Imperial Journal of Interdisciplinary Research (IJIR) Vol-2, Issue-5, 2016 ISSN: 2454-1362, http://www.onlinejournal.in

Imperial Journal of Interdisciplinary Research (IJIR) Page 1362

Active Vibration Control of Smart Cantilever Beam

Shaikh Tauseef1 & R. K. Agrawal2

PG Student, Department of Mechanical Engineering, YTIET, Karjat 410201 Associate Professor, Department of Mechanical Engineering, YTIET, Karjat 410201

Abstract: In this paper vibration control of cantilever beam which serve as basic model for different type of structure used in various industry fields is simulated using ANSYS 16 software. For vibration control piezoelectric ceramic LEAD ZIRCONATE TITANATE (PZT) patches are attached near the fixed end at the top and bottom surface serving as Actuator. The work considers Active Vibration Control (AVC). Beam as well as PZT patches are considered as Euler-Bernoulli beam element. The cantilever beam along with bonded PZT patch is called smart cantilever beam. Different voltage is applied within limit to the PZT patches and amplitude of vibration is observed.

1. Introduction: Light weight, thin walled structures are applied in many fields such as aircraft, turbines, space and civil structures. Negative effects of these structures includes low damping and high vibration sensibility. One of the solution is to make structures smart using electrostrictives, piezoelectric, magnetostrictives for Active Vibration Control. Technique of controlling the vibration by applying force to the structure that is equal in magnitude that of original force but appropriately out of phase is known as “Active Vibration Control”. Suresh Venna et al [1] determined the electric potential that would be generated in the piezoelectric vibration absorber using FEA to determine optimal locations for damping of the first mode. Zhang Shunqi [2] and Saurabh Kumar [3] used PID algorithm and PID controller respectively. Deepak Chhabra [4] used Modified Control Matrix and Singular Value Decomposition Approach to find optimal position of Piezoelectric Actuators along length of beam. Fabio Botta et al [8] developed a procedure and proposed a new analytical solution for active vibration control. From all above mentioned research and experiments and many other as well it is found that maximum damping or reduction of vibration amplitude is obtained when PZT patch is attached near the fixed end of cantilever beam. So in this study, PZT patches are attached near the fixed end each at top and bottom surface of the beam.

2. Beam Theory Beam under study is considered as Euler-Bernoulli beam also PZT patches are considered as Euler-Bernoulli beam element. For a cantilever beam deflection along length of beam at any instant i.e. is given by

Where,

Solving Equation (1) gives

………… (2)

Where,

Values of is given in Table 1.

Table 1. Values of n L 1 1.875 2 4.694 3 7.854 4 10.995 5 14.137

3. FEA Modelling Of Smart Cantilever Beam

An aluminium cantilever beam bonded with collocated pair of Piezoelectric Actuator as shown in Figure 1.

Figure 1.

Imperial Journal of Interdisciplinary Research (IJIR) Vol-2, Issue-5, 2016 ISSN: 2454-1362, http://www.onlinejournal.in

Imperial Journal of Interdisciplinary Research (IJIR) Page 1363

Using Lagrange’s equation, stiffness matrix and mass matrix for regular and smart beam element is given as,

Where,

Moment of inertia of beam Moment of inertia of piezoelectric patch Length of beam Length of piezoelectric patch

4. Modelling of PZT Actuator Following Linear Piezoelectric Constitutive equations are used for deriving the Actuator equations.

Where, ε = Strain σ = Stress SE = Compliance when electric field is constant d31 = Piezoelectric Constant (Coulomb/N or m/V) E = Electric field (Volt/m)

D = Electric displacement i.e. charge per unit area (Coulomb/m2) ξ = Dielectric constant (Permittivity) under constant stress Stress developed in the Actuator is

Where,

= Young’s modulus of elasticity The resultant bending moment produced by the actuator is given by

= the applied actuator voltage

The force produced by the Actuator is given by

5. Smart Beam Simulation Under Harmonic Excitation

Simulation is done on ANSYS 16 using Ansys Parametric Design Language (APDL) wherein Three dimensional structural element (solid186) is used for the metal part and piezoelectric patches are modelled using three dimensional coupled elements (SOLID5) which is shown in Figures 2(a),2(b) & 2(c).The dimension and material properties of beam and PZT patch is described Table 2.

Figure 2 (a).

Figure 2 (b).

Imperial Journal of Interdisciplinary Research (IJIR) Vol-2, Issue-5, 2016 ISSN: 2454-1362, http://www.onlinejournal.in

Imperial Journal of Interdisciplinary Research (IJIR) Page 1364

Figure 2 (c).

Table 2. Material Properties Properties Aluminium PZT-5H

Length 400 mm 20 mm

Width 40 mm 40 mm

Thickness 2 mm 0.15 mm

Young Modulus 68e9 (N/m2) ----- Density 2800 (kg/m3) 7500

(kg/m3)

Poison’s ratio 0.32 ----- Compliance Matrix S11 16.5

x 10 -12 (m2/N)

S12 -4.78 S13 -8.45 S22 16.5 S23 -8.45 S33 20.7 S44 2.33 S55 43.5 S66 42.6

Piezoelectric strain matrix x 10-12 (m/V)

d13 -274 d23 -274 d33 593

d52 720 d61 720 Relative permittivity

(F/m)

x 1977.4 y 1977.4 z 2395.4

Beam is excited harmonically by applying external force of magnitude 1N in frequency range of 1 to 10 Hz as for the dimension of the beam first fundamental natural frequency is 10Hz.It is found that maximum amplitude of vibration occurs at first natural frequency as shown in Figure 3, So in this study amplitude is reduced for first mode of vibration of beam.

Figure 3.

6. Result By applying voltage of 25V and 50V in thickness direction of PZT patch i.e. polarization direction, amplitude of vibration is reduced as shown in Table 3.

Table 3. Amplitude of Vibration Frequency (Hz)

Amplitude of vibration (cm)

Without Patch

With Patch (25V)

With Patch (50V)

1.9 1.31 1.26 1.22

2.8 1.37 1.31 1.27

3.7 1.46 1.39 1.34

1.39 1.59 1.51 1.45

5.5 1.79 1.69 1.60

6.4 2.10 1.96 1.83

7.3 2.64 2.41 2.20

8.2 3.72 3.26 2.86

9.1 6.87 5.4 4.30

10.0 124.73 20.09 9.76

7. Conclusion Vibration of beam can be controlled by using PZT as done in this work. Different voltage is applied across PZT patch to obtain different result.it is observed that when voltage is high vibration reduction is more but it should be noted that maximum voltage applied must be within the

Imperial Journal of Interdisciplinary Research (IJIR) Vol-2, Issue-5, 2016 ISSN: 2454-1362, http://www.onlinejournal.in

Imperial Journal of Interdisciplinary Research (IJIR) Page 1365

capacity of PZT patch which depends on Type, Material and dimension of PZT patch.

8. References Research Papers:

[1] Suresh Venna, Yueh-Jaw Lin,An Effective Approach for Optimal PZT Vibration Absorber Placement on Composite Structures, Modern Mechanical Engineering, Vol.3 No.1, February 27, 2013

[2] Shunqi Zhang, Rüdiger Schmidt, Xiansheng Qin, Active vibration control of piezoelectric bonded smart structures using PID algorithm, Chinese Journal of Aeronautics, Volume 28, Issue 1, February 2015, Pages 305-313, ISSN 1000-9361

[3] Saurabh Kumar, Rajeev Srivastava, R.K.Srivastava, Active Vibration Control Of Smart Piezo Cantilever Beam Using PID Controller, International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308

[4] Deepak Chhabra, Gian Bhushan, Pankaj Chandna, Optimal Placement of Piezoelectric Actuators on Plate Structures for Active Vibration Control Using Modified Control Matrix and Singular Value Decomposition Approach, International Journal of Mechanical, Aerospace, Industrial and Mechatronics Engineering Vol:7 No:3, 2013

[5] Fabio Botta, Daniele Dini, ChristophSchwingshackl, Luca di Mare, and Giovanni Cerri, Optimal Placement of Piezoelectric Plates to Control Multimode Vibrations of a Beam, Advances in Acoustics and Vibration Volume 2013, Article ID 905160, 8 pages

[6] M. Kerboua, A. Megnounif, M. Benguediab, K.H. Benrahou, F. Kaoulala, “Vibration control beam using piezoelectric-based smart materials,” Composite Structures, Volume 123, May 2015, Pages 430-442, ISSN 0263-8223

[7] Zhi-chengQiu, Hong-xin Wu, Chun-de Ye, “Acceleration sensors based modal identification and active vibration control of flexible smart cantilever plate,” Aerospace Science and Technology, Volume 13, Issue 6, September 2009, Pages 277-290, ISSN 1270-9638

[8] Ting Zhang, Hong Guang Li, Guo Ping Cai, “Hysteresis identification and adaptive vibration control for a smart cantilever beam by a piezoelectric actuator,” Sensors and Actuators A: Physical, Volume 203, 1 December 2013, Pages 168-175, ISSN 0924-4247

[9] Riessom W,Prasad Krishna,K.V. Gangadharan, “Displacement Feedback for Active Vibration Control of Smart Cantilever Beam,” International Journal of Engineering Research & Technology (IJERT)Vol. 3 Issue 4, April - 2014ISSN: 2278-0181

[10] Ruo Lin Wang, H. Gu, and G. Song, “Adaptive Robust Sliding Mode Vibration Control of a Flexible Beam Using Piezoceramic Sensor and Actuator,”Mathematical Problems in Engineering Volume 2014, Article ID 606817, 9 pages Website:

[1] www.nptel.ac.in [2] http://emweb.unl.edu/Mechanics-

pages/scottWhitney/325hweb/beams.htm