A Comparative Study of Vibration Analysis of Piezoelectrically-Actuated Cantilever Beam Systems under Different Modeling Frameworks

Transcription

1 A Comparative Study of Vibration Analysis of Piezoelectrically-Actuated Cantilever Beam Systems under Different Modeling Frameworks A Thesis Presented By Payman Zolmajd To The Department of Mechanical and Industrial Engineering in partial fulfillment of the requirements for the degree of Master of Science in the field of Mechanical Engineering Northeastern University Boston, Massachusetts August 2015

2 Contents Abstract Piezoelectric Materials Overall Concept History General Categories Materials How it works Practical applications Tennis Racquet Case Study Wind Power Generator Knock Sensors Tuned Mass Damper Nano-Mechanical Cantilever (NMC) probes Energy Method for Piezoelectric Materials Electrical Potential Energy Definition of Material Constants Piezoelectric Constants Piezoelectric-Based System Modeling Modeling Assumptions and Preliminaries Modeling Piezoelectric Actuators in Transverse Configuration Piezoelectric-Based Cantilever Beam Modeling Euler Bernoulli Theory Piezoelectric-Based Cantilever Beam Modeling Rayleigh Theor Piezoelectric-Based Cantilever Beam Modeling Timoshenko Theory Numerical Results Calculation of and values Calculation of eigenfunction coefficients Mode shapes Time and frequency domain Conclusions 85 List of Symbols. 86 Appendix A1- Finding equations (3.15) and (3.17) Appendix A2- Finding equations (3.37a-l) Appendix A3- Finding equations (3.69) and (3.71) Appendix A4- Solution of equation (3.82). 102 Appendix A5- Finding equations (3.88a-l)

3 Appendix A6- Finding equation (3.89). 106 Appendix A7- Finding equations (3.123), (3.125) and (3.128a) Appendix A8- Solution of equation (3.141) Appendix A9- Finding equations (3.150a-l) Appendix A10- Finding equation (3.151) Appendix A11- Solution of equation (3.155) Appendix A12- Finding equations (3.164a-l) Appendix B- MATLAB codes References

4 Abstract In this research a comprehensive modeling framework for a piezoelectrically-actuated cantilever beam is developed and a detailed model and vibration analyses is performed. To achieve this goal, the governing dynamics for the system as well as boundary conditions are derived using the extended Hamilton s principle. The equations of motion of cantilever beam are derived according to the Euler-Bernoulli, Rayleigh and Timoshenko theories separately. The Euler-Bernoulli theory neglects the effects of rotary inertia and shear deformation and is only applicable to analysis of thin beams. The Rayleigh theory considers the effect of rotary inertia, while the Timoshenko theory considers the effects of both rotary inertia and shear deformation for thick beams. It is evident from the nature of discontinuous geometry of system, equation of stress-strain relationship are modified as shown in theory subsection meanwhile the natural surface in the composite (beam-piezoelectric layer) portion of the cantilever beam must be considered in this stage of calculations. Then the first five natural frequencies of this composite system are obtained by those three different theories and the results are compared. Relevant mode shapes are also drawn and effects of including rotary inertia and shear deformation are discussed for slender and stocky beams. Then, the forced vibration problem is solved and the cantilever tip deflection is obtained in which applied voltage to the piezoelectric layer is considered to be a unit-step input. The results are compared again for slender and stocky beams. 1. Piezoelectric Materials 1.1 Overall Concept Piezoelectricity, also called the piezoelectric effect, is the ability of certain materials to generate an AC (alternating current) voltage when subjected to mechanical stress or vibration, or to vibrate when subjected to an AC voltage, or both,[26]. The piezoelectric effect is understood as the linear electromechanical interaction between the mechanical and the electrical state in crystalline materials with no inversion symmetry. The piezoelectric effect is a reversible process in that materials exhibiting the direct piezoelectric effect (the internal generation of electrical charge resulting from an applied mechanical force) also exhibit the reverse piezoelectric effect (the internal generation of a mechanical strain resulting from an applied electrical field),[25]. In other words, Piezoelectricity refers to an electromechanical phenomenon in particular solid-state materials that demonstrate a coupling between their electrical, mechanical, and thermal states generated by applying mechanical stress to dielectric crystals. The word piezoelectricity means electricity resulting from pressure. It is derived from the Greek piezo or piezein, which means to squeeze or press, and electric or electron, which means amber, an ancient source of electric charge,[25]. 1.2 History Piezoelectricity was discovered in 1880 by French physicist Jacques and Pierre Curie. They combined what they knew about pyroelectricity and about structures of crystals to demonstrate the effect with tourmaline, quartz, topaz, cane sugar and Rochelle salt. The converse effect however was discovered later by Gabriel Lippmann in 1881 through the mathematical aspect of 3

5 theory. These behaviors were labeled the piezoelectric effect and the inverse piezoelectric effect respectively. 1.3 General Categories Piezoelectric devices fit into four general categories, depending of what type of physical effect is used: generators, sensors, actuators, and transducers. Generators and sensors make use of the direct piezoelectric effect, meaning that mechanical energy is transformed into a dielectric displacement. This, in turn, is measurable as a charge or voltage signal between the metallized surfaces of the piezoelectric material. Actuators work vice-versa when transforming electrical energy into mechanical by means of the inverse piezoelectric effect. Finally, in transducers both effects are used within one and the same device. For all of these basic functionalities, different designs are available,[28]. 1.4 Materials The most commonly known piezoelectric material is quartz. But piezoelectric materials are numerous, the most used are: Aluminium nitride Apatite Barium titanate Bimorph Gallium phosphate Lanthanum gallium silicate Lead scandium tantalate Lead zirconate titanate Lithium tantalate Piezoelectric accelerometer Polyvinylidene fluoride Potassium sodium tartrate Quartz Unimorph 1.5 How it works 1) Normally, the charges in a piezoelectric crystal are exactly balanced, even if they're not symmetrically arranged. 2) The effects of the charges exactly cancel out, leaving no net charge on the crystal faces. (More specifically, the electric dipole moments vector lines separating opposite charges exactly cancel one another out.) 3) If you squeeze the crystal, you force the charges out of balance. 4

6 4) Now the effects of the charges (their dipole moments) no longer cancel one another out and net positive and negative charges appear on opposite crystal faces. By squeezing the crystal, you've produced a voltage across its opposite faces and that's piezoelectric effect,[27]. (Fig. 1.1) 1.6 Practical applications Fig. 1.1 Piezoelectric effect in quartz,[24] with permission One of the first applications of the piezoelectric effect was an ultrasonic submarine detector developed during the First World War. A mosaic of thin quartz crystals glued between two steel plates acted as a transducer that resonated at 50MHz. By submerging the device and applying a voltage they succeeded in emitting a high frequency 'chirp' underwater, which enabled them to measure the depth by timing the return echo. This was the basis for sonar and the development encouraged other applications using piezoelectric devices both resonating and non-resonating such as microphones, signal filters and ultrasonic transducers. However many devices were not commercially viable due to the limited performance of the materials at the time,[20]. The continued development of piezoelectric materials has led to a huge market of products ranging from those for everyday use to more specialized devices. Some typical applications can be seen below: Automotive: Air bag sensor, air flow sensor, audible alarms, fuel atomiser, keyless door entry, seat belt buzzers, knock sensors. Computer: Disc drives, inkjet printers. Consumer: Cigarette lighters, depth finders, fish finders, humidifiers, jewellery cleaners, musical instruments, speakers, telephones. Medical: Disposable patient monitors, foetal heart monitors, ultrasonic imaging. Military: Depth sounders, guidance systems, hydrophones, sonar,[20]. 5

7 1.6.1 Tennis Racquet Case Study A more recent innovation using piezoelectric technology is in the sports industry. Tennis manufacturers, Head, were requested by players to design racquets with comfort as well as power. Previously, racquets had been designed to be stiff so that they return maximum energy to the ball when it is hit but this means that the racquet transmits shock vibration to the players arm. In an attempt to reduce vibration, piezoelectric fibers have been embedded around the racquet throat and a computer chip embedded inside the handle (Fig. 1.2). The frame deflects slightly when the ball is hit so that the piezoelectric fibers bend and generate a charge (by the direct effect) which is collected by the patterned electrode surrounding the fibers. The charge and associated current is carried to an embedded silicon chip via a flexible circuit containing inductors capacitors and resistors, which boost the current and send it back to the fibers out of phase in an attempt to reduce the vibration by destructive interference. The fibers then bend (by the converse effect) to counter the motion of the racket and reduce vibration. The current generated is said to be only a couple of hundred micro amps generating 600 to 800 volts in only 2 to 3 milliseconds. The manufacturers claim 50% reduction in vibration compared with conventional rackets and the International Tennis Federation have approved them for tournament play,[20]. Fig. 1.2 Piezoelectric fibers on tennis racquet,[20] with permission Wind Power Generator This low cast solid state wind power generator turns the flexing of an omnidirectional shaft directly into electricity, using piezoelectric materials (Fig. 1.3). 6

8 Fig. 1.3 Wind power generator,[21] with permission A tall flexible stalk is surrounded with many embedded piezoelectric discs that are alternately sandwiched in-between rigid backup plates. These piezoelectric structures (toroids) compress and stretch when flexed in any direction, converting any motion directly into electricity with no intermediary mechanical generators, transmissions or propellers. A weighted wind-capturing tip can sustain the energy output from a single gust of wind by the continuing oscillation of this inverted pendulum after the gust fades. In light winds the power extraction would be maximized while remaining robust in high winds,[21] Knock Sensors Knock sensors are placed near the engine in order to detect irregular combustions. The measurement principle is the one also used in accelerometers. The piezoelectric material is placed between the vibrating structure and a seismic mass introducing the vibration forces into the piezo element. The piezo element itself converts the vibrations into an electric charge proportional to the applied force. Usually, piezoelectric ceramics (PZT) with specially tailored properties are used. The material has to withstand high temperatures (up to 200 C) as well as rapid temperature changes. Also, the piezoelectric coefficient of the material must be almost independent of the temperature and remain stable over the vehicle s lifetime. Only recently, first attempts were made to replace PZT by thin PVDF foil sensors,[28] Tuned Mass Damper A tuned mass damper (TDM) is composed of a mass, a spring and a damper and is supposed to reduce the dynamic response of the structure to which it is attached. The values of the mass, spring and damper of TDM are calculated so that the TDM will resonate out of phase with the structure when exited by an external loading. It is basically a damping system that minimizes the displacement of the main mass with a combination of both its spring and its viscous damping. 7

9 1.6.5 Nano-Mechanical Cantilever (NMC) probes NMC probes have recently attracted widespread attention in variety of applications including atomic force and friction microscopy, biomass sensing, thermal scanning microscopy and MEMS switches. For instance, in the Atomic Force Microscopy (AFM), the NMC oscillates at or near its resonant frequency (Fig. 1.4). The shift in the natural frequency due to the tip-sample interaction is used to quantitatively characterize the topography of the surface. In the biosensing applications, the NMC surface is functionalized to adsorb desired biological species which induce surface stress on the NMC. In this application, the added mass of species is estimated from the shift in the resonant frequency of the system away from that of the original NMC. An Active Probe is typically covered by a piezoelectric layer (e.g., ZnO) on the top surface. To develop an accurate dynamic model for NMCs with jump discontinuities in cross-section, a comprehensive framework has been recently developed. It has been shown that the effects of added mass and stiffness on the beam mode shapes and natural frequencies are significant. Also, results from forced vibration analysis indicate that the system frequency response is affected by geometrical discontinuities of the structure. It is experimentally shown that assuming uniform geometry and configuration for the dynamic analysis of the current NMC Active Probes is not a valid assumption since it oversimplifies the problem and creates significant error in measurements,[1]. Fig. 1.4 Nano-Mechanical Cantilever (NMC) probes,[1] with permission 8

10 2 Energy Method for Piezoelectric 2.1 Electrical Potential Energy The total electrical potential energy of the electrostatic filed є (with electrical potential ), while neglecting losses, is equal to the work needed to move total charge in this field. This relationship in variational form can be simply given by: (2.1) where is the total electric potential energy. The total accumulated charge for a continuum of volume V is defined as: (2.2) where the total charge is measured in Coulombs (C) and q is the charge density in C/. The current intensity, I, is defined as the rate of change of total charge as: Since Maxwell equation is defined as: (2.3) (2.4) Insertion of Maxwell equation (2.4) for charge density q into the definition (2.2) for total charge and substitution of resultant expression in (2.1) yields: (2.5) Using the following relationship for divergence: φ (2.6) expression (2.5) can be rewritten as: ( ( ) ( ) ) (2.7) The application of divergence theorem on the first term in (2.7) results in,[1]: 9 *The materials and procedure in this section come directly from: Jalili N (2010) Piezoelectric-Based Vibration Control: From Macro to Micro/Nano Scale Systems, Springer, New York

11 ( ( ) ) (2.8) Assuming either high-frequency applications or taking into account the fact that potential energy decreases at least with 1/r (where r is the distance) while also dielectric displacement D decreases at least with 1/, the first term in expression (2.7) can be safely ignored. Since The electric field ϵ with unit V/m, the analog of a conservative force field in mechanics, can be related to electric potential φ, similar to the relationship between conservative potential function and force in mechanics is defined as: φ (2.9) Considering this fact and the definition of electric field ϵ in (2.9), the electrical energy (2.8) reduces to: (2.10) The electrical potential energy (2.8) can be recast in indicial notation form as: (2.11) Since the total potential energy can be rewritten in its compressed notation as: (2.12) This electrical potential energy can now be augmented with the developed strain energy (2.12) to form the total potential energy as: ( ) (2.13) It is clear that the total energy (2.13) does not take into account other coupled and interacting fields such as magnetomechanical, electromagnetic, thermoelectric, thermomagnetic, and thermomechanical couplings. Since our primary objective here is to derive the constitutive relationships for standard piezoelectric materials in which the magnetic effects can be safely assumed negligible. It is also assumed that the thermal effects may be neglected; that is, either the heat exchange with the environment is assumed to be negligible (an adiabatic process) or the temperature is constant (an isothermal process). Although this is not a good assumption as most piezoelectric materials are virtually pyroelectric, this is a common practice and could save a lot of undue complications,[1]. 10

12 Representing the total energy density (energy per volume V ) as the following density form:, one can write (2.13) in (2.14) which, in comparison with (2.13), implies that total variation can be described as: ( ) ( ) (2.15) where the subscripts and imply that those values are measured at constant stress ( ) or constant electrical field ( 0). By comparing (2.14) and (2.15), one can relate the conjugated and dependent variables and as functions of independent variables and that is: ( ) ( ) (2.16a) ( ) ( ) (2.16b) i,j = 1,2,3 and p,q = 1,2,,6 Alternatively, the conjugated and dependent variables and can be related to independent variables and as: ( ) ( ) (2.17a) ( ) ( ) (2.17b) i,j = 1,2,3 and p,q = 1,2,,6 Equations (2.16) and (2.17) are called linear constitutive equations. These constitutive relationships can be recast in the following more useful form: + (2.18) where the indices i,j = 1,2,3 and p,q = 1,2,,6 refer to different directions within the material coordinate systems. It must be noted that in constitutive relationships (2.18) or subsequent configurations, the differentials in (2.16) or (2.17) have been replaced by the variables themselves. To justify this action, we have assumed that the nominal values of the variables used in either (2.16) or (2.17) are zero. Hence, the differentials are defined as the comparison between the variables themselves to these zero-value states,[1]. 11

13 In matrix form of (2.18), S is the strain vector, is the stress vector, is the electrical field vector measured in V/m, and D is the displacement vector measured in C/. The first relationship in (2.18) refers to converse piezoelectric effect (actuation), while the second equation describes the direct piezoelectric effect (sensing). These equations can be alternatively rewritten in the following form, which is mainly used for sensing applications: + (2.19) where (in a similar manner to ) is the compliance coefficients matrix under constant dielectric displacement (D = 0). Similar to constitutive relationships (2.18), the first equation in (2.19) refers to converse effect (i.e., actuation mechanism) while the second equation denotes the direct effect (i.e., sensing mechanism). Alternatively, (2.19) can be manipulated to arrive at the following more suitable form for actuation applications: + (2.20a) + (2.20b) where is the elasticity coefficients matrix under constant dielectric displacement (D = 0), and and are the piezoelectric constants matrices (the superscript S in refers to constant or zero strain condition for the impermittivity constants matrix). It is worthy to note that a set of relationships between material constants defined in (2.18) and (2.19) can be obtained by simple cross-insertion of these equations in each other,[1]. 2.2 Definition of Material Constants ( ) ( ) ( ) ( ) Table 2.1 Deffinition of material constant,[1] with permission Material Constant Notation Units Compliance coefficients matrix (inverse of elastic coefficient matrix) under constant electric field Matrix of piezoelectric strain constants relating electric displacement (measured in C/ ) to stress under zero electric field (short-circuited electrodes) Dielectric or permittivity constants matrix under constant stress 12 m/v or C/N F/m (Farad, F=C/V)

14 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Matrix of piezoelectric voltage constants relating strain to electric filed under zero stress Impermittivity constants matrix under constant stress Matrix of piezoelectric constants Elastic stiffness coefficients matrix under constant dielectric displacement Matrix of piezoelectric constants Vm/N or m/f V/m 2.3 Piezoelectric Constants To better visualize the material constants defined in the preceding subsection, the piezoelectric constitutive relations (2.18) can be written in matrix form as: { } (2.21a) { } ( ) { } ( ) { } ( ) ( ) { } (2.21b) { } The matrix forms (2.21) are in the most general form; however, when the material s elastic properties are invariant with respect to rotation of any angle about a given axis, the total number of compliance coefficients reduces to 5. These materials are referred to as transversely isotropic. Piezoceramics belong to this class of materials. It is commonly assumed that the third axis or direction 3 is along the polarization direction which also coincides with the axis of transverse isotropy. Hence, (2.21) for these materials (piezoceramics) reduces to,[1]: 13

15 { } (2.22a) { } ( ) { } ( ) { } ( ) ( ) { } (2.22b) { } Equations (2.22) imply that for transversely isotropic piezoceramics, there are five elastic constants, three iezoelectric strain constants, and two dielectric or permittivity constants. One can clearly see the transversely isotropic assumption for piezoceramics where an electric field applied in direction of polarization vector ( for instance) will result in same strains in 1 and 2 directions (see (2.22a) where ). This assumption, however, is not valid for nonisotropic piezoelectric materials such as PVDF where their piezoelectric strain constants matrix takes the form: ( ) (2.23) Equation (2.23) clearly demonstrates that the application of an electric field in the polarization direction for these piezoelectric materials results in different strains in directions 1 and 2 since. As a matter of fact, PVDF films are highly anisotropic with. Also, the dielectric strength of PVDF polymers is about 20 times higher than that of PZT, and hence, can endure much higher electric field compared to PZT materials. For both PZT and PVDF materials, piezoelectric strain constant implies that the application of electric field (normal to the polarization direction 3) produces a shear deformation or ( ). Since typically has the largest values among all piezoelectric constants, this property can be utilized to design effective shear actuators and sensors. As defined in previous section, the piezoelectric strain (or sometime referred to as charge) constant is the ratio of the induced electric polarization per unit applied mechanical stress. Alternatively, it is defined as produced mechanical strain per unit applied electric field. Therefore, representing this definition using the indicial notation, the piezoelectric strain constant can be defined as the generated strain along j -axis due to a unit electric field applied along i -axis, provided that all external stresses are kept constant. For example, is the 14

16 induced strain in direction 1 due to a unit electric field in direction 3 (polarization direction), while the system is kept under a stress-free field,[1]. 3 Piezoelectric-Based System 3.1 Modeling Assumptions and Preliminaries As discussed in previous section, the total potential energy of a linear piezoelectric material is expressed as: ( ) (3.1) By substituting the piezoelectric constitutive relationships (2.18), (2.19), or (2.20), the potential energy (3.1) reduces to appropriate relationships depending on the nature of the problem at hand. For instance, for actuator applications, substituting constitutive equation (2.20a) into energy (3.1) results in: ( ) (3.2) Equation (3.2) can be further simplified to: ( ) (3.3) Clearly, (3.3) can be separated into three parts: a purely mechanical term (elastic energy), a purely electrical term (dielectric energy), and a combined term (coupled energy). It must be noted that the electrical kinetic energy is still ignored in the calculations. However, the electrical virtual work due to application of electrical voltage in piezoelectric material will be considered in Hamilton s formulation as discussed in the next section (section 3.2),[1]. 3.2 Modeling Piezoelectric Actuators in Transverse Configuration Many structural vibration-control systems utilize piezoceramic materials that are typically implemented in the form of monolithic wafers. The term monolithic refers to a piezoceramic material which is free from added materials or augmenting structural components. While the axial configuration is mainly used for positioning applications, the laminar configuration is typically utilized in structural vibration control and sensing applications. This mode of actuation or sensing relies on in-plane actuation and sensing, i.e., induced stresses and strains parallel to the structure s surfaces (see Fig. 3.1). As a result, the piezoceramic wafers operate in mode. Using this approach, in-plane strains can be readily measured with an attached piezoceramic. Note that the values of are typically lower than those of,[1]. 15 *The procedure in this section comes to some extend from: Jalili N (2010) Piezoelectric-Based Vibration Control: From Macro to Micro/Nano Scale Systems, Springer, New York

17 Fig. 3.1 Piezoelectric Patch Actuator,[1] with permission For the purpose of model development and undue complications, a uniform flexible beam with piezoelectric patch actuator bonded on its top surface is considered. As shown in Fig. 3.2, the beam has total thickness, and length L, while the piezoelectric film possesses thickness and length and, respectively. It is assumed that beam has width and piezoelectric has width. It is also assumed that the piezoelectric actuator is perfectly bonded on the beam at distance measured from the beam support and the input voltage (t) applied to the piezoelectric actuator is considered to be independent of spatial coordinate x and serves as the only external effect. Fig. 3.2 Coordinate System and Detailed Descriptions of the Attachment,[1] with permission To establish a coordinate system for the beam, the x-axis is taken in the longitudinal direction and the z-axis is specified in the transverse direction of the beam with mid-plane of the beam to be z = 0 as shown in Figure 3.2. The simplified version of the constitutive equation (2.20a) for this configuration can be expressed as,[1]: 16

18 (3.4) Notice that for this configuration, the strain-displacement relationship is utilized as: where is the transverse displacement of the neutral axis. Before utilizing (3.4) and (3.5) into the potential energy (3.3), care must be taken for the multi-material and non-uniform nature of the system. For this, the original form of the potential energy (3.1) is a better choice when dealing with this variable geometry structure, i.e., this energy is written for three parts: the section before piezoelectric patch starts (0 to ), the zone where patch is affixed ( ), and the zone after patch ( to L). It must also be noted that in the segment of the beam where the piezoelectric patch is attached, the material properties change along the height (or z-axis); hence, both strain equation (3.5) and potential energy (3.1) need to be modified. That is, wherever the piezoelectric patch is not attached on the beam (i.e., ), the neutral surface is the geometric center of the beam (z = 0) and strain equation (3.5) holds. For the portions where the piezoelectric patch is attached (i.e., ), the strain equation (3.5) is modified to: (3.5) (3.6) where is the neutral surface (see Figure 3.2). This new neutral surface can be calculated by setting the sum of all forces in x-direction over the entire cross-section zero as: (3.7) where and are referred to as stresses induced in beam and piezoelectricmaterial segments, respectively. Utilizing Hook s law ( ) for each segment, while substituting strain relationship (3.6), yields : (3.8) where and are the respective Young s moduli of elasticity for beam and piezoelectric materials. Upon implifying (3.8), the neutral axis can be readily obtained as,[1]: (3.9) 17

19 3.3 Piezoelectric-Based Cantilever Beam Modeling Euler Bernoulli Theory In order to deal with the material dissimilarity and geometrical non-uniformity, the integral for the potential energy (3.1) for this configuration is also broken into several integrals, based on the location of the piezoelectric actuator. Hence, (3.1) is recast in the following form: (( ) ) (3.10) Note that in (3.10), strain from (3.5) is used in the first two and last integrals, while strain from (3.6) is used in the third integral. Similar to potential energy, the kinetic energy associated with this non-uniform configuration can be expressed as (notice that the electrical kinetic energy is neglected): { ( ) ( ) ( ) ( ) } ( ) where ( ) (3.11) (3.12) and H(x) is the Heaviside function, and are the respective beam and piezoelectric volumetric densities. Considering both viscous and structural damping mechanisms for beam material, the total mechanical virtual work can be given by,[1]: ( ) ( ) (3.13) 18

20 where and are the viscous and structural damping coefficients, respectively. The electrical virtual work due to input voltage to piezoelectric patch is given by: (3.14) Notice that for generalization, we again assume that the input voltage to piezoelectric actuator is a function of both spatial and temporal coordinates as presented in (3.14). At this stage, all the intermediate steps in deriving different expressions for use in the extended Hamilton s principle ( ) have been completed. By insertion of (3.5) and (3.6) into energy equation (3.10), and inserting the results along with kinetic energy (3.11) and total virtual works (3.13) and (3.14) into, and after some manipulations, we get (see Appendix A1) : [ ( ) { ( ) ( )} ( ) ( ) ] (3.15) where: [ ( ) ( )] (3.16) After some manipulations, one can simplify (3.15) as follows (see Appendix A1),[1]: 19

21 [ {( ( ) ) ( ) } ( ) ( ) (3.17) ( ( ) ) ] For (3.17) to vanish regardless of independent variations and, the integrant must vanish, and for the integrant to vanish we must have: For : ( ) (3.18a) For (3.18b) along with the boundary conditions : ( ) ( ) ( ( ) ) (3.18c) Equation (3.18a) represents the distributed-parameters equation of beam coupled with the dielectric displacement, (3.18b) indicates a static coupling between piezoelectric actuator and structure and finally (3.18c) presents the boundary conditions that need to be satisfied. Substituting the dielectric displacement from (3.18b) into both (3.18a) and boundary conditions (3.18c), one can obtain the PDE governing this type of actuator in response to input voltage as,[1]: 20

22 (( ) ) (3.19a) [( ) ] ( ) (3.19b) [ (( ) ) ] (3.19c) Due to geometrical non-uniformity arising from piezoelectric patch attachment, the coupling term appearing in (3.19) cannot be further simplified at this stage, but for some special or simple arrangements this expression can be further simplified. However, expression in (3.19) can be simplified to: ( ) ( ) ( ) ( ) (3.20) where as: is the equivalent stiffness of the piezoelectric actuator in laminar configuration defined (3.21) A widely accepted assumption for laminar piezoelectric actuator is to assume a uniform input voltage wherever the actuator is attached on the beam, and naturally zero input voltage elsewhere. To mathematically describe this voltage profile, can be expressed as: (3.22) where was defined earlier in (3.12) and is the input voltage to the actuator. Inserting the input voltage profile (3.22) and property (3.20) into (3.19), while noticing that in boundary equations (3.19b and 3.19c) yields,[1]: ( ) (3.23) 21

23 ( ) ( ) (3.24a) [ ( )] (3.24b) where: ( ) (3.25) Equation (3.23) and boundary conditions (3.24a) and (3.24b) represent the governing equations describing piezoelectric laminar actuators. They form the fundamental ground from which many vibration-control systems can be designed for these types of actuators. Now Consider a piezoelectric-based cantilever beam system which the beam has total thickness, and length L, while the piezoelectric film possesses thickness and length and, respectively. It is assumed that beam has width and piezoelectric has width. It is also assumed that the piezoelectric actuator is perfectly bonded on the beam at distance measured from the cantilevered end of the beam and the input voltage (t) applied to the piezoelectric actuator is considered to be independent of spatial coordinate x and serves as the only external effect. For this configuration, the variable mass per unit length, stiffness, and moment of inertia are given as: { (3.26) where: (3.27a) 22

24 { ( ) ( ( ) )} ( ) (3.27b) (3.27b) and: { (3.28) where is the neutral axis of the beam on the composite portion, and are the densities of the beam and piezoelectric layer, respectively, and the rest of the parameters were defined before. Moreover, the distribution of damping can be safely assumed to be uniform in the entire length of the cantilever. So free and undamped conditions associated with the transverse vibration of the beam are given by: ( ) (3.29) Assuming that the solution of (3.29) is separable in the form of be rewritten in the form of: (3.29) can ( ) (3.30) where is the natural frequency of the system. In order to obtain an analytical solution for (3.30), the entire length of the beam is divided into three uniform segments with two sets of continuity conditions at stepped points. Therefore, (3.30) can be divided into three equations given by: (3.31) where are mode shapes, flexural stiffness, and mass per unit length of beam at the nth segment, respectively,[1]. 23

25 The general solution for (3.31) can be written as: (3.32) where and are the constants of integration to be obtained by solving the characteristics equation of the system. So the three eigenfunctions for the three-section cantilever can be written as: (3.33) For this purpose, the boundary conditions for the beam as well as the continuity conditions at the stepped points must be applied. The clamped-free boundary conditions of the beam require: (3.34a) (3.34b) and the respective conditions for the continuity of deflection, slope of the deflection, bending moment, and shear force of the beam at the nth stepped point, where are given by: (3.35a) (3.35b) (3.35c) (3.35d) Applying 12 boundary conditions (6 geometric and 6 natural) (3.33) and (3.34) into (3.32), the characteristics matrix equation of system can be written as: [ ] [ ] [ ] (3.36) where the components of Appendix A2),[1]: matrix can be obtained from boundary conditions as follows (see 24

26 (3.37a) (3.37b) (3.37c) (3.37d) (3.37e) (3.37f) (3.37g) 25

27 (3.37h) (3.37i) (3.37j) (3.37k) (3.37l) 26

28 where according to following formulation can be utilized as functions of to make as a function of : ( ) ( ) [ ( ) ] [ ] (3.38) Setting the determinant of the characteristics matrix to zero leads to finding the system natural frequencies. The mode shape coefficients at each natural frequency can be obtained by solving the characteristics equation and using a normalization condition with respect to mass as follows: ( ) (3.39) where is the Kronecker delta, and and are rth and sth mode shapes corresponding to the rth and sth natural frequency of beam. For instance, is expressed as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { ( ) ( ) ( ) ( ) (3.40) The obtained natural frequencies and mode shapes are utilized to derive the governing equations of motion for the forced vibration of the system. According to the eigenfunctions expansion method, the response of system can be expressed in the form of,[1]: (3.41) 27

29 where and are the eigenfunction and generalized time-dependent coordinates for the rth mode for each section. Substituting (3.41) into (3.23) and carrying out the forced vibration analysis, the equation of motion of the system can be expressed in the following form: (3.42) or in indicial form: (3.43) On the other hand, problem: are the eigenfunctions and satisfy the free and undamped vibration (3.44) Now, substituting (3.44) into (3.43), premultiply the resulting expression by eigenfunction and integrating over the domain while utilizing the orthogonality conditions between eigenfunction and yields: ( ) ( ) ( ) ( ) ( ) (3.45) In this stage we assume which means no structural damping, and also using a normalization condition with respect to mass (3.39) we obtain,[1]: 28

30 ( ) (3.46) If we get and then change we obtain: { } (3.47) { } (3.48) ( ) ( ) (3.49) For the second distributional derivative of the Heaviside function used in (3.49), we can write: ( ) ( ) (3.50) where represents the Dirac delta function. In this simplification, we have utilized the following property of Dirac delta function: (3.51) Now substituting (3.50) into (3.49) yields,[1]: 29

31 ( ) ( ) (3.52) The truncated p-mode description of the beam model of (3.47) can now be presented in the following matrix form: (3.53) where: [ ] [ ] [ ] [ ] (3.54) Consequently, the state-space representation of (3.53) can be expressed as: (3.55) Where,[1]: [ ] [ ] (3.56) { } 3.4 Piezoelectric-Based Cantilever Beam Modeling Rayleigh Theory In this case similar to Euler Bernoulli s Theory, the total potential energy of a linear piezoelectric material can be expressed as: ( ) (3.57) or ( ) (3.58) 30

32 In order to deal with the material dissimilarity and geometrical non-uniformity, the integral for the potential energy (3.58) for this configuration is also broken into several integrals, based on the location of the piezoelectric actuator. Hence, (3.58) is recast in the following form : (( ) ) (3.59) Note that in (3.59), strain from (3.5) is used in the first two and last integrals, while strain from (3.6) is used in the third integral. Note that the effect of rotary inertia was ignored so far in the derivation. That means, the kinetic energy due to rotation of the beam was ignored. If this can t be ignored (Rayleigh s Beam Theory), then the kinetic energy must be modified as: ( ) ( ) (3.60) or: ( ) ( ) (3.61) where: (3.62) is the mass moment of inertia for unit length and the neutral axis (bending), and for small vibration: is the radius of gyration, both about (3.63) So similar to potential energy, the kinetic energy associated with this non-uniform configuration can be expressed as (notice that the electric kinetic energy is neglected): 31

33 { ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) } ( ) ( ) (3.64) So we can say: ( ) ( ) (3.65) where: ( ) ( ) (3.66) and H(x) is the Heaviside function, and are the respective beam and piezoelectric volumetric densities. Considering both viscous and structural damping mechanisms for beam material, the total mechanical virtual work can be given by: ( ) ( ) (3.67) where C and B are the viscous and structural damping coefficients, respectively. The electrical virtual work due to input voltage to piezoelectric patch is given by: (3.68) Notice that for generalization, we again assume that the input voltage to piezoelectric actuator is a function of both spatial and temporal coordinates as presented in (3.68). At this stage, all the intermediate steps in deriving different expressions for use in the extended Hamilton s principle 32

34 ( ) have been completed. By insertion of (3.5) and (3.6) into energy equation (3.59), and inserting the results along with kinetic energy (3.64) and total virtual works (3.67) and (3.68) into, and after some manipulations, we get (see Appendix A3) : [ ( ) ( ) { ( ) ( )} ( ) ( ) ] (3.69) where: [ ( ) ( )] (3.70) After some manipulations, one can simplify (3.69) as follows (see Appendix A3): [ {( ( ) ( ) ) ( ) } ( ) ( ) ( ( ) ) ] (3.71) 33

35 For (3.71) to vanish regardless of independent variations and, the integrant must vanish, and for the integrant to vanish we must have: For ( ) ( ) (3.72a) For along with the boundary conditions : (3.72b) ( ) ( ) ( ( ) ) (3.72c) Equation (3.72a) represents the distributed-parameters equation of Rayleigh beam coupled with the dielectric displacement, (3.72b) indicates a static coupling between piezoelectric actuator and structure and finally (3.72c) presents the boundary conditions that need to be satisfied. Substituting the dielectric displacement from (3.72b) into both (3.72a) and boundary conditions (3.72c), one can obtain the PDE governing this type of actuator in response to input voltage as: ( ) (( ) ) (3.73a) [( ) ] ( ) (3.73b) [ (( ) ) ] (3.73c) Due to geometrical non-uniformity arising from piezoelectric patch attachment, the coupling term appearing in (3.73) cannot be further simplified at this stage, but for some 34

36 special or simple arrangements this expression can be further simplified. However, expression in (3.73) can be simplified to: ( ) ( ) ( ) ( ) (3.74) where as: is the equivalent stiffness of the piezoelectric actuator in laminar configuration defined (3.75) A widely accepted assumption for laminar piezoelectric actuator is to assume a uniform input voltage wherever the actuator is attached on the beam, and naturally zero input voltage elsewhere. To mathematically describe this voltage profile, can be expressed as: where was defined earlier in (3.66) and is the input voltage to the actuator. (3.76) Inserting the input voltage profile (3.76) and property (3.74) into (3.73), while noticing that in boundary equations (3.73b and 3.73c) yields: ( ) ( ) (3.77) ( ) ( ) (3.78a) [ ( ) ] (3.78b) where: ( ) (3.79) 35

37 Equation (3.77) and boundary conditions (3.78a) and (3.78b) represent the governing equations describing piezoelectric laminar actuators. They form the fundamental ground from which many vibration-control systems can be designed for these types of actuators. Now for free-undamped case we can say: ( ) ( ) (3.80) Assuming that the solution of (3.80) is separable in the form of be rewritten in the form of: (3.80) can ( ) ( ) (3.81) where is the natural frequency of the system. In order to obtain an analytical solution for (3.81), the entire length of the beam is divided into three uniform segments with two sets of continuity conditions at stepped points. Therefore, (3.81) can be divided into three equations given by: (3.82) where are mode shapes, flexural stiffness, mass moment of inertia per unit length and mass per unit length of beam at the nth segment, respectively. The general solution for (3.82) can be written as (see Appendix A4): (3.83) where have been presented in App. A4 and are the constants of integration to be obtained by solving the characteristics equation of the system. So the three eigenfunctions for the three-section cantilever can be written as: (3.84) For this purpose, the boundary conditions for the beam as well as the continuity conditions at the stepped points must be applied. The clamped-free boundary conditions of the beam require: (3.85a) 36

38 (3.85b) ( ) (3.85c) and the respective conditions for the continuity of deflection, slope of the deflection, bending moment, and shear force of the beam at the nth stepped point, where are given by: (3.86a) (3.86b) (3.86c) (3.86d) Applying 12 boundary conditions (6 geometric and 6 natural) (3.85) and (3.86) into (3.84), the characteristics matrix equation of system can be written as: [ ] [ ] [ ] (3.87) where the components of Appendix A5): matrix can be obtained from boundary conditions as follows (see (3.88a) (3.88b) 37

39 (3.88c) (3.88d) (3.88e) (3.88f) (3.88g) (3.88h) 38

40 (3.88i) (3.88j) (3.88k) (3.88l) Setting the determinant of the characteristics matrix to zero leads to finding the system natural frequencies. The mode shape coefficients at each natural frequency can be obtained by solving the characteristics equation and using a normalization condition with respect to mass as follows (see Appendix A6): 39

41 ( ) ( ) (3.89) where is the Kronecker delta, and and are rth and sth mode shapes corresponding to the rth and sth natural frequency of beam. For instance, is expressed as: { (3.90) The obtained natural frequencies and mode shapes are utilized to derive the governing equations of motion for the forced vibration of the system. According to the eigenfunctions expansion method, the response of system can be expressed in the form of: (3.91) where and are the eigenfunction and generalized time-dependent coordinates for the rth mode for each section. Substituting (3.91) into (3.77) and carrying out the forced vibration analysis, the equation of motion of the system can be expressed in the following form: (3.92) On the other hand, problem: are the eigenfunctions and satisfy the free and undamped vibration 40

42 ( ) (3.93) Now, substituting (3.93) into (3.92), premultiply the resulting expression by eigenfunction and integrating over the domain while utilizing the orthogonality conditions between eigenfunction and yields: ( ) ( ) ( ) ( ) ( ) ( ) (3.94) In this stage we assume which means no structural damping, and also using a normalization condition with respect to mass (3.89) we obtain: ( ) (3.95) If we get and then change we obtain: { } (3.96) 41

43 { } (3.97) ( ) ( ) (3.98) For the second distributional derivative of the Heaviside function used in (3.98), we can write: ( ) ( ) (3.99) where represents the Dirac delta function. In this simplification, we have utilized the following property of Dirac delta function: (3.100) Now substituting (3.99) into (3.98) yields: ( ) ( ) (3.101) The truncated p-mode description of the beam model of (3.96) can now be presented in the following matrix form: (3.102) where: [ ] [ ] 42

44 [ [ ] ] (3.103) Consequently, the state-space representation of (3.102) can be expressed as: (3.104) where: [ ] [ ] { } (3.105) 3.5 Piezoelectric-Based Cantilever Beam Modeling Timoshenko Theory In this case curve is now is the angle for which internal moment is acting. The slope of the deflection which is expressed as: (3.106) where is rotation of cross-section from vertical axis and is shear distortion of the cross-section (Fig. 3.3). Fig.3.3 A clear description of equation (3.106) So in this special case, equations (3.5) and (3.6) will be modified to: 43

45 (3.107) (3.108) So the total potential energy equation (3.1) will be changed due to effects of rotation and shear distortion of cross section to: ( ) (3.109) or: ( ) (3.110) Now the shear force could be related to by multiplying shear stress by area. To account for the fact that shear is parabolically distributed on a cross-section, a constant shape-dependent variable s is defined which depends on the shape of the cross-section, hence: ( ) (3.111) is also called reduced section and is computed from classical beam theory. For example, for a plane rectangular cross-section, and for a plane circular section,. So we can say: ( ) (3.112) where: 44

46 ( ) (3.113) So the total potential energy (3.110) will be: (( ) ) (3.114) Note that in (3.114), strain from (3.107) is used in the first two and fourth integrals, while strain from (3.108) is used in the third integral. In this case similar to previous beam theory (Rayleigh s beam), we should consider the effect of rotary inertia, so the kinetic energy must be written as: ( ) ( ) (3.115) or: where: ( ) ( ) (3.116) (3.117) As we mentioned before, is the mass moment of inertia for unit length and is the radius of gyration, both about the neutral axis (bending). So similar to potential energy, the kinetic energy associated with this non-uniform configuration can be expressed as (notice that the electric kinetic energy is neglected): 45

47 { ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) } ( ) ( ) (3.118) So we can say: ( ) ( ) (3.119) where: ( ) ( ) (3.120) Considering both viscous and structural damping mechanisms for beam material, the total mechanical virtual work can be given by: ( ) ( ) (3.121) where C and B are the viscous and structural damping coefficients, respectively. The electrical virtual work due to input voltage to piezoelectric patch is given by: (3.122) Notice that for generalization, we again assume that the input voltage to piezoelectric actuator is a function of both spatial and temporal coordinates as presented in (3.122). At this stage, all the intermediate steps in deriving different expressions for use in the extended Hamilton s principle ( ) have been completed. By insertion of (3.107) and (3.108) into energy equation (3.114), and inserting the results along with kinetic energy (3.118) and total 46

48 virtual works (3.121) and (3.122) into, and after some manipulations, we get (see Appendix A7): [ ( ) ( ) { ( ) ( )} ( ) ( ) ( ) ( ) ] (3.123) where: [ ( ) ( )] (3.124) After some manipulations, one can simplify (3.123) as follows (see Appendix A7): 47

49 [ {( ( ( )) ) ( ( ) ( ) ( )) ( ) } (3.125) ( ) ( ( )) ] For (3.125) to vanish regardless of independent variations, and, the integrant must vanish, and for the integrant to vanish we must have: For : ( ( )) (3.126a) For : ( ) ( ) ( ) (3.126b) For : (3.126c) 48

50 along with the boundary conditions : ( ) ( ( )) (3.126d) Substituting the dielectric displacement from (3.126c) into both (3.126b) and also boundary conditions (3.126d), one can obtain the PDE governing this type of actuator in response to input voltage as: ( ( )) (3.127a) (( ) ) ( ) ( ) (3.127b) (( ) ) (3.127c) ( ( )) (3.127d) Now if we assume constants values for manipulations one can simplify (3.127) as follows (see Appendix A7): after some ( ) ( ) [ ( )] ( ) ( ) ( ) ( ) ( ) (3.128a) 49

51 (( ) ) (3.128b) ( ( )) (3.128c) Due to geometrical non-uniformity arising from piezoelectric patch attachment, the coupling term appearing in (3.127) cannot be further simplified at this stage, but for some special or simple arrangements this expression can be further simplified. However, expression in (3.127) can be simplified to: ( ) ( ) ( ) ( ) (3.129) where as: is the equivalent stiffness of the piezoelectric actuator in laminar configuration defined (3.130) A widely accepted assumption for laminar piezoelectric actuator is to assume a uniform input voltage wherever the actuator is attached on the beam, and naturally zero input voltage elsewhere. To mathematically describe this voltage profile, can be expressed as: (3.131) where was defined earlier in (3.120) and is the input voltage to the actuator. Inserting the input voltage profile (3.131) and property (3.129) into (3.127), while noticing that in boundary equation (3.127c) yields: ( ( )) (3.132) ( ) ( ) (3.133) 50