Location Optimization of Sandwiched Piezoelectric Shear Actuator for Maximum Deflection using ANSYS


Call for Papers Engineering Research Journal June 2019

Download Full-Text PDF Cite this Publication

Text Only Version

Location Optimization of Sandwiched Piezoelectric Shear Actuator for Maximum Deflection using ANSYS

Ramanand Dadhich

Department of Mechanical Engineering Rajasthan Technical University,

Rawatbhata Road, 324010 Kota, Rajasthan, India

Sandeep Kumar Parashar Department of Mechanical Engineering Rajasthan Technical University,

Rawatbhata Road, 324010 Kota, Rajasthan, India

AbstractThe commonly used configuration of bonding the extension actuators at the top and the bottom of a substrate layer to induce the flexural vibrations suffers the inherent disadvantage of de-bonding and high strain due to the location of the piezoelectric patches being far away from the neutral layer. The sandwiched piezoelectric shear actuator found to address this issue adequately. The specific location of piezoelectric patch for different mode of excitation plays a vital role in maximizing the deflection obtained.

This paper presents the finite element modeling for location optimization of cantilever piezoelectric sandwich beam excited by shear actuation. The sandwich beam is formed by sandwiching the piezoelectric patch of a certain length and rest by foam between the two metal substrates. The finite element software ANSYS is used for modeling, meshing and analysis. The piezoelectric patch is polarized along the axial direction, while the sinusoidal electric field is applied along the thickness to induce shear vibrations. The optimum location of the piezo patch for the maximum tip deflection/central deflection of the beam is determined for the first four resonant frequencies. A comparison is also performed with the cantilever beam utilizing extension actuators. The methodology and the numerical results presented herein shall be helpful in developing the new applications based on sandwiched shear actuators.

KeywordsSandwich beam, Shear actuation, ANSYS, Location optimization.

I.INTRODUCTION

When stress is applied on some of the materials they produce electric charge generally voltage and vice versa, this phenomenon is called piezoelectricity. The materials may be some specific crystals, ceramics and biological variants like bones, DNA, proteins. In crystal physics the evolution of piezoelectricity as a part of research was introduced by brothers Jacques Curie (18561941) and Pierre Curie (18591906) [1, 2]. The observed effect was named as piezoelectricity by Hankel [3]. On the basis of fundamental thermodynamic principle Lippman [4] has provided the existence of the piezoelectric converse effect just after the year of discovery of direct piezoelectric effect and the converse effect was verified by the curie brothers in the end of 1881. The use of piezoelectric materials was limited until World War I and at that time for detecting submarines with the use of echolocation, quartz was used in sonar for ultrasound sources as a resonator. These materials are now extensively used in

form of actuator and sensor for various purposes. The evolution of the easily manufactured piezoelectric ceramics with amazing performance naturally created a big and intense development of piezoelectric devices.

For obtaining a high displacements, piezoelectric materials generally actuators are stacked in some layers. This stacking provides high displacements but also results in the failure of the system as stacked layers will produce large displacements and accordingly the stress will also increase and will cause stress concentration at the interfaces. As a result crack will form and propagate. End results are failure of actuator due to debonding. Another reason for actuator debonding in case of extension actuator is stated by Benjeddou et al.[5]; that the surface mounted actuators working on d31 piezoelectric constant produces boundary concentrated forces and moments in the structure.

The disadvantage of actuator debonding is resolved by the use of shear actuator [5]. The sandwich shear actuators working on d15 piezoelectric constant, produces distributed moments in the structure. Hence the chances of the actuators debonding are very rare for the shear actuation mechanism. I t is also common known fact that for almost all the piezoceramic materials the value of shear coupling coefficient d15 is much higher as compared to d31 and d33. Later Benjeddou et al. [6] also demonstrated that for stiff structure and thick piezoelectric actuators shear actuation mechanism is better. In [7] the theoretical and numerical comparison of extension and shear actuated mechanism for dynamic and static control of beams was presented. Sun and Zhang [8] constructed an adaptive sandwich structure using piezoelectric material in shear mode. They also performed comparative study of sandwich structure and surface mounted actuation structure by the help of finite element analysis. They concluded that sandwich structure given many advantages as compared to conventional surface mounted actuation structures. Zhang and Sun [9] constructed a new adaptive sandwich structure using piezoelectric material in shear mode. Variational principle was used to derive the Governing equation of proposed beam. Finite element analysis was used to verify theoretical formulation. Khdeir and Aldraihem [10] investigated the piezoelectric actuation performance of the smart beams in shear and extension mode. The beam models were based on first order and higher order beam

deflection theories. They studied the effect of actuator location and length on the deflection shape of the beam and concluded that large deflection in case of shear patches are near the support and for extension patch it is at center. Raja et al. [11] derived FE formulation using quasistatic equation of piezoelectricity for modeling two different types of actuator in composite sandwich beam. They concluded that for same control effort shear actuator is more efficient than extension actuator for controlling vibration. Vel and Baillargeon [12] presented an exact analysis and active vibration reduction of laminated composite plate with piezoelectric shear actuators and sensors embedded in it. Positive position feedback and velocity feedback was implemented for suppression of active vibrations. They concluded with significant reduction in tip acceleration of beams and setting time. Baillargeon and Vel [13] assessed experimentally and numerically the vibration suppression of smart structure with piezoelectric actuator and sensor. Vibration suppression was achieved by using positive position feedback and strain rate feedback. Wang and Quek [14] provided the basic mechanics model of sandwich beam embedded with piezoelectric layer for flexural analysis. In the formulation Maxwell equations was used to obtain the distribution of piezoelectric potential.

The present paper aims with the determination of optimum location of piezoelectric shear actuator which is sandwiched in between metal substrate for maximum deflection on cantilever beam. The analysis will be performed on the Finite Element software ANSYS. The previous results will be verified for ensuring the accuracy of the method used in the present analysis. The results of extension actuator are also considered in order to provide a better comparison between the extension and the shear actuators. To the authors best knowledge, the location optimization of sandwiched piezoelectric shear actuator for maximum deflection using ANSYS has not been reported in literature before.

TABLE 1. GEOMETRIC SPECIFICATION OF PIEZOELECTRIC SANDWICHED BEAM.

Component

Length

(mm)

Width

(mm)

Height (mm)

Eachsubstrate layer

400

25

0.5

PZT patch

70

25

1

Foam

330

25

1

  1. CONSTITUTIVE EQUATIONS

    The constitutive equations tells about how a material is stressed when subjected to strain and for electrical sense it tells that when a dielectric material is subjected to electrical voltage then how charges moves in it. The piezoelectricity is defined mathematically as the interaction between material stress [T], strain [S], charge density displacement [D] and electrical field [E].

    Generally a relation given by Hooks law is

    (1)

    For dielectrical material constitutive equation is given as

    (2)

    For piezoelectric material the coupled field equations are:

    In Strain-Charge form as

    (3)

    (4)

    and in Stress-Charge form as

    (5)

    (6)

    Where

    s compliance coefficients matrix electric permittivity matrix

    d Strain-Charge form piezoelectric coupling coefficients matrix

    c stiffness coefficients matrix

    sE tells that the measured compliance data is under at least a constant and a zero electric field preferably.

    T tells that the measured permittivity data is under at least a constant and a zero electric field preferably.

    t matrix transpose.

  2. MATERIAL PROPERTIES

    The properties of the material utilized in this paper are given here under:

    1. Metal Beam [16]

      Youngs modulus: E=2.089e11 (N/m2); Poissons ratio=0.3; Density=7800(Kg/m3)

    2. Piezoceramic

      Flexibility compliance matrix (m2/N):

      Fig. 1. Geometry of shear actuated piezoelectric sandwich beam.

      II. GEOMETRIC SPECIFICATIONS

      0.15e 10

      0.45e 11

      0.5e 11

      0.45e 11 0.15e 10

      0.5e 11

      0.45e 11 0

      0.45e 11 0

      0.19e 10 0

      0 0

      0 0

      0 0

      0

      0

      0

      0

      0

      0

      Fig. 1 shows a shear actuated sandwiched piezoelectric beam.

      0

      0

      0

      0

      0.39e 10

      0

      The typical dimensions of the beam and piezo patch utilized

      0

      0

      0

      0

      0 0

      .39e

      10

      Fig. 1 shows a shear actuated sandwiched piezoelectric beam.

      0

      0

      0

      0

      0.39e 10

      0

      The typical dimensions of the beam and piezo patch utilized

      0

      0

      0

      0

      0 0

      .39e

      10

      0.39e 10 0 0

      in the work are given in Table I.

      Density=7790 (Kg/m3);

      Relative permittivity 11 = 1980

      22 = 2400

      33 = 1980

      Piezoelectric strain matrix [d] (C/N):

      of patch from the fixed end as Dc=11 mm. A similar beam is shown in Fig. 3, where the metal substrate portion is shown as aluminum (sky-blue), rigid foam (blue) and PZT-5H actuator (pink). The results of modal analysis are presented in Table 2. It can be seen that the results of the present model matches with the Ref. [5] fairly well.

      x y z

      x 0

      y 0

      0 2.1e 10

      0 2.1e 10

      z

      xy

      yz

      zx

      0

      0

      5.8e 10

      0

      0

      5.8e 10

      0

      0

      5e 10

      0

      0

      0

      Fig. 3. Schematic diagram for harmonic excitation.

      Mode

      no.

      Benjeddou et al. [5]

      (Hz)

      Present

      study (Hz)

      % error

      1.

      989

      967.29

      2.195

      2.

      3916

      3854.4

      1.573

      3.

      8374

      7983.6

      4.662

      4.

      17416

      16389

      5.896

      5.

      26025

      24640

      5.321

      Mode

      no.

      Benjeddou et al. [5]

      (Hz)

      Present

      study (Hz)

      % error

      1.

      989

      967.29

      2.195

      2.

      3916

      3854.4

      1.573

      3.

      8374

      7983.6

      4.662

      4.

      17416

      16389

      5.896

      5.

      26025

      24640

      5.321

      TABLE 2. NATURAL BENDING FREQUENCIES FOR SHEAR ACTUATED CANTILEVER BEAM.

    3. Rigid Foam [5]

    Youngs modulus E=35.3e6 (N/m2); Shear modulus G=12.76 e9(N/m2); Density=32 (Kg/m3)

  3. FINITE ELEMENT MODELING

    For modeling a 3-D geometry of beam and foam, 20 noded brick element SOLID186 with three degrees of freedom is used and for the piezoelectric patch SOLID226 element with four degrees of freedom (including electric) is used. After modeling tetrahedron free mesh with 10 (coarse) refinement is used for meshing. The meshed model of cantilever beam with mid location of piezo patch at a distance of 200 mm from the left end is shown in Fig. 2.

    Fig 2. Meshed model

  4. VALIDATION

    A comparison with the previous works on sandwich structure is carried out to validate the model.

    1. Modal Analysis for Frequency Comparison

      Benjeddou et al. [5] performed the modal analysis to get the first five bending natural frequency of a shear actuated sandwich cantilever beam. In order to validate the present study the natural frequencies are compared by taking the beam with same dimensions as of [5] i.e., length L=50 mm, height h=2 mm and with height of foam and patch t=0.5 mm with patch length a=20 mm and center distance

      1. Static Analysis for Deflection Comparison

    Zhang and Sun [9] did the static analysis to get the transverse displacement of a shear actuated sandwich cantilever beam. In order to validate the present study the transverse displacements at multiple axial locations of beam are determined and compared by taking the beam dimensions as length L=100 mm, height t=8 mm and height of patch tc=2 mm. Here, patch occupies the whole beam length and sandwiched between two metal substrate layers. The metal substrate portion is taken as aluminum with PZT-5H actuator. The voltage applied to the piezoelectric core for shear actuation has a value of V=20. The results are presented in Table 3. Again a very good match between the present study and the ref. [9] can be observed here.

    TABLE 3. STATIC ANALYSIS FOR SHEAR ACTUATED CANTILEVER BEAM.

    td>

    4.3

    Axial distance x- direction(m)

    Zhang and Sun [9] (m)

    Present study (m)

    % error

    0

    0.001e-7

    0

    1

    0.02

    0.21 e-7

    0.2 e-7

    4.76

    0.04

    0.43 e-7

    0.44 e-7

    -2.32

    0.06

    0.7 e-7

    0.67 e-7

    4.28

    0.08

    0.93 e-7

    0.89 e-7

    0.1

    1.18 e-7

    1.14 e-7

    3.38

  5. CONVERGENCE STUDY

    To ensure the accuracy a convergence study is done for the present work where the convergence is achieved by using h refinement where the mesh size is defined on a scale of 1 to 10 where 1 stands for super fine and 10 stands for extra coarse. The convergence results in terms of natural frequency are tabulated in Table 4.

    The results obtained provides an information that there is very less or in some cases no impact on natural frequency so extra coarse mesh size i.e. 10 is used.

    TABLE 4. CONVERGENCE RESULTS WHEN PIEZOPATCH PLACED AT 200 MM FROM FIXED END BY THE USE OF H REFINEMENT.

    Mode

    Refinement index

    Natural frequency (Hz)

    10

    12.76

    I

    8

    12.766

    6

    12.764

    10

    59.12

    II

    8

    59.117

    6

    59.1055

    10

    16593

    III

    8

    165.904

    6

    165.838

    10

    250.26

    IV

    8

    250.174

    6

    249.987

  6. MODAL ANALYSIS

    To determine the location of piezo-patch for maximum deflection at each resonant frequency it is necessary to determine the natural frequencies corresponding to the first few bending modes of the actuator for all the nine location of piezo patch which are obtained by dividing the beam into nine equal segments. The values of the first four natural frequencies on a cantilever piezoelectric sandwich beam for various locations are given in Table 5 Fig. 4 shows the first four mode shapes at a particular location of 200 mm that is mid location of piezo from fixed end.

    TABLE 5. NATURAL FREQUENCIES(HZ) FOR CANTILEVER

    BEAM WITH PIEZOPATCH.

    Locat ion

    Midpoint location of piezo from

    fixed end

    First mode

    Second mode

    Third mode

    Fourth mode

    I

    35 mm

    13.49

    74.47

    175.01

    281.26

    II

    76.25 mm

    13.46

    73.13

    150.14

    244.17

    III

    117.5 mm

    13.34

    64.23

    142.47

    263.05

    IV

    158.75 mm

    13.11

    59.32

    159.78

    257.06

    V

    200 mm

    12.76

    59.12

    165.93

    250.26

    VI

    241.25 mm

    12.31

    62.49

    149.83

    278.02

    VII

    282.5 mm

    11.78

    66.82

    144.85

    255.34

    VIII

    323.75 mm

    11.22

    66.91

    155.42

    245.78

    IX

    365 mm

    10.64

    62.26

    150.89

    247.74

    Fig. 4. Mode shapes of cantilever sandwich beam with piezopatch at 5th position i.e. 200mm from fixed end.

  7. NUMERICAL RESULTS

    To obtain the numerical results the modal and the harmonic analysis are performed for nine different locations of piezo patch.

    After getting the natural frequencies for each location harmonic analysis is done in order to get the tip deflection at first four frequencies of every location. For harmonic analysis the value of applied voltage taken is V=20 volts and the proportional damping with value of mass matrix multiplier =0.15 and stiffness matrix multiplier = 3.165e-6 is used as given in Parashar et al. [15]. In order to get the optimal location of piezo patch for maximum tip deflection at every mode various numerical experiments are performed. The values of natural frequencies and tip deflection which are calculated for all nine position of piezo patches are listed in Table 6.

    TABLE 6. NATURAL FREQUENCY AND TIP DEFLECTION FOR EACH SELECTED LOCATION OF PIEZOPATCHES ON

    CANTILEVER SANDWICH BEAM.

    Locat ion

    Mid location of piezo from fixed end

    Mode

    Natural frequency (Hz)

    Tip deflection (mm)

    First

    13.49

    0.42

    I

    35 mm

    Second

    74.47

    0.275

    Third

    175.01

    0.071

    Fourth

    281.26

    0.0336

    First

    13.46

    0.449

    II

    76.25 mm

    Second

    73.13

    0.3

    Third

    150.14

    0.054

    Fourth

    244.17

    0.00014

    First

    13.34

    0.475

    III

    117.5 mm

    Second

    64.23

    0.205

    Third

    142.47

    0.021

    Fourth

    263.05

    0.0205

    First

    13.11

    0.44

    IV

    158.75 mm

    Second

    59.32

    0.1125

    Third

    159.78

    0.06

    Fourth

    257.06

    0.0165

    First

    12.76

    0.37

    V

    200 mm

    Second

    59.12

    0.0085

    Third

    165.93

    0.0485

    Fourth

    250.26

    0.0141

    Locat ion

    Mid location of piezo from fixed end

    Mode

    Natural frequency (Hz)

    Tip deflection (mm)

    First

    13.49

    0.42

    I

    35 mm

    Second

    74.47

    0.275

    Third

    175.01

    0.071

    Fourth

    281.26

    0.0336

    First

    13.46

    0.449

    II

    76.25 mm

    Second

    73.13

    0.3

    Third

    150.14

    0.054

    Fourth

    244.17

    0.00014

    First

    13.34

    0.475

    III

    117.5 mm

    Second

    64.23

    0.205

    Third

    142.47

    0.021

    Fourth

    263.05

    0.0205

    First

    13.11

    0.44

    IV

    158.75 mm

    Second

    59.32

    0.1125

    Third

    159.78

    0.06

    Fourth

    257.06

    0.0165

    First

    12.76

    0.37

    V

    200 mm

    Second

    59.12

    0.0085

    Third

    165.93

    0.0485

    Fourth

    250.26

    0.0141

    For location optimization the key concern is to determine the location of node and antinode for each mode. Table 7 shows the natural frequency and maximum tip deflection

    TABLE 7. NATURAL FREQUENCY AND TIP DEFLECTION FOR EACH MODE OF CANTILEVER SANDWICH BEAM FOR NODE AND ANTINODE POSITION.

    Mode

    Mid location of piezo from fixed end

    Natural frequency (Hz)

    Tip deflection (mm)

    II

    191.5 mm(AN)

    58.8312

    0.03125

    311.31 mm(N)

    67.517

    0.137

    131.5 mm(AN)

    147.155

    0.029

    III

    206.66 mm(N)

    163.754

    0.045

    276.6 mm(AN)

    144.307

    0.0015

    344.242 mm(N)

    155.876

    0.0374

    103.5 mm(AN)

    255.244

    0.0156

    154.5 mm(N)

    260.0014

    0.0184

    IV

    207.4 mm(AN)

    255.89

    0.0215

    260.5 mm(N)

    270.2

    0.0153

    311.5 mm(AN)

    244.70

    0.000225

    360.45 mm(N)

    249.04

    0.0141

    The graphs shown in Fig 5 shows the variation in maximum tip deflection corresponding to the mid location of piezo from fixed end for all four modes. The graph in Fig. 5 represents the present shear actuated beam and graph in Fig. 6 represent the extension actuated beam. The triangle on the curve represents the antinode location whereas the inverted triangle represents the node location. Initially all nine values of tip deflection are plotted against the mid location of piezo patch from fixed end and then the best fitting curve is plotted. Further the values of tip deflection for antinodes and nodes are superimposed over these graphs.

    Fig. (a). 1st mode for shear actuated cantilever beam.

    First

    12.31

    0.33

    VI

    241.25 mm

    Second

    62.49

    0.096

    Third

    149.83

    0.026

    Fourth

    278.02

    0.0246

    First

    11.78

    0.23

    VII

    282.5 mm

    Second

    66.82

    0.155

    Third

    144.85

    0.009

    Fourth

    255.34

    0.0104

    First

    11.22

    0.19

    VIII

    323.75 mm

    Second

    66.91

    0.125

    Third

    155.42

    0.0475

    Fourth

    245.78

    0.0097

    First

    10.64

    0.12

    IX

    365 mm

    Second

    62.26

    0.075 Fig. (b). 2nd mode for shear actuated cantilever beam.

    Third

    150.89

    0.0255

    Fourth

    247.74

    0.0129

    First

    12.31

    0.33

    VI

    241.25 mm

    Second

    62.49

    0.096

    Third

    149.83

    0.026

    Fourth

    278.02

    0.0246

    First

    11.78

    0.23

    VII

    282.5 mm

    Second

    66.82

    0.155

    Third

    144.85

    0.009

    Fourth

    255.34

    0.0104

    First

    11.22

    0.19

    VIII

    323.75 mm

    Second

    66.91

    0.125

    Third

    155.42

    0.0475

    Fourth

    245.78

    0.0097

    First

    10.64

    0.12

    IX

    365 mm

    Second

    62.26

    0.075 Fig. (b). 2nd mode for shear actuated cantilever beam.

    Third

    150.89

    0.0255

    Fourth

    247.74

    0.0129

    Fig. (c). 3rd mode for shear actuated cantilever beam.

    Fig. (d). 4th mode for shear actuated cantilever beam.

    Fig. 5. Graph between tip deflection of beam and mid position of piezopatch for all four modes of shear actuated cantilever sandwich beam.

    Fig. (a). 1st mode for extension actuated cantilever beam [16].

    Fig. (b). 2nd mode for extension actuated cantilever beam [16].

    Fig. (c). 3rd mode for extension actuated cantilever beam [16].

    Fig. (d). 4th mode for extension actuated cantilever beam [16].

    Fig. 6. Graph between tip deflection of beam and mid position of piezopatch for all four modes of extension actuated cantilever beam [16].

    A common conclusion made here is that all shear actuated graph shows almost the same trend that the location of one of the antinode is having a minimum value of tip deflection and maximum deflection is seen nearer to the first location for first, second and third mode and apart from this the node location is having more tip deflection than antinode location but less than maximum deflection. But in case of extension actuation the node location is showing the minimum deflection and antinode location is showing the maximum deflection for any mode.

  8. CONCLUSION

    In the present work the optimum location of the piezo patch is determined to provide maximum tip deflection of a cantilever sandwiched beam. Finite element software ANSYS is utilized to study the piezoelectric actuator behavior for the shear mode of vibration. The three dimensional geometric model is produced and meshed in ANSYS. To get the maximum tip deflection at resonance condition for beam, a 20 volt sinusoidal electric field is assumed to be applied.

    For cantilever sandwiched beam the minimum deflection for every mode is given by the approximate location of one of the antinode whereas approximate node location gives the second maximum deflection values. Unlike the case of extension actuator [16] where the node location shows a minimum value of deflection and antinode shows the maximum deflection. Here in for shear actuator the antinodes are always the point of minimum deflection.

    The present work can be extended to study other boundary conditions of the beam.

  9. REFERENCES

  1. Curie, J., Curie, P. Development, by pressure, of the polar electricity in the hemispherical crystals inclined faces. C R Acad Sci Gen, 91, 1880a, pp. 294-295.

  2. Curie, J., Curie P. On the polar electricity in the semi-crystalline crystals' inclined faces. Acad Sci Gen, 91, 1880b, pp. 383-386.

  3. Hankel, W. On the actin and piezoelectric properties of the rock crystal and their relationship to the thermoelectric. Abh Sachs, 12, 1881, p. 457th

  4. Lippman, G., Principle of the conservation of electricity. Ann Chemie Phys 24, 1881, p.145.

  5. Benjeddou, A., M. A. Trindade, and R. Ohayon. "A unified beam finite element model for extension and shear piezoelectric actuation mechanisms." Journal of Intelligent Material Systems and Structures 8.12 (1997): 1012-1025.

  6. Benjeddou, A., M. A. Trindade, and R. Ohayon. "New shear actuated smart structure beam finite element." AIAA journal37.3 (1999): 378-383.

  7. Benjeddou, A., M. A. Trindade, and R. Ohayon. "Piezoelectric actuation mechanisms for intelligent sandwich structures." Smart Materials and Structures 9.3 (2000): 328.

  8. Sun, C. T., and X. D. Zhang. "Use of thickness-shear mode in adaptive sandwich structures." Smart Materials and Structures4.3 (1995): 202.

  9. Zhang, X. D., and C. T. Sun. "Formulation of an adaptive sandwich beam." Smart Materials and Structures 5.6 (1996): 814.

  10. Khdeir, Ahmed A., and Osama J. Aldraihem. "Deflection analysis of beams with extension and shear piezoelectric patches using discontinuity functions." Smart Materials and Structures 10.2 (2001): 212.

  11. Raja, S., R. Sreedeep, and Gangan Prathap. "Bending behavior of hybrid-actuated piezoelectric sandwich beams." Journal of Intelligent Material Systems and Structures 15.8 (2004): 611-619.

  12. Baillargeon, Brian P. "Active vibration suppression of smart structures using piezoelectric shear actuators." (2003).

  13. Baillargeon, Brian P., and Senthil S. Vel. "Active vibration suppression of sandwich beams using piezoelectric shear actuators: experiments and numerical simulations." Journal of intelligent material systems and structures 16.6 (2005): 517-530.

  14. Wang, Q., and S. T. Quek. "Flexural vibration analysis of sandwich beam coupled with piezoelectric actuator." Smart Materials and Structures 9.1 (2000): 103.

  15. Parashar, Sandeep Kumar, Utz Von Wagner, and Peter Hagedorn. "Finite element modeling of nonlinear vibration behavior of piezo- integrated structures." Computers & Structures 119 (2013): 37-47.

  16. Verma, Pramod., Location optimization of piezoelectric patches on cantilever and simply supported bimorph actuators using ANSYS. Mtech thesis, Rajasthan Technical University, Kota, India, 2017.

Leave a Reply

Your email address will not be published. Required fields are marked *