Modeling and Investigation of One-Dimensional Flexural Vibrating Mechatronic Systems with Piezoelectric Transducers

Piezoelectricity has found a lot of applications since it was discovered in 1880 by Pierre and Jacques Curie. There are many applications of the direct piezoelectric effect the production of an electric potential when stress is applied to the piezoelectric material, as well as the reverse piezoelectric effect the production of strain when an electric field is applied (Moheimani & Fleming, 2006). In this chapter analysis of mechatronic systems with both direct and reverse piezoelectric effects applications in mechatronic systems are presented. In considered systems piezoelectric transducers are used as actuators – the reverse piezoelectric effect application, or as vibration dampers with the external shunting electric circuit – the direct piezoelectric effect. In the first case piezoelectric transducers can be used as actuators glued on the surface of a mechanical subsystem in order to generate desired vibrations or also to control and damp vibrations in active damping applications (Kurnik et al., 1995; Gao & Liao, 2005). In this case electric voltage is generated by external control system and applied to the transducer. In the second case piezoelectric transducers are used as passive vibration dampers. A passive electric network is adjoined to transducer’s clamps. The possibility of dissipating mechanical energy with piezoelectric transducers shunted with passive electric circuits was experimentally investigated and described in many publications (Buchacz & Placzek, 2009a; Fein, 2008; Hagood & von Flotow, 1991; Kurnik, 2004). There are two basic applications of this idea. In the first method only a resistor is used as a shunting circuit and in the second method it is a passive electric circuit composed of a resistor and inductor. Many authors have worked to improve this idea. For example multimode piezoelectric shunt damping systems were described (Fleming et al., 2002). What is more there are many commercial applications of this idea (Yoshikawa et al., 1998).


Introduction
Piezoelectricity has found a lot of applications since it was discovered in 1880 by Pierre and Jacques Curie.There are many applications of the direct piezoelectric effect -the production of an electric potential when stress is applied to the piezoelectric material, as well as the reverse piezoelectric effect -the production of strain when an electric field is applied (Moheimani & Fleming, 2006).In this chapter analysis of mechatronic systems with both direct and reverse piezoelectric effects applications in mechatronic systems are presented.In considered systems piezoelectric transducers are used as actuators -the reverse piezoelectric effect application, or as vibration dampers with the external shunting electric circuit -the direct piezoelectric effect.In the first case piezoelectric transducers can be used as actuators glued on the surface of a mechanical subsystem in order to generate desired vibrations or also to control and damp vibrations in active damping applications (Kurnik et al., 1995;Gao & Liao, 2005).In this case electric voltage is generated by external control system and applied to the transducer.In the second case piezoelectric transducers are used as passive vibration dampers.A passive electric network is adjoined to transducer's clamps.The possibility of dissipating mechanical energy with piezoelectric transducers shunted with passive electric circuits was experimentally investigated and described in many publications (Buchacz & Płaczek, 2009a;Fein, 2008;Hagood & von Flotow, 1991;Kurnik, 2004).There are two basic applications of this idea.In the first method only a resistor is used as a shunting circuit and in the second method it is a passive electric circuit composed of a resistor and inductor.Many authors have worked to improve this idea.For example multimode piezoelectric shunt damping systems were described (Fleming et al., 2002).What is more there are many commercial applications of this idea (Yoshikawa et al., 1998).
Mechatronic systems with piezoelectric sensors or actuators are widely used because piezoelectric transducers can be applied in order to obtain required dynamic characteristic of designed system.It is very important to use very precise mathematical model and method of the system's analysis to design it correctly.It was proved that it is very important to take into consideration influence of all analyzed system's elements including a glue layer between piezoelectric transducer and mechanical subsystem (Pietrzakowski, 2001;Buchacz & Płaczek, 2010b).It is indispensable to take into account geometrical and material parameters of all system's components because the omission of the influence of one of them results in inaccuracy in the analysis of the system.This work presents the issues of modeling and testing of flexural vibrating mechatronic systems with piezoelectric transducers used as actuators or vibration dampers.Analysis method of the considered system will be presented, started from development of the mathematical model, by setting its characteristics, to determine the influence of the system's properties on these characteristics.
The discussed subject is important due to increasing number of applications, both simple and reverse piezoelectric phenomena in various modern technical devices.The process of modeling of technical devices with piezoelectric materials is complex and requires large amounts of time because of the complexity of the phenomena occurring in these systems.The correct description of the system by its mathematical model during the design phase is fundamental condition for proper operation of designed system.Therefore, in the work the processes of modeling, testing and verification of used mathematical models of onedimensional vibrating mechatronic systems will be presented.A series of discretecontinuous mathematical models with different simplifying assumptions will be created.Using created models and corrected approximate Galerkin method dynamic characteristics of considered systems will be designated.An analysis of influence of some geometrical and material parameters of system's components on obtained characteristics will be conducted.Mathematical model that provides the most accurate analysis of the system and maximum simplification of used mathematical tools and minimize required amount of time will be indicated.Identification of the optimal mathematical model that meets the assumed criteria is the main purpose of this work, which is an introduction to the task of synthesis of onedimensional vibrating continuous systems.

Considered system with piezoelectric transducer and assumptions
The main aim of this work is to designate dynamic characteristics of a mechatronic system with piezoelectric transducer used as an actuator or passive vibration damper.It is a cantilever beam which has a rectangular constant cross-section, length l , width b and thickness h b .Young's modulus of the beam is denoted E b .A piezoelectric transducer of length l p is bonded to the beam's surface within the distance of x 1 from a clamped end of the beam.The transducer is bonded by a glue layer of thickness h k and Kirchhoff's modulus G.The glue layer has homogeneous properties in overall length.The system under consideration in both cases (with piezoelectric actuator or vibration damper) is presented in Fig. 1.
In order to analyze vibration of the systems following assumptions were made:  material of which the system is made is subjected to Hooke's law,  the system has a continuous, linear mass distribution,  the system's vibration is harmonic,  planes of cross-sections that are perpendicular to the axis of the beam remain flat during deformation of the beam -an analysis is based on the Bernoulli's hypothesis of flat cross-sections,  displacements are small compared with the dimensions of the system.
where η b and η k denote structural damping coefficients of the beam and the glue layer that have time unit (Pietrzakowski, 2001).
It was assumed that the beam is made of steel and piezoelectric transducer is a PZT transducer.Geometric and material parameters of the system's elements: mechanical subsystem -the beam, the glue layer and the piezoelectric transducer are presented in tables 1, 2 and 3.

Geometric parameters
Material parameters where y(x,t) is the linear displacement of the beam's sections in the direction perpendicular to the beam's axis.In case of the system with piezoelectric vibration damper it is dynamic flexibility -relation between the external force applied to the system and beam's deflection (equation 3).In case of the system with piezoelectric actuator it is relation between electric voltage that supplies the actuator and beam's deflection (equation 4) (Buchacz & Płaczek, 2011).Externally applied force in the first system and electric voltage in the second system are described as: and they were assumed as harmonic functions of time.

www.intechopen.com
Modeling and Investigation of One-Dimensional Flexural Vibrating Mechatronic Systems with Piezoelectric Transducers 31

Approximate Galerkin method verification -Analysis of the mechanical subsystem
In order to designate dynamic characteristics of considered systems correctly it is important to use very precise mathematical model.Very precise method of the system's analysis is very important too.It is impossible to use exact Fourier method of separation of variables in analysis of mechatronic systems, this is why the approximate method must be used.To analyze considered systems approximate Galerkin method was chosen but verification of this method was the first step (Buchacz & Płaczek, 2010c).To check accuracy and verify if the Galerkin method can be used to analyze mechatronic systems the mechanical subsystem was analyzed twice.First, the exact method was used to designate dynamic flexibility of the mechanical subsystem.Then, the approximate method was used and obtained results were juxtaposed.The mechanical subsystem is presented in Fig. 2. A b and J b are the area and moment of inertia of the beam's cross-section.In order to determine the solution of the differential equation of motion (7) Fourier method of separation of variables was used.Taking into account the system's boundary conditions, after transformations the characteristic equation of the mechanical subsystem was obtained: Graphic solution of the equation ( 9) is presented in Fig. 3.The solution of the system's characteristic equation approach to limit described by equation: This solution is precise for n > 3.For the lower values of n solutions should be readout from the graphic solution (Fig. 3) and they are presented in table 4. Taking into account the system's boundary and initial conditions, after transformations the sequence of eigenfunctions is described by the equation: Assuming zero initial conditions and taking into account that the deflection of the beam is a harmonic function with the same phase as the external force the final form of the solution of differential equation ( 7) can be described by the equation: and dynamic flexibility of the mechanical subsystem can be described as: where: In the approximate method the solution of differential equation ( 7) was assumed as a simple equation (Buchacz & Płaczek, 2009b, 2010d): www.intechopen.com

33
where A is an amplitude of vibration.It fulfils only two boundary conditions -deflection of the clamped and free ends of the beam: The equation of the mechanical subsystem's vibration forced by external applied force can be described as: Distribution of the external force was determined using Dirac delta function (x-l).
Corresponding derivatives of the assumed approximate solution of the differential equation of motion (15) were substituted in the equation of forced beam's vibration ( 18).Taking into account the definition of the dynamic flexibility (3), after transformations absolute value of the dynamic flexibility of the mechanical subsystem (denoted Y) was determined: Taking into account geometrical and material parameters of the considered mechanical subsystem (see table 1), the dynamic flexibility for the first three natural frequencies are presented in Fig. 4. In this figure results obtained using the exact and the approximate The exact method The approximate method Table 4.The first three roots of the characteristic equation and shifts of values of the system's natural frequencies methods are juxtaposed.Inexactness of the approximate method is very meaningful for the first three natural frequencies.Shifts of values of the system's natural frequencies are results of the discrepancy between the assumed solution of the system's differential equation of motion in the approximate method and solution obtained on the basis of graphic solution of the system's characteristic equation in the exact method.These discrepancies are shown in table 4. So it is possible to identify discrepancies between the exact and approximate methods without knowing any geometrical and material parameters.Knowing the characteristic equation of the mechanical system with known boundary conditions and assumed solution of the differential equation of motion it is possible to determine whether the solution obtained using the approximate method differs from the exact solution.
The approximate method was corrected for the first three natural frequencies of the considered system by introduction in equation ( 19) correction coefficients described by the equation: where ω n and ω n ' are values obtained using the exact and approximate methods, respectively (Buchacz & Płaczek, 2010c).The dynamic flexibility of the mechanical subsystem before and after correction is presented in Fig. 5 separately for the first three natural frequencies.
Results of assumption of simplified eigenfunction of variable x (equation 15) are also inaccuracies of the system's vibration forms presented in Fig. 6.
The approximate Galerkin method with corrected coefficients gives a very high accuracy and obtained results can be treated as very precise (see Fig. 5).So it can be used to analyze mechatronic systems with piezoelectric transducers.The considered system -a cantilever beam was chosen purposely because inexactness of the approximate Galerkin method is the biggest in this way of the system fixing.

Before correction After correction
Fig. 5.The dynamic flexibility of the mechanical subsystem -exact and approximate method before and after correction.

Mechatronic system with broad-band piezoelectric vibration damper
The considered mechatronic system with broad-band, passive piezoelectric vibration damper was presented in Fig. 1.In this case, to the clamps of a piezoelectric transducer, an external shunt resistor with a resistance R Z is attached.As a result of the impact of vibrating beam on the transducer and its strain the electric charge and additional stiffness of electromechanical nature, that depends on the capacitance of the piezoelectric transducer, are generated.
Electricity is converted into heat and give up to the environment.Piezoelectric transducer with an external resistor is called a shunt broad-band damper (Buchacz & Płaczek, 2010c;Hagood & von Flotow, 1991;Kurnik, 1995).Piezoelectric transducer can be described as a serial connection of a capacitor with capacitance C P , internal resistance of the transducer R P and strain-dependent voltage source U P .However, it is permissible to assume a simplified model of the transducer where internal resistance is omitted.In this case internal resistance of the transducer, which usually is in the range 50 -100 Ω (Behrens & Fleming, 2003) is negligibly small in comparison to the resistance of externally applied electric circuit (400 kΩ), so it was omitted.Taking into account an equivalent circuit of the piezoelectric transducer presented in Fig. 7, an electromotive force generated by the transducer and its electrical capacity are treated as a serial circuit.The considered mechatronic system can be represented in the form, as shown in Fig. 1.So, the piezoelectric transducer with an external shunt resistor is treated as a serial RC circuit with a harmonic voltage source generated by the transducer (Behrens & Fleming, 2003;Moheimani & Fleming, 2006).A series of mathematical models of the considered mechatronic system with broad-band, passive piezoelectric vibration damper was developed.Different type of the assumptions and simplifications were introduced so developed mathematical models have different degree of precision of real system representation.A series of discrete -continuous mathematical models was created.The aim of this study was to develop mathematical models of the system under consideration, their verification and indication of adequate model to accurately describe the phenomena occurring in the system and maximally simplify the mathematical calculations and minimize required time (Buchacz & Płaczek, 2009b, 2010b).

Discrete -continuous mathematical model with an assumption of perfectly bonded piezoelectric damper
In the first mathematical model of the considered mechatronic system there is an assumption of perfectly bonded piezoelectric transducer -strain of the transducer is exactly the same as the beam's surface strain.Taking into account arrangement of forces and bending moments acting in the system that are presented in Fig. 8, differential equation of motion can be described as: T(x,t) denotes transverse force, M(x,t) bending moment and M P (x,t) bending moment generated by the piezoelectric transducer that can be described as: www.intechopen.comPiezoelectric materials can be described by a pair of constitutive equations witch includes the relationship between mechanical and electrical properties of transducers (Preumont, 2006;Moheimani & Fleming, 2006).In case of the system under consideration these equations can be written as: Symbols 33 T , d 31 , s 11 E are dielectric, piezoelectric and elasticity constants.Superscripts T and E denote value at zero/constant stress and zero/constant electric field, respectively.Symbols D 3 , S 1 , T 1 and E 3 denote electric displacement, strain, stress and the electric field in the directions of the axis described by the subscript.After transformation of equation ( 24), force generated by the transducer can be described as: ,, where: Symbol c 11 E denotes Young's modulus of the transducer at zero/constant electric field (inverse of elasticity constant).U C (t) is an electric voltage on the capacitance C p .Due to the fact that the piezoelectric transducer is attached to the surface of the beam on the section from x 1 to x 2 its impact was limited by introducing Heaviside function H(x).Finally, equation ( 21) can be described as: where: Equation of the piezoelectric transducer with external electric circuit can be described as: where: C P is the transducer's capacitance, U P (t) denotes electric voltage generated by the transducer as a result of its strain.Voltage generated by the transducer is a quotient of generated electric charge and capacitance of the transducer.After transformation of the constitutive equations ( 23) and ( 24) electric charge generated by the transducer can be described as (Kurnik, 2004): where: is an electromechanical coupling constant that determines the efficiency of conversion of mechanical energy into electrical energy and electrical energy into mechanical energy of the transducer, whose value usually is from 0,3 to 0,7 (Preumont, 2006).Equation (33) describes the electric charge accumulated on the surface of electrodes of the transducer with an assumption about uniaxial, homogeneous strain of the transducer.Assuming an ideal attachment of the transducer to the beam's surface its strain is equal to the beam's surface strain and can be described as: Finally, equation (31) can be described as: Using the classical method of analysis of linear electric circuits and due to the low impact of the transient component on the course of electric voltage generated on the capacitance of the linear RC circuit the electric voltage U C (t) was assumed as: www.intechopen.com

40
where |Z| and φ are absolute value and argument of the serial circuit impedance.
Equations ( 27) and ( 35) form a discrete-continuous mathematical model of the considered system.

Discrete -continuous mathematical model with an assumption of pure shear of a glue layer between the piezoelectric damper and beam's surface
Concerning the impact of the glue layer between the transducer and the beam's surface, the mathematical model of the system under consideration was developed.It will allow more detailed representation of the real system.First, a pure shear of the glue layer was assumed.
Arrangement of forces and bending moments acting in the system modeled with this assumption is presented in Fig. 9. .
Δl is a displacement of the lower and upper surfaces of the glue layer.Movements of the beam, the glue layer and the transducer are shown in Fig. 10.Uniform distribution of shear stress along the glue layer was assumed.The real strain of the transducer is a difference of the glue layer's upper surface strain and the free transducer's strain that is a result of electric field on the transducer's electrodes, so Δl can be described as: where: b and k are the beam's and the glue layer's upper surfaces strains.
Finally, obtained system of equations: is the discrete-continuous mathematical model of the system under consideration with the assumption about pure shear of the glue layer.

Discrete -continuous mathematical model taking into account a shear stress and eccentric tension of a glue layer between the piezoelectric damper and beam's surface
In the next mathematical model the system under consideration was modeled as a combined beam in order to unify parameters of all components (Buchacz & Płaczek, 2009c).Shear stress and eccentric tension of the glue layer were assumed.The substitute cross-section of considered system presented in Fig. 11 was introduced by multiplying the width of the piezoelectric transducer and the glue layer by factors: Symbol υ denotes the Poisson's ratio of the glue layer.
Taking into account the eccentric tension of the glue layer under the action of forces presented in Fig. 12 the stress on the substitute cross-section's surfaces was assigned.
F P (t) denotes force generated by the piezoelectric transducer and F b (t) denotes forces generated by the bending beam as a result of the beam's elasticity.The area, location of the central axis and moment of inertia of the substitute cross-section were calculated:

k p p h hh yA b h h h m h h m
were calculated and stress on the surfaces of the substitute cross-section was assigned: where subscript i denotes element of the composite beam (i=b,k,p).In case of the beam, value of the symbol m b is equal to one, while in case of the transducer and the glue layer m p and m k are described by equations ( 41) and ( 42).Using the basic laws and dependences from theory of strength of materials the real strain of the piezoelectric transducer was assigned: where: .
To determine the value of shear stress on the plane of contact of the transducer and beam the following dependence was used: where: S Z (y) is a static moment of cut off part of the composite beam's cross-section relative to the weighted neutral axis.Transverse force T(x,t) can be calculated as a derivative of bending moment acting on the beam's cross-section: where: Finally, the discrete-continuous mathematical model of the system can be described as: ,1 Obtained system of equations is a mathematical model with assumptions of shear stress and eccentric tension of the glue layer.

Discrete -continuous mathematical model taking into account a bending moment generated by the transducer and eccentric tension of a glue layer between the piezoelectric damper and beam's surface
Taking into account parameters of the combined beam introduced in section 4.1.4the discrete-continuous mathematical model with influence of the glue layer on the dynamic characteristic of the system was developed.However, in this model the impact of the piezoelectric transducer was described as a bending moment, similarly as in the mathematical model with the assumption of perfectly attachment of the transducer.Homogeneous, uniaxial tension of the transducer was assumed and its deformation was described by the equation ( 47).In this case the bending moment generated by the transducer can be described as: Obtained system of equations: is the discrete-continuous mathematical model of the system under consideration.

Dynamic flexibility of the system with broad-band piezoelectric vibration damper
Dynamic flexibility of the considered system was assigned using corrected approximate Galerkin method.Solution of the differential equation of the beam's motion with piezoelectric damper was assumed as a product of the system's eigenfunctions in accordance with the equation ( 15).For all mathematical models analogous calculations were done, therefore, an algorithm used to determine the dynamic flexibility of the system using the first mathematical model is presented.Obtained results for all mathematical models are presented in graphical form.
In the mathematical model of the considered mechatronic system with the assumption of perfectly bonded piezoelectric damper -equations ( 27) and ( 35) the derivatives of the approximate equation (15) were substituted.Assuming that the dynamic flexibility will be assigned on the free end of the beam (x=l), after transformations and simplifications a system of equations was obtained: .0 Using Cramer's rule amplitude of the system's vibration can be calculated as: where W is a main matrix determinant and W A is a determinant of the matrix formed by replacing the first column in the main matrix by the column vector of free terms.Obtained equation can be substituted in the assumed solution of the derivative equation of the beam's motion (15).Finally, in agreement with definition (3), the dynamic flexibility of the system under consideration can be described as: Taking into account geometrical and material parameters of the considered system presented in tables 1, 2 and 3, graphical solution of the equation ( 72) is presented in Fig. 13.Results obtained using the others mathematical models of the considered system are also presented in Fig. 13.
Using developed mathematical models and corrected approximate Galerkin method, very similar course of dynamic characteristics were obtained, except the second mathematical model with the assumption about pure shear of the glue layer.Shift of the natural frequencies in the direction of higher values of the mechatronic system in the direction of higher values can be observed.This shift is a result of increased stiffness of mechatronic system compared with the mechanical subsystem.

Mechatronic system with piezoelectric actuator
Developed mathematical models of the system with piezoelectric vibration damper were used to analyze the mechatronic system with piezoelectric actuator.In this case inverse piezoelectric effect is applied.Strain of the piezoelectric transducer is a result of externally applied electric voltage described by the equation ( 6).The considered system is presented in Fig. 1.Its parameters are presented in tables 1, 2 and 3.The aim of the system's analysis is to designate dynamic characteristic that is a relation between parameters of externally applied voltage and deflection of the free end of the beam (it was assumed that x=l), described by the equation ( 4).
In this case the internal capacitance C P and resistance R P of the piezoelectric transducer were taken into account, so transducer supplied by the external harmonic voltage source can be treated as a serial RC circuit with harmonic voltage source and was described by the equation (Buchacz & Płaczek, 2011): Equations of motion of the beam with piezoelectric actuator for all developed mathematical models were designated in agreement with d'Alembert's principle similarly as in the case of mechatronic system with piezoelectric vibration damper.Obtained absolute value of dynamic characteristics for all mathematical models of the system are presented in Fig. 14.
Final results are very similar for all mathematical models, except the second model with the assumptions about pure shear of the glue layer, as it was in case of analysis of system with piezoelectric vibration damper.

Analysis of influence of parameters of considered systems on dynamic characteristics
Developed mathematical models of considered systems were used to analyze influence of geometric and material parameters of systems on obtained dynamic characteristics.This study was carried out in dimensionless form in order to generalize obtained results.Results are presented in the form of three-dimensional graphs that show the course of the dimensionless absolute value of dynamic characteristic in relation to dimensionless frequency of externally applied force or electric voltage and one of the system's parameters dimensionless value.Dimensionless values of dynamic characteristics were introduced as: Influence of the other parameters on characteristics of considered systems were analyzed in other publications (Buchacz & Płaczek, 2009b, 2010a).

Conclusions and selection of an optimal mathematical model
Realized studies have shown that the corrected approximate Galerkin method can be used to analyze mechatronic systems with piezoelectric transducers.Verification of the approximate method proved that obtained results can be treated as very precise.Precision of the mathematical model of considered system has no big influence on the final results.
There are no significant differences between the values of natural vibration frequencies of considered systems and course of dynamic characteristics, except the second model.In case of the mathematical model with the assumption of pure shear of the glue layer a very significant shift of natural frequencies values and increase of piezoelectric damper or actuator efficiency were observed.These discrepancies are the results of the assumed System with piezoelectric damper System with piezoelectric actuator simplifications of the real strain of the transducer and resulting generated shear stress in the glue layer.There was also an assumption about pure shear of the glue layer, while, in the real system, this layer is under the influence of forces that cause the eccentric tension of it.
The simplest is the mathematical model with the assumption about perfectly bonded piezoelectric transducer.But taking this assumption it is impossible to define influence of the glue layer on the dynamic characteristic of the system.Using this model it is not possible to meet requirements undertaken in this work.To take into account properties of the glue layer and its real loads to which it is subjected, mathematical models, where an eccentric tension of glue layer was considered, were developed.Interactions between elements of the system were being taken into consideration and real strain of the transducer was determined.The third mathematical model is much more complex then the last one, while obtained results are very similar.It is therefore concluded that the optima, in terms of assumed criteria, is the last mathematical model where a bending moment generated by the transducer and eccentric tension of a glue layer between the piezoelectric transducer and surface of the beam were taken into account.Using this model it is possible to analyze influence of all components of the system, including glue layer between the beam and transducer, while it is quite simple at the same time.
Fig. 1.Shape of the considered system: a) with piezoelectric passive vibration damper, b) with piezoelectric actuator Structural damping of the beam and glue layer was taken into account in mathematical models of considered systems using Kelvin-Voigt model of material.It was introduced by replacing Young's modulus of the beam and modulus of elasticity in shear of the glue layer by equations:

Fig. 2 .
Fig. 2. Shape of the mechanical subsystem The equation of free vibration of the mechanical subsystem was derived in agreement with d'Alembert's principle.The external force F(t) was neglected.Taking into account equilibrium of forces and bending moments acting on the beam's element, after transformations a well known equation was obtained:     24 4 24 ,, , y xt y xt a tx    (7)

Fig. 3 .
Fig. 3.The graphic solution of the characteristic equation of the system (equation 7)

Fig. 6 .
Fig. 6.Vibration forms of the mechanical subsystem -the exact and approximate methods, a) the first natural frequency, b) the second natural frequency, c) the third natural frequency

Fig. 7 .
Fig. 7.The substitute scheme of the piezoelectric transducer with an external shunt resistor

Fig. 8 .
Fig. 8. Arrangement of forces and bending moments acting on the cut out part of the beam and the piezoelectric transducer with length dx

Fig. 9 .
Fig. 9. Forces and bending moments in case of the pure shear of the glue layer Shear stress was determined according to the Hook's law, assuming small values of pure non-dilatational strain:

Fig. 10 .
Fig. 10.Movements of the beam, the glue layer and the piezoelectric transducer in the case of pure shear of the glue layer

Fig. 11 .
Fig. 11.Position of the center of gravity of substitute cross-section of the beam

Fig. 13 .
Fig. 13.Absolute value of the dynamic flexibility of mechatronic system with piezoelectric vibration damper, for the first three natural frequencies (in a half logarithmic scale)

Fig. 14 .
Fig. 14.Absolute value of the dynamic characteristic of mechatronic system with piezoelectric actuator, for the first three natural frequencies (a half logarithmic scale) Dimensionless frequencyies of external force or electric voltage were introduced by dividing their values by the value of the first natural frequency of the mechanical subsystem.Dimensionless values of analyzed parameters were obtained by dividing them by their initial values.Obtained results for selected parameters are presented in Fig. 15 and Fig. 16.

Fig. 16 .
Fig. 16.Influence of piezoelectric constant of the piezoelectric transducer on the absolute value of the dimensionless dynamic characteristics

Table 1 .
Parameters of the mechanical subsystem

Table 2 .
Parameters of the piezoelectric transducer