maximum electric field that can be applied. Electromagnetic devices depend on rapid change of flux linkage to achieve strong damping, which is limited by the practical number of coil turns, the magnetic flux gradient, and the internal velocity, which is low for low-frequency operation. Piezoelectric devices have the additional problem that effective coupling of the electrical load is generally prevented because the output impedance of the piezoelectric element is dominated by its capacitance. Furthermore, all systems suffer from inefficiencies in the power conversion circuitry: electrostatic devices lose power easily to parasitic capacitance in the load circuit, and electromagnetic devices often lack efficient rectification because of their low output voltages. All these issues are extensively considered in the literature for linear motion devices [4, 5], and the conclusions are generally equally relevant to the rotational devices described here. Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
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RESONANT OSCILLATING ROTATIONAL DEVICES
As mentioned earlier, the non-resonant rotating mass energy harvester fails to gain a power advantage from the absence of an inherent constraint on the internal motion range. As in the linear motion devices, resonant operation is needed to exploit large internal motion amplitude. Such a resonant device will not, however, overcome the conclusion inherent in equation (18) that a linearly excited system cannot benefit from multiple rotations per excitation cycle. Therefore, only the case of excitation by a rotating source will be considered. To implement a rotationally resonant system, the proof mass must be coupled to the frame with a spring as well as a damper. This is illustrated in Fig. 4(b), in this case with a full circular mass, cut away to show the spring (damper not indicated). Using the previous notation, with the spring torque being Tk ¼ k(V2 u), and assuming the damping has the form of equation (5), the equation of motion is obtained ††
†
†
I u ¼ DðV uÞ þ kðV uÞ
ð21Þ
This is exactly equivalent to the linear motion case, the analysis of which has been extensively reported [4]. If operated at resonance, a system described by equation (21) achieves a power output which increases without limit as the damping magnitude is reduced towards zero. In practice one of two factors bounds the maximum power: the limit (if any) on the internal travel magnitude (the device size in the linear motion case), or the effects of additional, unwanted (parasitic) damping mechanisms. The strategy for designing and operating a linear motion device depends on which of these factors dominates, as well as on whether the chosen transducer can reach the damping strength required for optimization [5]. In the rotating case, however, the displacement constraint is no longer inherent (although there may be a practical limit caused by the spring), and so parasitic damping is taken as the key consideration. Typically, the parasitic effects are approximated as an
Fig. 4
Schematic of resonant rotating generator proof mass (a) and cut-away view showing coil spring (b) JMES701 # IMechE 2008
Energy harvesting from motion using rotating and gyroscopic proof masses
additional linear damping force proportional to the relative velocity, so that both can be represented in equation (21) according to D ¼ De þ Dp, with De and Dp the electrical (transduction) and parasitic components, respectively. The latter may be caused by viscous drag from the air adjacent to the moving body (an effect that can potentially be greatly reduced in microengineered structures by vacuum packaging). Then, following the analysis method in reference [4], it can be found that the internal displacement at resonance will be C 0 ¼ V0
Iv D
ð22Þ
and the useful power output will be I 2 v4 V20 D2e 2De ðDe þ Dp Þ2
Pe ¼
ð23Þ
This gives maximum power for De ¼ Dp, at which 2
PeðmaxÞ ¼
4
V20
I v 8Dp
Q¼
Iv Dp
ð25Þ
The mechanical Q of the structure can be used to recast the achievable power as PeðmaxÞ ¼
I v3 V20 Q 8
ð26Þ
and assuming a half disk mass with I ¼ mR 2/4 (for ease of comparison with the non-resonant case) PeðmaxÞ
mv3 V20 R2 Q ¼ 32
V0 Q 2
ð27Þ
ð28Þ
To compare these results with the non-resonant case in the previous section, it must be noted that the JMES701 # IMechE 2008
Prot V2 R ffi 0:11 0 Qrot Plin Y0
ð29Þ
Considering the case of a small wrist mounted device during walking motion, a linear amplitude Y0 of 10 cm, a rotational amplitude V0 of 0.3 rad, and a proof mass radius of 5 mm might be obtained. Then the ratio Prot/Plin ’ 51024Qrot. Thus in this case, with a fairly large ratio of linear excitation amplitude to device size, a Q greater than 2000 is needed to achieve higher power from the rotating device. Even if this could be achieved, it would imply an excessive sensitivity to drive frequency. It would also require an unrealistically large rotation range for the spring. As in the non-resonant case, then, it is concluded that linear excitation is preferred to rotational excitation if a linear drive amplitude greater than the device dimensions is available, which is likely to be the case on a host body which provides rotational oscillation of significant amplitude. 4
Also, for this damping level, since D ¼ 2Dp ¼ 2Iv/Q, the internal displacement will be C0 ¼
latter assumed there was no parasitic damping, which accounts for the lost factor of 2 between equations (27) and (11). Thus the achievable power is increased by a factor Q for the resonant case, as long as the internal motion range of equation (26) can be achieved. If, for example, a spring is implemented that allows a full rotation about equilibrium in either rotation, and the driving motion has an amplitude of 308, then a Q up to 24 (a modest level for a MEMS device) can be effectively exploited, with an equivalent increase in attainable power. Conversely, if the rotating mechanism has an inherent mechanical Q of 10, the full factor 10 power increase can be reached for an internal rotation range only five times the driving amplitude. Of course, just as in the linearly translating case, the power output of a high-Q device will drop rapidly as the drive frequency moves away from resonance. Finally, the potential power of the resonant rotating device is compared with the linearly driven one. Taking the ratio of equations (27) to (4), and using Rc ¼ 4R/3p
ð24Þ
A quality factor Q for the rotating mass can be defined as the ratio of angular kinetic energy stored in an undriven device, U ¼ Iv2u20/2, to the energy lost to parasitic damping per radian of rotation, DU ¼ Dpvu20/2. This yields
33
EXCITATION BY CONTINUOUS SOURCE ROTATION
Angular oscillation is a possible motion source for both body-mounted and machine-mounted devices. Continuous rotation is another possible source worth investigating, with possible applications including tire pressure sensors or condition monitoring on rotating machines. This application can be analysed with reference to Fig. 3(b). The excitation is in this case a constant frame rotation dV/dt ¼ vd. A velocity damper is assumed as in the oscillating Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
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E M Yeatman †
†
case, giving electrical torque Te ¼ DðV uÞ. This is counteracted by a gravitational torque Tg ¼ mgRc cosu with g the acceleration of gravity. The equation of motion is then ††
†
I u ¼ Dðvd uÞ mgRc cosðuÞ
ð30Þ
Extraction of power depends on a relative velocity, i.e. †
†
u , V. This could be provided by slippage, with a finite proof mass rotation, but because of the dependence of the gravitational torque on u, the motion would be highly non-linear in such a case. However, a straightforward solution can be obtained for the case where the proof mass remains static, such that ††
†
u ¼ u ¼ 0. Then D ¼ mgRc cos(u)/vd, which has a maximum value at u ¼ 0, i.e. D ¼ mgRc/vd. This †
†
gives a maximum power, using P ¼ DðV uÞ2 , of P ¼ mgRc vd
ð31Þ
For example, if there is a semi-circular mass of 1 g, and Rc of 2 mm, then taking the rotational speed in r/min, P (mW ) ¼ 2.1 fr/min. Thus, 1 mW power can be reached by using 500 r/min. For a tire pressure sensor application, this speed corresponds to a vehicle speed of about 50 – 60 km/h (dependent on tire diameter). Equation (31) can be compared with the linear case using equation (2), but the comparison is clarified by substituting in equation (2) the maximum acceleration for a harmonic linear excitation, a0 ¼ v2Y0, giving Plin ¼
ma0 zl v p
ð32Þ
For any given restraint on device size, zl/p and Rc will be of similar magnitude. Then the relative power levels attainable from linear oscillation and continuous rotation depend only on the vibrating source acceleration a0 versus the acceleration of gravity g. This suggests that continuous rotation is an attractive power source for scavenging, particularly for applications such as machine monitoring, and one that does not appear to have been exploited hereto. It should be noticed that devices based on this principle would not strictly be inertial scavengers, as they depend on the gravitational rather than the inertial proof mass, although from a practical point of view they are equivalent, requiring only a single point of contact to a moving host. A key practical difference is that devices exploiting gravitational torque, unlike properly inertial devices, have a performance strongly dependent on their orientation. Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
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GYROSCOPIC GENERATORS
A further option for rotating structures can be considered. The fundamental limit of equation (2) on translating devices contains two parameters of the source and two of the device; the latter are the internal displacement, discussed above, and the mass. The extracted power is proportional to the damping force of the transduction mechanism, but if this force exceeds ma, where a is the source acceleration, it will be too great and the mass will simply cease to move against the damper. The equivalent consideration in the rotating case is that the transducer torque cannot exceed the moment of inertia of the proof mass times the angular acceleration if internal motion is to occur. In both cases the result of excessive damping is that the proof mass simply moves along with the host. However, gyroscopic action provides a way to greatly enhance the ability of a mass to resist changes to its position (although only its angular, not its linear, position). This is the basis of its value in navigation instruments. Thus, it is worth examining the potential of gyroscopic structures for energy scavenging. This has been introduced for the first time in reference [10], and is examined here in more detail. A possible architecture is that shown schematically in Fig. 5. A disk shaped mass is actively driven to rotate around the z-axis at a speed vd. It is mounted in an inner frame, which is coupled to an outer frame via a torsional damper. If an external rotation is applied around the y-axis, as a variation of the angle V as shown, then this couples with the driven rotation to produce a gyroscopic torque Tg about the x-axis, inducing a change to u. This change is resisted by the electrical torque Te applied by the transducer. The gyroscopic torque, given by Tg ¼ Izvd dV/dt, sets a maximum on Te, which multiplied by the maximum angular deviation, 2u0, gives the maximum work per half cycle. If the excitation is harmonic, with amplitude V0 and frequency ve, then the maximum value of dV/dt will be veV0. Thus the
Fig. 5
Schematic of possible non-resonant gyroscopic inertial generator JMES701 # IMechE 2008
Energy harvesting from motion using rotating and gyroscopic proof masses
maximum power is straightforwardly obtained as Pmax ¼
2 V0 u0 Iz v2e vd p
ð33Þ
However, clearly power is needed to drive the gyroscopic mass, and this reduces the maximum net power of the device. If a mechanical Q of the driven mass is defined as the stored energy U(v) divided by the energy lost per radian of rotation through parasitic damping, then the drive power needed to compensate this parasitic loss will be Pd ¼ Uvd/Q, which taken with U ¼ 12Izv2d gives Pd ¼
Iz v3d 2Q
ð34Þ
The net power is given by the difference between equations (33) and (34), i.e. Pnet ¼ Pmax 2Pd. Since the drive power increases more quickly than output power with drive frequency vd, it can be inferred that there is an optimal value of vd to maximize Pnet. By taking the derivative of Pnet with respect to vd and setting this equal to zero
vdðoptÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ¼ ve u 0 V0 Q 3p
ð35Þ
Substituting this in equations (33) and (34) and taking the difference, the maximum net power can be obtained
Pnet
4 V0 u 0 ¼ 3p
3=2
pffiffiffiffi Iz v3e Q
ð36Þ
This is similar in form to the resonant rotating case given bypequation (26), except that the power increases only as Q, rather than as Q. However, there is a key difference. In the resonant case, exploiting a high Q value necessarily results in a device with very narrow input frequency response, a severe disadvantage in most applications unless dynamic tuning can be implemented. However, in the gyroscopic case the Q only implies a low loss driven rotation; the responsiveness to excitation frequency remains broadband, as in the passive non-resonant case of equation (11). Compared with that case, then, gyroscopic energy scavenging offers a potentially large power advantage without loss of operating range. If we compare the gyroscopic case with the linearly excited case of equation (2), using Iz ¼ mR 2/2 for the gyro Pgyr R pffiffiffiffi Q ffi 0:22ðV0 u0 Þ3=2 Plin Y0 JMES701 # IMechE 2008
ð37Þ
35
Even more than in the case of the resonant device, if the ratio R/Y0 is small, very large Q factors appear to be needed to obtain an advantage. However, good rotational spinning bearings are beginning to appear in MEMS technology. In reference [11], a magnetic bearing system is described for a 3-mm disk having a moment of inertia of 7.7210215 kg m2, and air drag is estimated at 2.7410211 Nm at 200 Hz (12 000 r/min). From these, a Q of 445 is obtained. Since the torque due to air drag is estimated to vary as v1.5, Q should increase with increasing speed. The need for a high Q could be partly offset by implementing a structure that allows a large internal reaction u0. For example, the inner and outer frames could be made to allow continuous rotation within each other without limit, either resonantly or non-resonantly. Although this requires two sets of rotating bearings on orthogonal axes, and is mechanically rather complex, it is not inherently infeasible. It is tempting to consider driving the gyroscopic device by linear motion. However, this would require us to convert linear motion of the source into a rotational disturbance, which can be done using the use of an offset pivot, as in Fig. 3. This depends for its functioning on the torque produced by the linear inertia of the proof mass. Thus, the maximum excitation torque will simply be product of the mass m, the source linear acceleration, and the lateral offset between the pivot and the centre of mass. If a transducer torque larger than this is applied, the mass will simply not rotate with respect to the frame, and no energy will be obtained. This differs from the rotating source in that in the latter case, it is assumed that the outer device frame always follows the host structure’s movement and doesn’t significantly impede it, i.e. the host can provide limitless torque (or force in the linear case). As stated above, MEMS devices typically use flexures to support moving parts, because of the high mechanical resistance, low precision, and low lifetime of sliding bearings at submm size scales. For this reason, high angular rate, low loss rotating gyros are not currently feasible in MEMS. However, another option has long been known and exploited for MEMS gyroscopic sensors: namely the vibrating gyro [12]. This structure (Fig. 6) typically has a proof mass vibrating in two orthogonal directions in-plane, one the sense direction and one the driven direction. Rotation about the out-of-plane axis couples these oscillations, allowing the sense signal to be obtained. This is a well-developed structure, highly suited to MEMS fabrication, that provides a promising basis for a gyroscopic energy scavenger. Analysis of vibrating MEMS gyroscopes has been extensively reported Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
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gyroscopic proof masses is introduced. This technique is shown to offer potentially enhanced power densities along with broad frequency response, although very-low-parasitic drag levels are shown to be needed to achieve these power enhancements.
REFERENCES Fig. 6
Schematic of MEMS vibrating gyro. After reference [13]
Rotational approaches to inertial energy scavenging methods have been analysed in detail for the first time. These offer the possibility of enhancements in attainable power levels for motion energy scavengers, but low mechanical resistance is a critical requirement. For excitation by linear source motion, the ultimate power levels for rotational and linear internal motion are similar, so the choice between them will depend mainly on practical constraints in particular applications. Resonant rotating-mass devices have the possibility of high power densities, but depend on the achievement of springs with large angular ranges, in addition to low-parasitic drag effects. A method to exploit continuous source rotation is introduced, and attainable power levels as a function of rotation speed are derived. Finally, the use of
1 Starner, T. and Paradiso, J. A. Human generated power for mobile electronics. In Low-power electronics design (Ed. C. Piquet), 2004, pp. 1–35 (CRC Press, Boca Raton, FL) 2 Roundy, S., Steingart, D., Frechette, L., Wright, P., and Rabaey, J. Power sources for wireless sensor networks. In Proceedings of Wireless Sensor Networks, Lecture Notes in Computer Science, 2004, vol. 2920, pp. 1–17 (Springer Berlin, Heidelberg, Germany). 3 Shenck, N. S. and Paradiso, J. A. Energy scavenging with shoe-mounted piezoelectrics. IEEE Micro, 2001, 21, 30 –42. 4 Mitcheson, P. D., Green, T. C., Yeatman, E. M., and Holmes, A. S. Architectures for vibration-driven micropower generators. J. Microelectromech. Syst., 2004, 13, 429– 440. 5 Mitcheson, P. D., Reilly, E. K., Toh, T., Wright, P. K., and Yeatman, E. M. Performance limits of the three MEMS inertial energy generator transduction types. J. Micromech. Microeng., 2007, 17(9), S211 –S216. 6 Mitcheson, P. D., Miao, P., Stark, B. H., Yeatman, E. M., Holmes, A. S., and Green, T. C. MEMS electrostatic micropower generator for low frequency operation. Sens. Actuators A, Phys., 2004, 115, 523 –529. 7 Chapuis, A. and Jaquet, E. The history of the selfwinding watch 1770– 1931, 1956 (Editions du Griffon: Neuchatel). 8 Holmes, A. S., Hong, G., and Pullen, K. R. Axial-flux permanent magnet machines for micropower generation. J. Microelectromech. Syst., 2005, 14, 54– 62. 9 Frechette, L. G., Jacobson, S. A., Breuer, K. S., Ehrich, F. F., Ghodssi, R., Khanna, R., Wong, C. W., Zhang, X., Schmidt, M. A., and Epstein, A. H. High-speed microfabricated silicon turbomachinery and fluid film bearings. J. Microelectromech. Syst. J., 2005, 14, 141 –152. 10 Yeatman, E. M. Rotating and gyroscopic MEMS energy scavenging. In Proceedings of 3rd International Workshop on Wearable and Implantable Body Sensor Networks, Boston, 3–5 April 2006, pp. 42 –45 (IEEE, Piscataway, NJ, USA). 11 Coombs, T. A., Samad, I., Ruiz-Alonso, D., and Tadinada, K. Superconducting micro-bearings. IEEE Trans. Appl. Supercond., 2005, 15, 2312–2315. 12 Xie, H. and Fedder, G. K. Integrated micromechanical gyroscopes. J. Aerospace Eng., 2003, 16(2), 65– 75. 13 Xie, H. and Fedder, G. K. Fabrication, characterization, and analysis of a DRIE CMOS-MEMS gyroscope. IEEE Sens. J., 2003, 3, 622 –631. 14 Pourkamali, S., Hashimura, A., Abdolvand, R., Ho, G. K., Erbil, A., and Ayazi, F. High-Q single crystal silicon HARPSS capacitive beam resonators with self-aligned sub-100-nm transduction gaps. J. Microelectromech. Syst., 2003, 12, 487 –496.
Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science
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in the literature, and detailed analysis for the purpose of energy scavenging is beyond the scope of this paper. However, some broad conclusions can be stated. In particular, achieving a large power enhancement depends on having a high mechanical Q. It should also be noted that the displacement of the proof mass in this case is constrained, unlike the rotating case. High kinetic energy must instead be obtained by increasing the drive frequency. To do this while retaining a low-drive power also depends on having a high Q. Although what can be achieved depends strongly on the nature of the device, it is worth noting that very high Q values have been reported for MEMS components. For example, in reference [14] a Q of 177 000 was measured for a single crystal silicon beam vibrating in vacuum at 19 kHz. In section 2, the possible mechanisms by which to extract power were briefly discussed. The mechanism in the vibrating device would be the sense axis electrostatic comb drive, and this presents another interesting advantage of the gyroscopic device concept. The extracted energy comes out not at the low source frequency, but at the higher drive frequency. This could make the design of efficient power conversion circuits more straightforward. 6
CONCLUSIONS
