CHAPTER 4 PIEZOELECTRIC MATERIAL BASED VIBRATION CONTROL

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1 36 CHAPTER 4 PIEZOELECTRIC MATERIAL BASED VIBRATION CONTROL 4.1 INTRODUCTION This chapter focuses on the development of smart structure using piezoelectric patches in order to control the vibration actively. More advanced technology and materials in industry lead to the implementation of lightweight components for miniaturization and efficiency. Lightweight components and certain materials, however, are susceptible to vibrations. (Autur KK 1997) The flexible structures that make up these systems pose a great problem to vibration control. Flexible structures are extensively used in many space applications, for example, space-based radar antennae, space robotic systems, and space station, etc. The flexibility of these space structures results in problems of structural vibration and shape deformation, etc. Active control methods have to be developed to suppress structural vibration and improve the performance of these flexible structures. In this current study, the main focus is to analyze the effect of Lead Zirconium and Titanate (PZT) in vibration control over GFRP composite and aluminium structures. Also, the position of PZT along with the length of the structure is studied with various input voltage and control gains. The settling time of the structures with the above parameters has also been studied. 4.2 PIEZOELECTRIC MATERIAL Piezoelectric materials are active materials generally with high bandwidths. The two properties that piezoelectric materials have are the direct effect

2 37 and the converse effect. The direct effect of a piezoelectric material is an electric polarization that occurs as the material is stressed by producing an electrical charge at the surface of the material. The converse piezoelectric effect results in a strain in the material when placed within an electric field. These properties make piezoelectric materials the most popular smart materials. Lead Zirconium Titanate (PZT) and Polyvinylidene Fluoride (PVDF) are two piezoelectric materials that are most widely used in actuation and sensing. Differences in the composition of these materials allow them to be used as actuators and sensors, respectively. PZT is roughly 4 times as denser, 40 times stiffer and has a relative permittivity of 100 times greater as that of PVDF. The rigidity of PZT makes this material a perfect candidate for actuators and, on the other hand, the flexibility and extreme sensitivity of PVDF makes it a perfect candidate for sensing. In this current study, PZT has been used for vibration control of cantilever structures. The specifications of PZT are presented in Table 4.1. Table 4.1 Specification of PZT patch Length (mm) 76.2 Width (mm) 25.4 Thickness (mm) 2 modulus (GPa) 63 Density (kg/m 3 ) Damping constants Max. Input voltage (V) MODELING OF THE STRUCTURE In this study, the GFRP composite of (0 /0 /0 ) s ply orientation and aluminium cantilever beams have been taken for analysis of vibration control.

3 38 The size of the beam is 500 mm x 50 mm x 2mm. The properties of the beams are shown in Table 4.2. Table 4.2 Properties of aluminium and GFRP Property Aluminium GFRP Density (kg/m 3 ) Pa) Figure 4.1 shows the cantilever beam modelled using ANSYS. The PZT patches have been placed at various positions along the length of the beam such as 50 mm, 250 mm and 450 mm from the fixed end. The influences of these positions over the settling time of the beams have been studied. (a) 50 mm from the fixed end (b) 250 mm from the fixed end

4 39 (c) 450 mm from the fixed end Figure 4.1 Position of PZT from the fixed end of the beam The typical finite element used in the modeling and analysis of piezoelectric crystal was (SOLID5), which has piezoelectric capacity in three dimensional couple field problem. Like other structural solid elements, this element has three displacement degrees of freedom per node. In addition to this degree of freedom, the element has also potential degree for analysing of the electromechanical coupling problems. Piezoelectric actuator inherently exhibits anisotropic and yield three-dimensional spatial vibration in their response to the piezoelectric actuation. The models developed for the passive portion includes consistent degree of freedom at the location where these elements interface. For modeling the passive portion of the smart structure solid element used is (SOLID45). The passive portion is made of aluminum and GFRP. 4.4 MODAL ANALYSIS AND DEVELOPMENT OF CONTROL LAWS Modal analysis was performed on both the aluminium and GFRP beam to find out the natural frequency of the structure. The analysis was furthur carried out for both passive and active structures. Table 4.3 presents the first four natural

5 40 frequencies of aluminium and composite beams or structures. From this table, it can be inferred that the addition of PZT patch increases both the mass and stiffness of the system. But the increase was not proportional, causing the natural frequency to increase. If they had proportionally increased, the natural frequency would have remained constant. The natural frequency of the beams can be validated analytically by using the Equation 4.1(Rao SS 2002). Table 4.3 Natural frequencies of aluminium and GFRP beams Modes Natural frequency of aluminium (Hz) Passive Beam Active Beam Natural frequency of GFRP (Hz) Passive Beam Active Beam First Mode Second Mode Third Mode Fourth Mode , f = (4.1) The harmonic response analysis was used to determine the steady response of the linear structure under the harmonic loads. Under normal circumstances, the PZT patches were actuated by a sine-wave power from the power supply. This kind of PZT-structure coupled analysis accorded with the conditions of the harmonic response analysis. Figure 4.2 shows the response of harmonic analysis of the aluminium and composite beams. It can be noticed that the peak occurs in the frequencies corresponding to the frequencies found by using modal analysis.

6 41 (a) Aluminium beam (b) GFRP beam Figure 4.2 Harmonic response of cantilever beam From these figures, it can be inferred that only the vibration modes corresponding to first, second and fourth modes have been obtained. This is due to the fact that they correspond to the bending loads, since bending load is only applied. Vibration modes corresponding to the third and fifth natural frequencies would rise while applying the torsion loads. Only, when bending loads are applied, their corresponding natural frequencies are validated.

7 42 Closed loop simulation for active vibration control in smart structure has been performed by using ANSYS. Control actions have been incorporated into the finite element model by using APDL (ANSYS Parametric Design Language) codes. K s, K c and K v are the sensor, control and power amplification factors, respectively. K s and K v are taken as 100 and K c is changed in the analyses by selecting the values starting from 10 with the step increase of 10. Only the proportional control has been applied. The multiplication of K s, K c and K v is the proportional constant for the actuator voltage V a. Therefore, changing the values of K s, K c and K v and keeping the same multiplication do not seem to affect the results. In this control system, the controller used is that of a proportional controller with a displacement feedback. The strain rate feedback control has also been used for vibration control. From the literatures reviewed, displacement feedback seems to enable better controlling action with higher actuation voltages when compared to strain rate feedback. Modal analysis have been performed to find the undamped natural as 1/(20f h ), where f h is the highest frequency. In the transient analysis, the which taken in this study. The displacement has been calculated at the tip of the beam and it is multiplied by K s and then subtracted by zero. The zero value is the reference input value. The difference between the input reference and the sensor signal is called the error signal. The error value is multiplied by K c and K v to determine V a at a time step. The part of the macro which enables the calculations for the closed loop analysis for ts=4! settling time *DO,t,2*dt,ts,dt

8 43 DDELE,N370,VOLT Vmax=270 ref=0 *get,u1,node,346,u,z!tip displacement measurement err=ref-ks1*u1 Va=kc*kv*err D,N370,VOLT, Va D,N359,VOLT,0 time,t solve *enddo The actuation voltage to be applied for the piezoelectric actuator is found by multiplying error signal by the gain K c and K v. The analysis continues step by step for a specific duration after vibration amplitudes reach a steady-state. From the results, calculated and then used to determine the damping ratio for both aluminium and GFRP beams they are used to compare the vibration characteristics of both aluminium and GFRP beams. To find out the influence of position of the PZT patch over the settling time, the control gain of 10 and 20 has been considered as an input to the PZT. The damping ratio and settling time for each position are found out. The position where settling time is minimum has been selected as an optimum position of the PZT patch. After selecting the optimum position, the influence of input voltage to PZT over settling time is determined by applying different control gain values. The maximum input voltage of 270V for the selected PZT patch is taken into account.

9 DETERMINATION OF OPTIMUM POSITION OF PZT The position of PZT over the aluminium and GFRP composite beam is simulated by using finite element code to find out optimum location of PZT, resulting in improved vibration control. Figures 4.3 to 4.5 show the settling time of aluminium beam for the positions of piezoelectric 50 mm, 250 mm and 450 mm respectively. From these, it can be clearly noticed that the settling of the beam is minimum when piezoelectric patch is located at the distance of 50 mm from the fixed end of the beam. Figure 4.3 Settling time of aluminium beam when PZT at 50 mm Figure 4.4 Settling time of aluminium beam when PZT at 250 mm

10 45 Figure 4.5 Settling time of aluminium beam when PZT at 450 mm This is due to the position of piezoelectric in the high strain region resulting in more controlled reduction in the tip displacement of the structure. Similarly, the positions of piezoelectric at 250 mm and 450 mm have more settling time. Settling time of aluminium structure for the various locations from the fixed end of the beam is shown in Table 4.4. Table 4.4 Settling time of aluminium beam with various positions of PZT Distance of the PZT patch from the fixed end (mm) Settling time(s) Similar to the aluminium cantilever beam, the GFRP beam has also been simulated for the optimum position of PZT. Figures 4.6 to 4.8 show the settling time of GFRP for the positions of PZT at 50 mm, 250 mm and 450 mm respectively.

11 46 Figure 4.6 Settling time of GFRP beam when PZT at 50 mm Figure 4.7 Settling time of GFRP beam when PZT at 250 mm Figure 4.8 Settling time of GFRP beam when PZT at 450 mm

12 47 Table 4.5 again proves that the position of PZT at the distance of 50 mm from the fixed end has a lesser settling time when compared to other positions via 250 mm and 450 mm respectively. Table 4.5 Settling time of GFRP beam with various positions of PZT Distance of the PZT patch from the fixed end (mm) Settling time(s) DETERMINATION OF OPTIMUM CONTROL GAIN Different gain values are given to the PZT patch in order to maximize damping effect without exceeding the maximum voltage range that could be applied to the PZT patch (270V). Since the optimum position of the PZT patch is found to be at a distance of 50 mm from the fixed end, the need for the optimum gain of the PZT patch arises. In order to obtain the optimum gain, values of 10, 20, 30 and 40 have been selected as an input to PZT patch and their corresponding voltage graph is plotted for the aluminium structure from the Figures 4.9 to From these figures, it is evident that an increase in control gains increases the voltage to PZT. When the control gain exceeds 20, the applied voltage to PZT exceeds the maximum value of 270 V of the selected PZT. Figure 4.9 Output voltage for the control gain of 10 Aluminium beam

13 48 Figure 4.10 Output voltage for the control gain of 20-Aluminium beam Figure 4.11 Output voltage for the control gain of 30-Aluminium beam Figure 4.12 Output voltage for the control gain of 40-Aluminium beam

14 49 From the results, it can be assured that the increase in control gain leads to an increase in applied voltage as shown in Figure The increase in applied voltage results in the structure settles earlier due to the increased control action offered by the PZT. Maximum voltage of actuator(v) Control gain Figure 4.13 Actuator voltages for aluminium beam When the PZT is located in the high strain region of the structure i.e near the root of the structure, it provides better control action when compared to the location at the free end. The same can be extended to GFRP structure as well. Figures 4.14 to 4.16 present the settling time of the beam with applied voltage to PZT. Figure 4.14 Output voltage for the control gain of 10 - GFRP beam

15 50 Figure 4.15 Output voltage for the control gain of 20 - GFRP beam Figure 4.16 Output voltage for the control gain of 30 - GFRP beam for the GFRP. Similar to the aluminium beam, Figure 4.17 shows the actuator voltage Maximum voltage of actuator(v) Control gain Figure 4.17 Actuator voltages for GFRP beam

16 LOGARITHMIC DECREMENT The logarithmic decrement is to be found in order to find the damping ratio. Logarithmic decrement is found out for both the aluminium and GFRP beam at their optimum position by using Equation 4.2 (Rao SS, 2002). Damping ratio should not exceed more than 1 because it results in an over damped system. = ln (x 1 /x 2 ) (4.2) where x 1 = first maximum displacement, x 2 = second maximum displacement = logarithmic decrement. 4.8 DAMPING RATIO Damping ratio is a dimensionless measure which describes how oscillations in a system decay after a disturbance and is calculated by using Equation 4.3 (Rao SS, 2002). It characterizes the frequency response of a second order ordinary differential equation which is important in the study of system damping and system control. (4.3) Logarithmic decrement value is calculated both for aluminium and GFRP beams with different control gains and position of PZT at 50 mm from the fixed end. Table 4.6 shows the value of logarithmic decrements. Table 4.6 Logarithmic decrement of aluminium and GFRP at 50 mm from the fixed end Control gain Material Aluminium GFRP

17 52 Damping ratio = = 0.16 The Damping ratio for Aluminium and GFRP beams with PZT patch at a distance of 50 mm from the fixed end with various control gains are shown in Table 4.7. From this, it can be inferred that the damping ratio of the aluminium beam is higher than that of GFRP composite beam. Table 4.7 Damping ratio of aluminium and GFRP at 50 mm from fixed end Control gain Material Aluminium GFRP Figure 4.18 depicts the damping ratio for aluminium and GFRP composite structure. From this, it is clearly noticed that for the position of PZT 50 mm from the fixed end and the control gain of 20, aluminium has a better damping ratio leading to earlier settling of the structure. This is due to the fact that GFRP structure has less stiffness when compared to aluminium Damping ratio Aluminium GFRP Control gain Figure 4.18 Damping ratio for the PZT patch at a distance of 50 mm from the fixed end

18 CONCLUDING REMARKS Finite element modelling for the closed loop control system has been developed by using ANSYS. Modal analysis and harmonic analysis have been carried out to find the undamped natural frequencies of the system and it is compared with the analytical results. With the modal frequencies as input, time step is calculated for closed loop transient analysis. Transient analysis and closed loop control laws are incorporated into the finite element models using APDL (ANSYS Parametric Design Language) for different control gains and for different positions of PZT patch on the flexible beam. From the results, it can be noticed that attaching the piezoelectric patch at a distance of 50 mm from the fixed end of the beam has a minimum settling time and it is found to be the optimum position for placing the PZT patch on the beam compared to the other positions. Similarly, an increase in input voltage tends to have minimum settling time. For the control gain of 20, the maximum actuation voltage falls within 270V, which is the maximum exciting voltage for PZT. Also, from the analysis carried out for two different materials such as GFRP and aluminium, it can be inferred that aluminium settles more quickly than GFRP due to its high damping ratio.