energy harvesting

International CAE Conference 2013

21-22 Oct. 2013

Finite element analysis of optimized piezoelectric bimorphs for vibrational “energy harvesting”

Marco Gallina, Denis Benasciutti University of Udine, DIEGM Udine, Italy Email: [email protected], [email protected]

Summary The concept of “energy harvesting” is to design smart systems to capture the ambient energy and to convert it to usable electrical power to supply small electronics devices and sensors. The goal is to develop autonomous and self-powered devices that do not need any replacement of traditional electrochemical batteries. Piezoelectric devices are commonly used, due to their high conversion efficiency and easy of manufacture. The purpose of this paper is to numerically analyze the electromechanical response of piezoelectric bimorphs subjected to vibrations. The bimorph is made up of two layers of piezoelectric material glued on a stainless steel shim, which form a cantilever beam with a tip mass that has the capability to convert the mechanical bending strain within the piezoelectric layers into electric charges on its external surface The strategy here used to increase the average mechanical strain, and hence the generated power output, is to modify the geometry of rectangular piezoelectric beam, which is traditionally used in applications. Optimized configurations with trapezoidal shapes (direct and reversed), with either constant width or constant volume, have been proposed and numerically analyzed. A detailed 3D finite element model is used to evaluate and to compare the electromechanical response of the proposed optimized bimorphs, in terms of resonant frequency, harmonic transfer function, output voltage and power. The electromechanical vibration response has been studied with a modal analysis and a harmonic coupled simulation with imposed base acceleration. The obtained results confirm an increment in the electric performance of the proposed optimized bimorphs, with a net increase in specific volumetric power compared to the traditional rectangular configuration.

Keywords Energy harvesting, piezoelectric material, vibrations, FEM

Introduction Among the available energy sources (solar, heat, electromagnetic, vibrations) and transduction mechanisms, great attention has been focused on vibrational energy harvesting with piezoelectric materials, due to their excellent electromechanical coupling, high conversion efficiency and favorable frequency response. In this paper, piezoelectric energy harvesters are considered like bimorph cantilevers structures that transduce ambient vibrations energy to electrical energy for supply connected devices (e.g. sensors, RF transmitters), especially when batteries replacement could be unfeasible.

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The standard layout of the bimorph is the rectangular shape (Fig.1a). The cantilever is exited at the fixed end at a given frequency and loaded at the free end with a proof mass. Piezoelectric layers are surface bonded to the metallic shim and poled for series operation. These structures are generally designed to vibrate at resonance [3]. With the purpose of electric charge collection, on the top and bottom of the device are deposed a nickel electrodes (Fig.1b). In this paper, the dimensions of a single rectangular piezoelectric layer are: le=25 mm length, w=14 mm width, tp=0.2 mm thickness. The metallic layer thickness is ts=0.1 mm.

Figure 1: a) Rectangular bimorph layout; b) Close view of bimorph through thickness arrangement; c) Stress distributions along the bimorph length for rectangular and direct trapezoidal shapes. In this paper, a modeling approach is proposed to impose a constant base acceleration to the vibrating bimorph during the frequency sweep over a wide frequency range. The anisotropic mechanical (stiffness matrix) and electrical (piezoelectric coupling, permittivity matrix) properties of the piezoelectric material, poled along the direction transverse to bending, have also been characterized by suitable electromechanical constitutive matrices. The value of the mechanical damping ratio, which represents a parameter that is generally difficult to assess, is evaluated with reference to literature data. Coupled harmonic simulations have been carried out with a wide set of electrical resistances connected to the piezoelectric bimorph, to investigate the electromechanical response in a range of electrical loads ranging from closed to opened circuit conditions. The strain in piezoelectric material produced by vibrations is converted into an electric charge distribution and collected in the upper and lower electrodes. An external electric circuit made up of a pure resistive load is then connected for electric power consumption. A resistance is not a realistic circuitry, but it is used for comparison purposes. The design parameters and materials used in this paper for finite element analysis are given in (Tab.1). Piezoelectric material Shim material Proof mass material Base acceleration Resistance range Proof mass weight Structural damping Resonance frequency range

PSI-5A4E (Piezo Systems, Inc.) Stainless steel Structural steel 280 [mm/s2] 5 – 775 [kΩ] 7.46 [g] 0.02 54.5-56 [Hz]

Table1: Design parameters and materials. In this paper , the geometry of piezoelectric devices is designed to obtain the highest power extraction (Fig.1c). In (Fig.2) are shown the top view drawings of the optimized bimorphs. These are designed so as to keep constant the first natural frequency. Optimized geometries having either the same maximal width or the same total volume of the piezoelectric material are compared. The performance of the bimorphs will depend on the applied resistive load.

Figure 2: a) Rectangular; b-c)Trapezoidal direct and reversed at same total volume of PZT; d-e) Trapezoidal direct and reversed at same maximal width.

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Finite element analysis A coupled-field electromechanical finite element model is developed by means of the commercial ANSYS® code. A modal analysis of all the bimorph shapes is firstly performed to calculate the natural vibration frequencies (Fig.5a). The electric power generated by varying the resistance is implemented next with an harmonic analysis. A finite element model is then generated (Fig.3a-b). Linear mapped brick elements (SOLID45) are used for the metallic shim and the proof mass and linear mapped couple field brick elements (SOLID5) for the piezoelectric layers, which are connected with an electric circuit element (CIRCU94) for the resistor. Grid dimension with a side of 1 mm and 2 through thickness elements for every layer were used.

Figure 3: a) Finite element model; b) Close view of through thickness mesh. Finite element modeling has two different types of constraints: electrical and mechanical. For the electric point of view (Fig.4a), the nodes positioned on top and bottom surfaces of the piezoelectric layer are coupled together simulating a series connection and top surface of the upper layer is constrained to zero voltage. In the mechanical view (Fig. 4b), the cantilever base is clamped on nodes belonging on top and bottom surfaces. Using the large mass method, a point masses (MASS21) are located at those nodes, which are several orders of magnitude larger than the total mass. The sinusoidal acceleration is applied as a force acting on that masses. This not conventional approach is necessary to apply a constant acceleration into the finite element model.

Figure 4: a) Electrical constraints; b) Mechanical constraints and loads. To calculate the power generated, a forced dynamic analysis is implemented. Excitation of sinusoidal form is applied at the cantilever base at constant acceleration amplitude. By varying the resistive loads, it is possible to calculate the displacements and the charge distributions along the cantilever against the frequency (Fig. 5b-c).

Figure 5: a) First vibration mode; b) Base-extremity displacement transfer function vs. frequency for various resistances; c) Voltage vs. frequency for various resistances. For all load resistance is possible to draw a curve that presents a peak point. The value assumed at maximum point represents the structure when is vibrating at his own natural frequency. A positive shift in the natural frequency occurs when the resistance is increased from closed to open circuit. This data points can be collected for every resistance and a different graphic form can be presented to evaluate the maximum power generation.

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Results The resulting power values versus the resistive load of the different bimorphs are shown in (Fig. 6a-b) in terms of (µW) and (µW/mm3). For maximum power, the difference between the geometries is rather small. Instead, if specific volumetric power is analyzed, the reduced geometries (trapezoidal direct and reversed at same maximal width) shown a large increasing in performances. This analysis have not included the material strength limits, that can be achieved near the clamped beam end.

Figure 6: a) Power vs. resistance; b) Specific volumetric power vs. resistance.

Conclusions Proposed geometries with equal maximum width compared to rectangular shape, allow considerable gains in the achieved maximum specific volumetric power levels (µW/mm3). Best results are obtained with reversed trapezoidal. In this case the only limiting factor is the piezoelectric material strength limit. Minor differences are reached in the case of maximum power levels (µW). It can be noticed that resistance values associated to maximum power levels of reduced geometries are larger than the others. Rectangular Trapezoidal Trapezoidal reversed Reduced trapezoidal Reduced trapezoidal reversed

µW 5.57 5.58 5.62 5.45 5.29

µW/mm3 0.039 0.039 0.040 0.062 0.091

Comparison of the maximum value of power and specific volumetric power extracted.

References [1] Baker J, Roundy S and Wright P: “Alternative geometries for increasing power density in vibration energy scavenging for wireless sensors networks”. 3rd Int. En. Conv. Eng. Conf. AIAA, 2005, 5617. [2] Goldschmidtboeing F, Woias P: “Characterization of different beam shapes for piezoelectric energy harvesting”. J. Micromech Microeng, No. 18, 2008, 104013. [3] Benasciutti D et al: “Vibration energy scavenging via piezoelectric bimorphs of optimized shapes”. J. Microsyst. Tech., No. 16, 2010, pp. 657-668.

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