# A Device for Harvesting Energy From Rotational Vibrations

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3 Input Displacement Mechanical Losses b i α Relative Displacement φ J K Vibration ource Torsion pring Rotor Inertia Air Gap π p r m r oc r ic v M = M eˆ v r M = M eˆ r Permanent Magnets y φ Back Iron (Rotor) urface v Current i = I eˆ z i v = I ˆ e z θ x Back Iron (tator) Electrical Transducer T e Fig. Cross-sectional view of the magnetic circuit to scale Fig. 1 Modeling Lumped parameter model of the harvester Conceptual operation of the device is relatively straightforward. Figure 1 is a lumped parameter model describing the operation. The system acts like a standard spring-mass-damper with base excitation. The device is built inside a casing that is attached to a vibrating source and thus acts as the vibrating reference frame. A spring element is provided by a torsion rod, an inertial mass element by a rotor core, and a damping element by a combination of unavoidable losses in the support bearings and the energy removed from the system represented in the model as an electromagnetic torque. The second order governing differential equation describing the relative motion of the proof mass in terms of the system parameters and the vibration input can be written as + b i J + 1 J T e + K = J 1 This system is well studied in textbooks on vibration, and ideally the frequency of vibration input is matched with the resonant frequency, thus amplifying the magnitude of the relative motion and providing exponential gains in power. ince the energy is harvested in an electromagnetic transducer, the model consists of two main components: a mechanical model and an electromagnetic model. As the magnets move relative to a coil attached to the base, they create a varying magnetic flux field relative to the base that generates an emf voltage in the coil. If a load is attached to the emf voltage potential, energy is removed from the system through the load; as a result, the current flowing through the coil to power the load generates a reaction torque that acts to dampen the vibration. The electromagnetic model calculates the varying flux based on the elements in the magnetic circuit magnets, air gap, and back iron. Based on the flux calculation, the expected voltage in a coil attached to the base is calculated. Assuming a resistive electrical load, the reaction torque is returned to the mechanical model as an additional damping element. Key model components affecting the power output are the mechanical quality factor Q, natural frequency n, rotational inertia J, and electrical damping coefficient b e..1 Magnetic Model. Electromagnetically, the device is similar to a standard rotary generator Fig.. An outer casing acts as focusing back iron for a surface wound coil attached to the inside of the casing or stator. Continuing concentrically inward, an air gap separates the coil from a set of magnets attached to a rotor that acts as focusing back iron for the magnets. Rare earth permanent magnets on the rotor provide a magnetic field in the air gap between the magnets and steel casing. As the rotor moves relative to the case, the magnetic field thorough the coils changes and generates an emf voltage. According to Faraday s law of induction, the emf voltage can be written as emf = = d dt = d d d dt = d d Under some simplifying assumptions, the quasi-static versions of Maxwell s equations can be used to solve for analytically. First, ignoring fringing fields at each end of the rotor, the D solutions for Maxwell s equations can be used. econd, comparing the magnetic permeabilities and relative dimensions, the reluctance of the focusing iron is sufficiently less than the reluctance in the air gap so that the iron can be assumed to be infinitely permeable and a perfect conductor, which simplifies the boundary conditions. Under these assumptions, the solution for the magnetic flux density is shown by Ref. 8 to be B B a = R k=1 k 1 r ar sinpk 1ê r a r 3 r a cospk 1ê Although Eq. 3 is an infinite summation as k, the magnitude of the higher order harmonics tends to zero. As can be seen in Fig. 3, for the dimensions of the proposed prototype, the relative error between truncated estimates of the flux density amplitude evaluated at the casing surface quickly tends to zero. In fact, the relative error between an estimate using only the first term, k=1, and an estimate using the first and second terms, k=1,, is less B r RM[B rk B rk 1 ] Radial flux density.5 k= 1 k= k= 3 k= θ x1 3 Relatve RM error ummation term (k) Fig. 3 Plot of the calculated radial flux density, B r, evaluated at r oc. Although B r is an infinite summation, as can be seen, the influence of higher order terms is small and can be neglected / Vol. 13, EPTEMBER 1 Transactions of the AME

5 A G F C D H B E G A F Voltage [V] Calc. Meas. Fig. 6 olid model exploded and cross section of the prototype design: A, end caps/bearing supports; B, casing; C, rotor; D, magnets; E, torsion rod; F, torsion rod clamps; G, bearings; H, coil 3 Prototype Design and Testing The theory was used to create a design code that enabled an experimental prototype to be designed, built, and tested. The device was required to fit into a defined cylindrical form factor of 1.5 in mm diameter and 5 in. 17 mm length. The device is comprised of a 1.5 in mm diameter,.1 in..5 mm thick steel cylinder casing, which provides the structural ground/attachment for the device as well as acts as the stator back iron. An electrical coil wound from a 3AWG magnet wire is attached to the inner surface of the steel. A.9 in..9 mm outer diameter rotor assembly comprised of a steel core with neodymium-boron permanent magnets attached around the outside is located in the center of the device by ball bearings at each end. At one end of the rotor assembly, a itinol piano wire is rigidly affixed to the steel core and passes axially through the hollow center of the core where it is rigidly affixed to the casing and acts as the torsion spring Fig. 6. Although in operation the electrical and mechanical models are coupled, the prototype is designed so the electromagnetic model can be tested separately from the mechanical model. Thus, each of the model parameters can be verified independently. tarting with the electromagnetic model, the model is verified by comparing the expected emf to the measured output from the device under a constant rotational velocity input the torsion spring is removed so Time [s] Fig. 8 Plot of the emf voltage versus time in open circuit operation for a single frequency input. As can be seen, the experimentally measured voltage is nearly identical to the predicted voltage. the rotor can spin freely with respect to the stator. As can be seen in Fig. 7, when operating the harvester like a generator, the predicted voltage output is nearly identical to the measured voltage, suggesting that the proposed electromagnetic model is correct. Turning now to the mechanical model, the model is verified piecewise. The moment of inertia of the rotor is determined from the solid model and is verified by comparing the predicted mass and dimensions of the solid model to the measured mass and dimensions of the device. The natural frequency of the system is verified by comparing the maximum amplitude of the open circuit voltage to a swept single frequency sinusoidal input. The mechanical Q of the system is experimentally determined by comparing the calculated open circuit voltage to the measured voltage for a single frequency sinusoidal input. The open circuit voltage is determined by solving the mechanical model Eq. 13 for the relative motion where in open circuit b e = and then using the calculated in the electrical model Eq. 6 to determine the voltage. Figure 8 is a plot of the voltage as a function of time for a single frequency harmonic input, and Fig. 9 is a plot of the voltage amplitude as a function of frequency for a single frequency harmonic input slowly swept through the frequency spectrum. The agreement of the experimental and calculated values in Figs. 8 and 9 suggests that the assumptions made to create the model are appropriate and represent the form of the equation and additionally suggests that the physical parameters have been properly calculated. Combining all the components for a single frequency input of the same typical frequency and amplitude as the Fourier decomposition of an example realistic acceleration input Fig. 1, the Voltage [V] 3 1 Calc. Meas Freq [Hz] Fig. 7 Verification of emf model V= sin pt+ and =.73We Fig. 9 Plot of the amplitude of emf voltage as a function of frequency for a single frequency input. Each of the x s represent an independent experiment at the input frequency shown. As can be seen, the experimentally measured voltage coincides with the predicted voltage / Vol. 13, EPTEMBER 1 Transactions of the AME

6 Rotational Accel. rad s s Time [s] Frequency [Hz] Fig. 1 Example real-world signal the device is expected to produce energy from. hown are the time and frequency decompositions of the desired signal and the actual signal. The input vibration was provided to the device without feedback control; however, as can be seen, the actual input signal is still representative of a wide-band vibration as commonly encountered. model predicts that mw can be harvested, and experimental tests verify a harvested power of about 5 mw Fig. 11. As mentioned previously, a primary challenge to resonant based energy harvesters in application is the wide, noisy nature of realistic vibrations. As an example, Ref. 11 presented a linear vibration spectrum that was excited by a rotating system. The spectrum presented shows multiple frequency peaks in close proximity, which will cause interference with resonance. Figure 1 is a plot of an example vibration spectrum. Given this input, the theoretical model predicts 85 mw average power during the 5 s input, and experimental tests verify an average harvested power of 7 mw. Figure 1 is a plot of the voltage as a function of time with the acceleration input from Fig. 1 and loaded by a 1 resistor. As can be seen, the model accurately predicts the voltage across the load and thus the power from the wide-band signal. otice that although the maximum acceleration amplitude of the vibration input is the same as the single frequency test, the harvested power is reduced by roughly a factor of. This illustrates one of the ongoing problems of resonant based harvesters. Possible solutions Power [W] Calc. Meas Freq [Hz] Fig. 11 Plot of the power versus frequency for a single frequency input at approximately the same amplitude as the example realistic signal shown in Fig. 1 Voltage [V] - umerically Calculated Measured Time [s] Fig. 1 Plot of the voltage across a load resistor as a function of time for the real-world input shown in Fig. 1 to this problem in the literature include multifrequency resonators such as those presented in Ref. 1 or tunable harvesters such as those presented in Refs. 13,1; however, these approaches only work if the input consists of highly separated distinct frequency peaks. For noisy or closely spaced frequency spectra, a viable solution is still needed. Conclusion As outlined, the aim of this work was twofold. First, due to the random nature of most real-world vibrations, the design approach was to create and verify a detailed model that can be used to accurately predict the power output of torsional vibration harvesters when subjected to random inputs. For a single frequency harmonic input, Ref. 1 shows that the maximum power that can be harvested is P max = J Q n which for the prototype harvester with an assumed quality factor of 6 would predict a maximum power output of 5 mw. However, when the prototype and the model are subjected to a wideband real-world input, the power falls off by nearly an order of magnitude. However, the more detailed model accurately predicts the reduced power. Based on the accuracy of the models when compared with the prototype design, the design tools can be used with confidence for evaluating the potential power that can be harvested from real world signals. Additionally, with some changes to the electromagnetic model, the basic structure can be easily adapted to linear systems for similar evaluations. econd, a macrosized harvester placed in the center of a rotating system designed to directly harvest torsional vibrations is a viable method to harvest energy from rotating systems. Although a direct comparison of a device attached to the exterior of the rotating system and harvesting the near linear acceleration inputs of a rotationally oscillating system at some distance from the centerline with a device similar to the one presented would be ideal, this comparison is not currently available. As a substitute, Eq. 15 shows that normalizing the power by the inertia, acceleration amplitude squared, and natural frequency provides a reasonable power density for comparison. ote that for harmonic inputs, this is the same as normalizing by the mass, velocity, and acceleration, which come directly from the definition of power. However, since volume is more commonly reported and mass and volume can be assumed to be proportional, an approximate normalization using the device volume is used. Table 1 shows the comparison of power densities. As can be seen, the torsional device has comparable power densities, suggesting that this approach is viable. omenclature B magnetic flux density B a magnetic flux density in air gap region B R remanence of permanent magnets H magnitude of the complex frequency response function relating input acceleration and output velocity J mass moment of inertia of proof mass K torsion spring constant number of turns in the coil Q mechanical quality factor R coil electrical resistance in coil R load electrical resistance of load power spectral density of the input acceleration T torque T e electrical transducer torque b e equivalent electrical damping coefficient b i internal damping coefficient h axial dimension of the coils and magnets Journal of Mechanical Design EPTEMBER 1, Vol. 13 / 911-5

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