Jeffrey L. Kauffman

A Low-order Model for the Design of Piezoelectric Energy Harvesting Devices

Piezoelectric energy harvesting devices are an attractive approach to providing remote wireless power sources. They operate by converting available vibration energy and storing it as electrical energy. Currently, most devices rely on mechanical excitation near their resonance frequency, so a low-order model which computes a few indicators of device performance is a critical design tool. Such a model, based on the assumed modes method, develops equations of motion to provide rapid computations of key device parameters, such as the natural frequencies, mode shapes, and electro-mechanical coupling coefficients. The model is validated with a comparison of its predictions and experimental data.

Piezoelectric-Based Vibration Reduction of Turbomachinery Bladed Disks via Resonance Frequency Detuning

Piezoelectric-based resonance frequency detuning can alleviate unwanted vibration of turbomachinery blades, thus reducing the dangers of high-cycle fatigue while also decreasing the blade weight. This semiactive approach applies to structures that are subjected to frequency-sweep excitation and involves altering the structural stiffness (here, by switching the piezoelectric electrical boundary conditions) to avoid a resonant condition, thus limiting the blade response. Detuning requires two switches per resonance/excitation frequency crossing, including a switch back to the original the original state,many fewer than other semiactive approaches that require four switches per cycle of vibration.Resonance frequencydetuningapplies to anymode of vibrationwith apositive electromechanical coupling coefficient, and it provides the greatest normalized vibration reduction for slow sweeps, low damping, and high coupling coefficient. Yet even for amoderate sweep rate 10 4 andmodal damping 0:1%, optimally detuning a structure with an electromechanical coupling coefficient k 10% provides the same vibration reduction as increasing either the sweep rate ormodal damping by an order of magnitude.With a lower sweep rate 10 5 and modal damping 0:01%, detuning with a coupling coefficient of only k 3% provides equivalent vibration reduction as an order of magnitude increase in sweep rate or modal damping.

Damping Models for Shear Beams With Applications to Spacecraft Wiring Harnesses

Spacecraft wiring harnesses can fundamentally alter a spacecraft’s structural dynamics, necessitating a model to predict the coupled dynamic response of the structure and attached cabling. Although a beam model including first-order transverse shear can accurately predict vibration resonance frequencies, current time domain damping models are inadequate. For example, common proportional damping models result modal damping that depends unrealistically on the frequency. Inspired by a geometric rotation-based viscous damping model that provides frequency independent modal damping in an Euler–Bernoulli formulation, a viscous damping model with terms associated with the shear and bending angles is presented. The model provides modal damping that is approximately constant in the bending-dominated regime (low mode numbers), increasing by at most 6% for a particular selection of bending and shear angle-based damping coefficients. In the shear-dominated regime (high mode numbers), damping values increase linearly ...

Switch Triggers for Optimal Vibration Reduction Via Resonance Frequency Detuning

Resonance frequency detuning (RFD) is a piezoelectric-based vibration reduction approach that applies to systems experiencing transient excitation through a system resonance. Particularly, this vibration reduction technique can be applied to turbomachinery experiencing changes in rotation speed, such as on spool-up and spool-down. This technique relies on the inclusion of piezoelectric material and manipulation of its electrical boundary conditions, which control the stiffness of the piezoelectric material—the open-circuit condition corresponds to the high stiffness state of the material and the short-circuit condition corresponds to the low stiffness state. When placed in a region of high strain, the altered stiffness of the piezoelectric material causes a global stiffness change in the system. Resonance frequency detuning takes advantage of this effect by switching from the opento the short-circuit stiffness state as the excitation approaches resonance, subsequently detuning the structure from the excitation and reducing the vibratory response. Although other piezoelectric vibration techniques exist that allow for a greater reduction of the response (spanning the range from passive to active approaches), these techniques suffer drawbacks when applied to systems with tight size and power requirements, such as a turbomachinery environment. Resonance frequency detuning simplifies these approaches by relaxing some of these requirements by creating a large broadband vibration reduction approach with limited power, circuitry, and signal processing requirements. For this approach, the peak response dynamics are

'Geometric' Viscous Damping Model for Nearly Constant Beam Modal Damping

This paper considers the damped transverse vibration of flexural structures. Viscous damping models available to date, such as proportional damping, suffer from the deficiency that the resulting modal damping is strongly frequency dependent, which is a situation not representative of experiments with built-up structures. The focus model addresses a viscous geometric damping term in which an internal resisting shear force is proportional to the time rate of change of the slope. Separation of variables does not lead directly to a solution of the governing partial-differential equation, although a boundary-value eigenvalue problem for free vibration can nevertheless be posed and solved. For small damping the method of weighted residuals provides an alternate approach to the development of approximate modal equations of motion and estimation of modal damping. In a discretized finite-element context the resulting damping matrix resembles the geometric stiffness matrix used to account for the effects of membran...

Vibration Reduction of Turbomachinery Bladed Disks with Changing Dynamics using Piezoelectric Materials

Piezoelectric-based vibration reduction can alleviate the unwanted vibration levels of turbomachinery, thus reducing the dangers of high-cycle fatigue while also decreasing the blade weight, drag, and jet noise. Most passive approaches provide limited benet in turbomachinery due to the changing blade dynamics and excitation conditions that make optimally tuning a shunt circuit dicult. Active control typically provides large vibration reduction but requires a power source in the rotating frame. Semi-active approaches seek to balance the advantages of passive and active systems, outperforming the passive approaches while signicantly reducing the power required. This research presents a semiactive approach that applies to excitations with swept frequencies. It involves detuning the structural resonance frequency from the excitation frequency by altering the structural stiness (here by switching the electrical boundary conditions of a piezoelectric element), thus limiting the structural dynamic response. Including a switch back to the original stiness state, detuning requires two switches per resonance / excitation frequency crossing, orders of magnitude fewer than other state switching approaches that require four switches per cycle of vibration. The detuning method applies to any mode of vibration with a positive electromechanical coupling coecient and provides the greatest normalized vibration reduction for slow sweeps, low damping, and high coupling coecient. Yet even for a moderate sweep rate = 10 4 and modal damping = 0:1%, detuning a structure with a coupling coecient k 2 = 10% provides the same vibration reduction as increasing either

Optimal Resonance Frequency Detuning Switch Trigger Determination Using Measurable Response Characteristics

For systems experiencing transient vibration associated with passage through resonance, resonance frequency detuning (RFD) offers a source of vibration reduction by altering the stiffness state of the structure as the swept excitation frequency approaches any resonance frequency. The peak response dynamics and overall maximum amplitude reduction is governed by the sweep rate, modal damping ratio, electromechanical coupling coefficient, and the point at which this stiffness switch occurs, and the optimal switch trigger has been established when all system information is readily available. This study investigates a method of determining the optimal switch trigger using only the open-circuit piezoelectric voltage time response. For the purposes of simulation, a curve fit is employed and a subsequent optimal trigger control law is extracted. This empirical control law agrees well with and produces comparable response to that of the optimal control determined using perfect and complete knowledge of the system.

Performance of piezoelectric-based damping techniques for structures with changing excitation frequencies

The performance of piezoelectric-based damping and vibration control techniques has been studied and analyzed extensively under impulse response or harmonic steady state conditions. Considered here is their performance when subjected to an excitation whose frequency is close to a structures resonance frequency but varies sufficiently quickly to preclude a harmonic analysis. Although a rapidly-varying excitation frequency will reduce the peak response amplitude, additional vibration reduction is often desired. The current research investigates the performance of several common passive and semi-active (state switching) vibration reduction techniques. In many cases, particularly for high electromechanical coupling, a system provides sufficient vibration reduction to approximate a steady state condition. Special attention is paid to turbomachinery bladed disks and the feasibility of implementing a particular vibration reduction approach. Semi-active switching approaches are more robust for vibration reduction of multiple frequencies than passive systems which require optimal tuning to the excitation condition. State switching, synchronized switched damping, and resonance frequency detuning provide the most realistic embedded package. Of these three approaches, synchronized switched damping delivers the greatest performance, although all provide significant vibration reduction. With far fewer and less stringent switching requirements, resonance frequency detuning requires significantly less power than other semi-active approaches.

A Low-Order Model for the Design of Energy Harvesting Piezoelectric Devices

The use of energy harvesting devices is an attractive method of utilizing available mechanical energy by converting it into usable electrical energy. Numerous potential applications exist for energy harvesting devices, including structural health monitoring, discrete actuation systems, and wireless sensor networks, which are considered here. A piezoelectric element is employed to convert mechanical energy from a vibration environment to electrical energy, which is then converted to a regulated power source through an attached circuit. While the devices considered vary in configuration, the essential component is the piezoelectric unimorph or bimorph annular plate with an associated proof mass designed to be mechanically driven near its resonance frequency. The development of a low-order model is a critical step in predicting the device behavior for both the current and future generations of devices. Such a model, based on the assumed modes method, is developed and presented. As employed here, the assumed modes method uses Lagrange’s equations and a computation of the (both electrical and mechanical) potential and kinetic energy and virtual work of the device. The model provides a rapid computation of key parameters such as open- and short-circuit natural frequencies, device coupling coecient, and mode shapes for a device of circular geometry, as well as the ability to produce frequency response functions and time-varying responses to arbitrary forcing functions. Model predictions are compared with experimental data and the model is also used to analyze various physical connections of such plates in a manner like component mode synthesis. A particular strength of the model is the ease with which device parameters, such as the piezoelectric element thickness or device radius, can be changed to evaluate their impact on the device performance. As such, the model is useful when designing the next generation of devices or optimizing a particular configuration, an example of which is presented.

A continuous switching model for piezoelectric state switching methods

Piezoelectric-based, semi-active vibration reduction approaches have been studied for over a decade due to their potential in controlling vibration over a large frequency range. Previous studies have relied on a discrete model when switching between the stiffness states of the system. In such a modeling approach, the energy dissipation of the stored potential energy and the transient dynamics, in general, are not well understood. In this paper, a switching model is presented using a variable capacitance in the attached shunt circuit. When the switch duration is small in comparison to the period of vibration, the vibration reduction performance approaches that of the discrete model with an instantaneous switch, whereas longer switch durations lead to less vibration reduction. An energy analysis is then performed that results in the appearance of an energy dissipation term due to the varying capacitance in the shunt circuit.