Sensitive analysis on added damping of shunted piezoceramic damping: cantilever and simply supported beam

. The current research investigates the prediction of piezoelectric damping of resistively shunted beams caused by resistors via joule heating. In order to maximize the extra damping of the piezoelectric shunted beam system, a sensitivity analysis was done. The geometrical impacts on the maximum additional damping simulation are investigated for different length and thickness ratios with the position of the PZT-5H from the base of the cantilever and simply supported beams. Prior to doing sensitivity analysis, a mathematical model for estimating extra damping from voltage produced. Validation experiments are also carried out.


Introduction
In recent years, the use of smart materials-related technology in vibration control has become an alternative to a traditional vibration control technique. The vibration reduction is facilitated by using a piezoceramic patch attached to an electrical shunt network. The Lead Zirconium-Titanate (PZT), a piezoceramic material, converts mechanical energy into electrical energy and vice-versa. Since the first experimental demonstration and model proposed by Hagood et al. [1], to describe the damping capacity of piezoceramic material is based on the change of stiffness of a PZT, stiffness varies with short to open circuit. The loss factor due to shunting of the piezo was found from complex modulus and the damping behaviour of piezo material was considered similar to viscoelastic material. The shunted damping mainly depends upon the generalized electro mechanical coefficient . It is the function of open, shorted frequencies. Implicitly depending on the open and short stiffness of the structure. The resulting change in resonance frequencies is 0.65 % from short to open and validation was done for the only first mode of vibration experimentally. Davis et al. [2], developed the strain energy approach to the prediction of resistively shunted piezoceramic damping. Two sets of experiments address the ability to predict the added damping for 2 nd , 3 rd and 4 th modes, but do not accurately predict the magnitude of added damping due to shunting of PZTs at strain nodal positions. S. M. Yang et al. [3] and Y. J. Lee et al. [4] investigated added damping due to PZT considering inertia and stiffness of the surface-bonded piezoelectric materials. The model includes the effect of actuation in close circuit conditions including shear and rotary inertia and generalized stiffness due to PZT being either positive or negative. N. W. Hagood et al. [5] modelled the effect of dynamic coupling between a structure and electrical network through the piezo effect by using the Hamilton principle and validated it experimentally on an actively controlled cantilever beam with voltage drive. Fig. 1 shows a cantilever beam structure with PZT patch. Bending deformation of the beam is considered and shear and rotation deformations are ignored. During vibration, the structure length ( ), Modulus of elasticity (C), Poisson ratio , structural damping is assumed constant and independent of vibration amplitude. The equation of motion for the piezoceramic coupled electro mechanical system and voltage generation across PZT can be derived from the procedure followed by Hagood et al. [5]:

Modeling of voltage generation across piezoceramic
where is the potential energy, is the kinetic energy, is the electrical energy and is the external work applied to the system, is the strain, is stress, is the electric field, is a vector of electrical displacement, is the volume, ( ) is the vector of mechanical displacement, ( ) is the vector of applied forces at and is applied charge, is the density and subscripts and , represent the piezoelectric material and base substance, respectively. The linear constitutive relations of the piezoelectric to be introduced into the potential energy term and the variation of both the potential and kinetic energy must be found: where is the dielectric constant and the superscript ' ', signifies the parameter was measured at constant strain and the superscript ' ', indicates the parameter was measured at the constant electric field (short circuit). These constitutive equations relate to the electrical and mechanical properties of the piezoelectric element. The specifications of these relationships will allow electromechanical interaction to be included in the model. The term ' ' is the piezoelectric coupling coefficient and relates the stress to the applied electric field. The piezoelectric material constant can be written as shown in Eq. (6) in terms of the more commonly specified coupling coefficient by: where is the piezoelectric coupling coefficient with the subscript and referring to the direction of the applied field and the poling respectively. The piezoelectric properties are incorporated in the variation equation. This equation can now be used to solve for the equations of motion of any mechanical system containing piezoelectric elements. The Rayleigh-Ritz procedure is used to solve Eq. (1), where displacement of the beam can be written as the summation of modes in the beam and a temporal coordinate: where ( ) is the assumed mode shapes of the structure which can be set to satisfy any combination of boundary conditions, ( ) is the temporal coordinate of the displacement and is the number of modes to be included in the analysis. The Euler-Bernoulli beam theory allows the strain in the beam to be written as the product of the distance from the neutral axis and the second derivative of displacement for position along the beam: where is the differential operator for the particular elastic problem. The electric potential across the piezoelectric element is assumed as indicating that no field is applied to the beam: where ( ) is the assumed electric potential, ( ) is the voltage across the PZT: Substituting Eqs.
Eqs. (13), (14) are called actuator and sensor equations, where , , , and are the mass and stiffness for structure and piezoelectric is a clamped capacitance, and Θ is the coupled matrix: The stiffness matrix can be written as: The electromechanical coupling matrix Θ and capacitance matrix are defined as: In a cantilever beam the forcing function used to model the inertia of the beam is: where is base excitation amplitude and Ω is forcing frequency at open conditions. The voltage at the piezoelectric electrodes is measured when the applied charge is zero, = 0 in Eq. (14), which yields: Eq. (22) is substituted in Eq. (13): where is dynamic stiffness from PZT, the amount of stiffness added to the electro mechanical coupling. For PZT attached at one side of the beam, Eq. (19), and ( ) is the temporal coordinate of the displacement is substituted in Eq. (22) to get: is the distance between shifted neutral axis to the PZT surface as shown in Fig. 2. If PZT is attached on both the sides of the beam and the electromechanical coupling, Θ is defined as: And Voltage to be:  Fig. 3. shows an equivalent circuit for PZT and connected to the load resistor. Mechanical energy is converted to strain energy, and in turn, into electrical energy, and is dissipated via joule heating. Energy dissipated by the resistor per cycle is estimated, as stated by author Law et al. [11].

Fig. 3. Equivalent circuit
The impedance of the capacitor, The total impedance of the circuit, Current in the circuit, Voltage across the resistor: Power dissipated across the resistor ( ) = * , here * is complex conjugate. The energy dissipated by the resistor per cycle is: The equivalent added damping ratio can be determined through this energy dissipation relationship: where is beam maximum temporal displacement can be determined by solving the above equation in a steady state condition. By simplification of Equations, added damping ratio, : Above equation when PZT is attached at one side of the beam, if pairs of PZT are attached on the beam: Differentiating Eq. (32) with respective to obtain an expression for optimal resistor for maximum added damping is:

Verification based on literature data for added damping
The accuracy of the model to predict generated voltage across the piezoceramic material and added damping due to shunt was compared with available data in the literature. Experimental results and analytical results verified the ability of the model to accurately predict the added damping when subjected to transverse vibration. Stepped beam configuration also considered for simulating analysis.

Piezoelectric patch attached to the cantilever and simply supported beams
Experimental tests were conducted on an aluminium cantilever beam and simply supported beam specimen with surface bonded piezoceramic patches. One end of the aluminium beam is clamped at the electrodynamics shaker to simulate clamped boundary conditions for cantilever and PZT used for simply supported beam, and PZT is attached on one side at 44.39 mm from the fixed end for cantilever beam. The capacitance measured is 58.7 nF. The cantilever beam specimen is shown in Fig. 5. Simply supported beam (497×25.4×1.988 mm) specimen, with surface bonded Piezoceramic patches. Resistive shunting induced damping at the strain nodal position of the PZT is measured for a simply supported condition. In the simply supported beam on the other hand, PZT is attached to one side at 223.1 mm from the left end and other piezo at 320 mm from the left end, which is used for excitation of the beam. The electrodynamics shaker (Derritron VP 4/4B) is used for testing as shown in Fig. 5, Fig. 7 Fig. 7(a), (b). Modal mass due to tip sensor and cable weight were considered in the analytical formulation for correlation experimental results. The piezoelectric accelerometer and tip accelerometer are connected to two separate charge amplifiers B K 2635. The output signal is connected to the data acquisition system NI PCI 6259. The frequency response function of the beam is shown in Fig. 6. The voltage at resonance frequency at a particular tip amplitude of vibration and damping of the structure at ambient conditions is estimated. The input voltage to the shaker is kept constant by the controller using the National Instruments Lab View program and excited several frequencies in the vicinity of resonance frequency for steady-state conditions. A resolution of 0.01, 0.05, and 0.1 Hz is set for 1 st , 2 nd and 3 rd modes with the data obtained at 50 frequencies in the vicinity of resonance. FRFs are obtained from the cantilever beam base and tip response. Inverse line fit is used to estimate the added damping for a beam with PZT short, shunted cases for particular amplitudes of vibration for the beam controlled by input voltage to the shaker. The voltage generated at the open circuit across the PZT at that resonance, tip displacement, and base acceleration are used to estimate base damping from the inverse line fit method. The resistors across PZT are varied in the range 1/10 th of optimal resister to 10 th of optimal resister and every resistance. The added damping was evaluated from the difference between the shunt damping to base damping. FRF of cantilever and simply supported beam open condition and at resistor is shown in Fig. 8.

Cantilever beam and Simply supported beam
To investigate the geometrical effects on the maximum added damping simulation is done for various length and thickness ratios with the position of the PZT from the base of the cantilever beam and simply supported beam. To describe the thickness effect on added damping, the maximum added damping at optimal resistance as a function of thickness ratio for bimorph beam system with geometrical properties listed in Table 1, where substance (beam) thickness fixed while PZT thickness is varied, assuming simulated system inherent damping is 0.0045.

Simulation procedure for finding maximum added damping at optimal thickness ratio
PZT is placed at 1 = 0 (at fixed end of beam), length of PZT is fixed, and thickness of PZT at initial value, evaluated maximum added damping at optimal resistance using Eq. (32) and Eq. (33). Now change the thickness value find maximum added damping at optimal resistance. Above process repeated till thickness of PZT reaches given maximum value. Among all maximum added damping select for thickness ratio to get maximum added damping (Maximum added damping at optimal thickness ratio) at particular fixed length and position fixed. Now length of PZT ( / ) is changed and repeat the same procedure as discussed above without changing position of PZT on beam. Find the maximum added damping at optimal thickness ratio, Next change the position of PZT ( 1 / ) on beam repeats the procedures for maximum added damping for optimal thickness ratio. Optimal thickness ratio vs ratio of length of PZT to beam is shown in Fig. 12 for 1 st mode, Fig. 14 for 2 nd mode, Fig. 16 for 3 rd mode in cantilever beam, for simply supported beam Fig. 18, Fig. 20, Fig. 22 for 1 st , 2 nd , 3 rd mode respectively, maximum added damping at optimal thickness ratio shown in Fig. 13 for 1 st mode, Fig. 15, Fig. 17 for 3 rd mode in cantilever beam, for SS beam Fig. 19, Fig. 21, Fig. 23 for 1 st , 2 nd , 3 rd mode respectively. Schematic diagram for sensitive analysis on maximum added damping as shown in Fig. 9.

Results and discussion
Added damping increases as the resistor value increases and reaches its maximum value further, added damping decreases as the resistor value increases as observed in Fig. 4. Impedance of resister and PZT capacitance are the two contribution parameters to the energy dissipation through resister. When PZT across the shunted resister value is small compared to the impedance of the capacitor, the capacitor dominates to produce current through the resistor. Current decreases and energy dissipation also decreases. If the resistor is large compared to the impendence of the capacitor, the resistor will dominate to energy dissipation and current is small, in both the cases energy dissipation across the resistor is small. When the resister value is equal to the impendence of the capacitor value will reach maximum energy dissipation at the optimal resistor. The optimal resister across the PZT for maximum added damping is obtained in E . 33 . Hagood's proposed model predicted 16 % more than the experimental, and the voltage generation model predicted 3.3 % deviation as compared to the experimentally added damping. The deviation is due to the Hagood proposed model assumption of uniform strain model and uniform mode shape for estimation of added damping. Experimental results show that the amplitude of vibration at its resonance influences the voltage generated at the open circuit and damping at ambient conditions. The voltage generated at the open circuit at a particular tip amplitude and maximum added damping is shown in Table 2 and Table 3 for the 1 st , 2 nd , and 3 rd modes cantilever and simply supported beam respectively. Base acceleration, and tip amplitude also shown for three modes and the short and open circuit resonance frequencies are measured from experiments to estimate in Table 2. In put voltage and tip displacement are shown in Table 3 for simply supported beam. Added damping estimated from . Experimental and analytical listed in Table 4, shows that for cantilever beam close argument between approaches, but for simply supported beam except 2 nd and 4 th modes very close argument between approaches. 2 nd and 4 th modes are at the strain modeshape of the beam, the voltage generation is zero considering the PZT placed exactly at the strain nodal point coinciding at the middle of the piezo and negative and positive electrodes placed up and below the PZT. Open and short frequencies being equal in magnitude results in = 0 (added damping = 0), leading to shunting damping which is equal to base damping. PZT (sparkler ceramics) wrapped-around configuration with a 3 mm negative electrode in the upper portion, and only a small amount of voltage is generated (0.70v 0-pk for 2 nd mode and 0.50v for 4 th mode). Only a small amount of voltage is available to enhance the added damping estimated by the analytical energy dissipation model. The deviation of experimentally added damping to analytical added damping at the strain nodal position of piezoceramic is due to the difference in damping (shunt to open circuit) of the order of 10 -5 . The frequency domain technique (Inverse FRF method) does not generally give more reasonable results of the order of 10 -5 . The time domain method is used for evaluating reasonably added damping. The percentage of deviation from experimental to analytical is high at 2 nd and 4 th nodal position due to the above-stated reasons. The frequency domain vibration attenuation of both beams at open and at resistor condition shown in Fig. 8 and time domain attenuation tip displacement are shown in Table 2 and Table 3.

Optimal geometry for maximum added damping: Cantilever beam
To investigate the geometrical effects on the maximum added damping simulation is done for various length and thickness ratios with the position of the PZT from the base of the cantilever beam. Fig. 10 shows that initially, the added damping increases sharply as the thickness ratio increases until it reaches a maximum of 0.0067 at a thickness ratio of 1.804. After the added damping decreases slowly and converges at 0.006. This behaviour can be explained from a force point of view. When the thickness ratio is small, increasing the ratio does not change the inertia force ̈ significantly, however, the converted damping force increases considerably. When the thickness ratio is very large through overall damping force due to resistive PZT not operating in the strain location due to inertia force, so energy dissipation is not efficient as thin layer configuration. Fig. 10. Maximum added damping at optimal resistance versus thickness ratio Now describe the thickness ratio on added damping for changing the length of PZT, position of PZT's first point is at 0 mm from the fixed end, as shown in Fig. 11:

Thickness ratio and length ratio effects on added damping for the first mode
(34) Fig. 11. Maximum added damping at optimal resistance versus thickness ratio for different lengths of PZT For the particular length of PZT, the position is fixed, and the thickness of PZT increases does not change (total strain energy) significantly, compared to electromechanical stiffness (Energy dissipation), if the thickness ratio of piezo increases the overall electromechanical stiffness and are large. The ratio of these two numbers will lead to a small number, that at higher thickness ratio added damping is less, as shown in Fig. 11. For increasing the length of the PZT, varying the thickness of the PZT above reason it holds. The optimal thickness ratio decreases due to the effect of increasing the total strain energy which will lead to settling down to the lower optimal thickness ratio compared to the lesser length of the piezo as shown in Fig. 12. The added damping increases first and will come to optimal then decreases due to increasing the length of PZT at optimal thickness ratio the total strain energy is not utilized for converting to the joule heating in that region, at the higher length of the PZT added damping decreases due to the total strain energy increases as shown in Fig. 13. For changing the position of the PZT, varies the length of PZT is added to damping curves and optimal thickness ratio curves plotted in Fig. 13 and 12 shows changing the position of PZT leads to a decrement in added damping due to the available strain being less. If the PZT is placed near the tip, and changing the length of the PZT added damping at optimal thickness curve slope is high due to the curvature between two endpoints of PZT is very less, and optimal thickness ratio is high to achieve maximum added damping. From the above graphs can conclude that for the cantilever first mode PZT is placed at the root position, the length ratio / ratio is 0.2253, and the thickness ratio ( / ) is 1.226 for maximum added damping.

Thickness ratio and length ratio effects on added damping for the second mode
For the second mode of the cantilever beam, the optimal thickness ratio is decreased for the length ratio increases due to the total stiffness increases, if the location is moved from the root to the tip of the cantilever beam depending upon the strain available for the PZT, the optimum thickness varies, as shown in Fig. 14 for 2 nd mode of the cantilever beam. At 1 / is 0.9 and 0.200 the strain is very less so, for getting max added damping at that location / requires more amount. To explain Fig. 15, added damping at optimal thickness ratio increases and decreases when particular length ratio due to the zero strain at 0.2 , PZT is placed at the near root. For 1 / is 0.2 and 0.9 (at the tip) the added damping curve decrement slope is more due to the unavailability of the strain. From above graphs can conclude that for cantilever second mode, PZT is placed 1 / = 0.4, length ratio / ratio is 0.31, thickness ratio ( / ) is 1.326 for maximum added damping.

Thickness ratio and length ratio effects on added damping for the third mode
From above graphs can conclude that for cantilever third mode, PZT is placed 1 / = 0.6, length ratio / ratio is 0.2, thickness ratio ( / ) is 1.553 for maximum added damping.

Thickness ratio and length ratio effects on added damping for the first mode for simply supported beam
For simply supported beam first mode, the optimal thickness ratio is high if PZT is placed near the supported two ends of the beam due to the available strain is less to maximize the added damping for particular length ratio, to increase length ratio optimal thickness ratio decreases due to the total stiffness increases other words inertia force increases. if change the position of the PZT depends upon strain availability optimal thickness varies as shown in Fig. 18. the added damping increases first and will come to optimal then decreases due to increasing the length of piezo at optimal thickness ratio the total strain energy is not utilized for converting to the joule heating in that region, at the higher length of the piezo added damping decreases due to the total strain energy increases as shown in Fig. 19. If the piezo is shifted towards another end of the beam depending upon the strain availability added damping will change, i.e 1 / is 0.2 added damping will more be compared to the 1 / is 0. As shown in Fig. 19.
From the above graphs can conclude that for simply supported the first mode PZT is placed 1 / = 0.3 positions, length ratio / ratio is 0.4229, thickness ratio ( / ) is 1.158 for maximum added damping.

Thickness ratio and length ratio effects added damping for the second mode
For the second mode simply supported beam PZT placed at near end and at 1 / = 0.5 the optimal thickness ratio for max added damping requires more compared to the other places of PZT due to the strain is zero at the point, both curves are lain due to the symmetry of the modal strain shape, if the length of the piezo increases optimal thickness decreases to achieve max added damping as shown in Fig. 20. The added damping increases first and will come to optimal then decreases due to increasing the length of piezo at optimal thickness ratio the total strain energy is not utilized for converting to the joule heating in that region, at the higher length of the piezo added damping decreases due to the total strain energy increases as shown in Fig. 21. if the piezo is shifted towards another end of the beam depending upon the strain availability added damping will change, i.e 1 / is 0.4 added damping decreases if the length of PZT increases due to the strain node point at 1 / is 0.5 for the second mode simply supported beam.
From the above graphs can conclude that for simple supported the second mode PZT is placed 1 / = 0.1 or 0.6 positions, length ratio / ratio is 0.27, thickness ratio ( / ) is 1.474 for maximum added damping.

Thickness ratio and length ratio effects added damping for the third mode
From the Figs. 22-23 can conclude that for simply supported the third mode PZT is placed 1 / = 0.1 positions, length ratio / ratio is 0.14, thickness ratio / is 1.921 for maximum added damping.

Conclusions
The paper discusses about analytical model based on the voltage generation method for estimation of added damping for structure. The new model is validated concerning the experimental data available in the literature. Experiments are also conducted on a beam and compared with the derived analytical model, results correlate and a analytical model based on the voltage generation method. To optimize the geometry of the cantilever shunted structure for maximum added damping is shown in Table 5 and Table 6 for cantilever and simply supported beam.