Dependences of force from displacement
1. Introduction
In recent years there is growing interest of materials, called smart materials. They have one or more properties that can be significantly changed. Smartness describes abilities of shape, size and state of aggregation changes. The main groups of smart materials are:
piezoelectric plates,
magnetorheostatic materials,
electrorheostatic materials,
shape memory alloys.
Those materials are widely used in technology and their numbers of applications still growing. Piezoelectric effect was discovered by French physicists Peter and Paul Curie in the 1880s. They described generation of electric charge on the surface with various shape during its deformation in different directions.
In their research, first of all, they focused on tourmaline crystal, salt and quartz. In 1881s Gabriel Lippman suggested the existence of the reverse piezoelectric phenomenon, which was confirmed experimentally by the Curie brothers. As a solution of research such two, unique properties of piezoelectric materials were assigned:
showing of simple piezoelectric effect, which rely on generating of voltage after deformation of material,
reverse piezoelectric effect, which rely on changing of sizes (by around 4%) after applying a voltage to piezoelectric facing.
Designing of technological systems, which contains piezoelectric elements should not be framed only to mechanical system analysis, but should be taken under consideration also electrical part. The entity should be considered as complex system, which contains independent subsystem.
Problem with mechanicalphysical systems synthesis, first of all electrical and mechanical ones, is well known and frequently published (Arczewski, 1988; Bellert 1981; Białas, 2012; Buchacz & Płaczek, 2012). In articles concerned theory and designing of filters not much space was devoted to mechanical systems with parameters distributed in continuous way. Determination tests of mechatronic systems characteristics, applications of graphs and structural numbers were carried out at Silesian Center repeatedly (Buchacz, 2004; Sękala & Świder, 2005; Wróbel, 2012). Those studies gave assumption to analysis of piezoelectric work. In many publications and papers, mechanical systems investigations on example of vibration beams and rods, were introduced. Moreover, rules of modellinig by nonclassical method and attempts of analysis by using hypergraph skeletons (Buchacz & Świder, 2000), graphs with signal flow and matrix methods (Bishop et at.,1972) were shown in these works.
Nowadays, numerous piezoelectric advantages caused its multiapplication in mechanics and in many replaced field of science (Shin et al.,2005; Ha, 2002). Many times beams configurations, with respect to different boundary conditions and during piezoelectric application in damping of vibrations, were analyzed. In the paper (Sherrit, 1999) capability of piezoelectric systems modelling using equivalent Manson models were presented. Analysis of longitudinal vibrations were made taking into consideration dielectric and piezoelement layer. Mason in (Mason, 1948) introduced onedimensional, equivalent system parameters widely used in modelling systems both free, and loaded. The main disadvantage of such approach is the equivalent of the mechanical system by discreet model. In article (Shin et at.,2005) author presents 4port equivalent system of piezoelectric plate, used to identification of system response on mechanical force. A matrix, size 5x5, inputoutput dependences, with different conditions of support, was also determined. Another type of piezoelectric transducer, which was based on Masons alternative systems of higher number of piezoelectric layer, were presented in many articles. Simulation was carried out in frequency domain, furthermore result was compared with values obtained by experimental method. Bellert in his volumes of chosen works (Bellert 1981) many times wrote about modelling of replacing systems, examined as 4ports. In work (Bolkowski, 1986) author provide chain method of connection electric 4ports. However, both: (Bolkowski, 1986) and (Bellert 1981) concerned primarily electric systems. In research work number N502 071 31/3719 attempts of active, mechanical systems, with damping in scope of graphs and structural numbers methods, were analyzed. In such, rich publications from field of vibration analysis, solution of piezoelectric plate itself with respect to dynamic characteristic was not undertaken, with the exceptions (Kacprzyk, 1995).
Previous presented solutions were conducted mainly in field of time and concerned single plate. Present paper is continuation of mentioned publications with stack of piezoelectric plates. This work is an author’s idea of calculations of complex systems with many elements. The base of calculation is matrix method and application of aggregation of graphs to determination characteristic parameters of bimorphic systems, as well as to drawing its characteristics.
2. Vibrating level sensor as a practical example of the application of piezoelectric stacks
An important characteristic of the designed and analyzed piezoelectric systems is the possibility of their practical application. This chapter presents options for further research related to the piezoelectric phenomenon of complex systems. Both a single piezoelectric plate, as well as complex systems, are often used in pressure, level, force and displacement sensors. As part of future research is proposed execution of laboratory stand for tests of piezoelectric plates used in vibration sensors. These sensors were used for level detection of loose materials in open or pressurized tanks. Output signal is a binary signal, transmitted to the automation systems via a relay. In Fig. 1 and Fig. 2 the level of vibration sensors manufactured by the “Nivomer” company from Gliwice were shown.
The application of stack tiles for intensification of the output signal. The sensors consist of two pairs of receiving plates and two or three supplying plates, connected in a bimorphic system (Fig. 3, Fig. 4). Variable voltage, which feeds the supply plates results in a change in their thickness proportionally to the value of applied voltage.
Changes in the plate thickness causing mechanical vibrations of the element, so called “fork”. When the “forks” are not covered by material, full deformation of supplying plates are transferred to the receiving ones. As a result of elongation of receiving plates, on its facing, there is a difference of potentials, proportional to the force. The value of this voltage is transformed by an electronic system (Fig. 5). In case of covered ”forks”, the receiving plates are no longer crushed and stretched. At the same time the potential is not generated on the facing of the plates.
Described sensor of "Nivomer" company is made of the body, ended with membrane which intensify a signal, to which are welded two identical vibrating rods (Fig. 6). In the presented sensor piezoelectric plates in the form of discs were used but there is a possibility of replacing the plates with shapes analyzed in previous chapters. Effect of plates stacks analysis in sensors designing The proposed analysis of the piezoelectric phenomenon of bimorphic systems allows at the design stage to determine the optimal parameters of piezoelectric plates. Wellchosen plate size and their number is crucial in the performance of other mechanical parts.
After the preliminary analysis of the construction of sensors with the company "Nivomer" from Gliwice found that by introducing a variable number of plates in the system is possible to choose the frequency of the generator and the maximum deflection of the fork carrying vibrations. The proposed methods and algorithms of work concern complex systems, can be used to design a stack of piezoelectric plates in the presented sensor level. In future work, it is proposed to conduct vibration test level sensors and a comparison of the algebraic method with the experimental method.
3. Object model under examination
Under consideration is vibrating piezoelectric plate with parameters distributed in a continuous way. The model has a section
In the analyzed example calculations are based on constitutive equations that include the assumed boundary conditions. In the study assumed that the test object is vibrating piezoelectric plate treated as a onedimensional system. Piezoelectric plate constitutive equations are as follows (Soluch, 1980):
where:
The equation of motion of a given element is as follows:
where:
It was assumed that Poisson equation is:
Rearranging equation (1) due to
Substituting expression (4) into equation (1), strain is given:
where:
From equation (3) result that
or assuming that the volume wave equation in piezoelement is equal to:
equation of motion (6) was presented in the form:
Assuming the expansion of the plate, mainly in the perpendicular plane to the axis, the following boundary conditions were defined:
Replaced mechanical stress by force, using the formula
The solution of equation (8) is harmonical displacement:
or assuming, that
written as:
Taking into account the boundary conditions included in the model (9) was obtained:
In order to determine the coefficients
Using the dependences between the trigonometric and exponential functions:
equation (17) was written as:
and assuming that the thickness of the plate
Determinant
so:
Using the record:
and substituting the expression (21), (23) into equation (24),
Furthermore, the determinant
so:
Analogously to (25)
Substituting (11) into equation (5) was written:
Using (10), the force was defined
substituting determined
Carrying out the multiplication of expressions in brackets (31):
and excluding the displacement
And taking into account the dependence:
was obtained:
Finally, introducing trigonometric functions, equation (35) was written as:
In (36) the ralationship of force
and substituting the determinants
Carrying out the multiplication of expressions in brackets:
and excluding displacements
and using dependences (18, 19) written:
Finally, taking into account the trigonometric functions,
was calculated.
Dependences of the forces and displacements acting on the system under consideration, taking into account the piezoelectric effect, are taking the following form:
Analyzing the effects occurring in the piezoelectric plate, also the electrical parameters such as voltage on the plates of piezoelectric and current value were took into account. Voltage
where:
Integrating (44):
where:
finally the voltage in a function of current was shown as:
where:
Capacitance, depends directly on the dimensions of the plates, aswell as physicochemical properties, written in the form:
Introducing (46) into equations (36), (42), (46), dependences of replacement set of piezoelectric plate were received, plate characterized by three equations:
Assuming signs :
and
obtained the relations between the electrical and mechanical values of piezoelectric plates:
which are also written in a matrix form:
4. Mapping matrix into graph
The values of susceptibility, admittance and characteristics were determined from the formula:
The matrix (58) was written as respond of system for operating extortion. The individual elements of matrix, presented as a flexibility, admittance and characteristics were recorded as follows: determined dependences of force from displacement are given in Table 1.

The dependences between mechanical and electrical parameters were shown in Table 2.

In table 3 electrical dependences: voltage and current, so called admittance of piezoelectric system, were listed.

In order to determine graph, representing modeled system of piezoelectric plates, as the symbols in the matrix were used. The elements of matrix (58) assigned to the edges of graph are presented as:
The graphical representation of mapping is shown as:
The symbol
Dependences according to the index
where
where
where
where
where
where
where
where
where
A set of drawings of the relation (fig. 9.) – (fig.17.), represents 4vertex graph, were created and presented in Fig. 18.
In the rest of the work earlier created 4vertex graph was replaced by structural number method to the 3vertex graph.
5. Construction of the replacement graph
Furthermore, the use of an extended 4vertex graph may prove to complicated calculations. In such case, a modelling of system using the replaced graph was performed. In order to maintain clearness of mapping, characteristics determined in paragraph 4 are indicated by Arabic numerals in parentheses, in accordance with (Bellert, 1981). As a consequence of introduction of the replaced graph, a graph presented in Fig. 19 was obtained. It is the basis for further network analysis methods.
As a result of insertion of replaced graph, replaced flexibility of the system was calculated by structural number method:
6. Chain equation of simple and complex system plate
On the figure 14 a piezoelectric plate with parameters distributed in the continuous way, the left and right end is free, was presented. The model of a single plate was marked by (i). Currently considered a model system is reduced system in the previous graph from 4vertex to 3vertex graph, as shown in Fig. 20.
Longitudinal vibrations of piezoelectric plate were considered, in the literature described also as thickness. The parameters specifying the system, in accordance with the previously accepted assumptions, were the sizes of input
where:
The relations between displacements of plate, and the forces acting on them, written in matrix form:
Transforming the matrix (26) to the chain form expects to receive in the form of matrices:
where:
In Fig. 21 the free system, consisting of two plates was presented. Superscript indicates the subsequent number of subsystem.
Finally, chain equation was written in general form:
After the operations carried out according to (79) it was found, that the chain matrix with cascade structure is the ratio of chain matrix of individual cells of the complex system. Obtained transition matrix is presented as:
Calculated coefficients (80) were substituted and the final form of the transition matrix was received:
In order to obtain the flexibility of the complex system, calculated coefficients of chain equation (79), was transformed to the basic form:
Equation (83) is the components of the complex characteristics of the matrix taking into account obtained chain parameters of complex system.
7. Charts of simple and bimorph system
In this paragraph, graphical charts of characteristics of piezoelectric plates were shown. The parameters adopted for graphs plotting was presented in table 4.
No.  Symbol  Value  Unit 
1  7.5  
2  150  
3  3.1  
4  1 
8. Conclusions
The chapter concerns the analysis of simple and complex piezoelectric systems, in order to determine the impact of piezoelectric plates parameters on the characteristics of the system. For a long time in the machine building are used subassemblies, whose operation is based on the piezoelectric phenomenon. In a researches of machine elements, on their surface piezoelectric sensors are glued, whereas to monitor the state plates are used transducers made from piezoelectric foil. Piezoelectric are often used in machine building also as assemblies, subassemblies or executive elements. Implementation of the piezoelectric system, which acts as a sensor or actuator is based on the selection of geometric dimensions of the plate, and their basic material parameters. In systems composed of several layers is also important piezoelectric plate number. Moreover, there are new problems at the design stage for the designers working in the field of machine building, concerning the application of both: single and stack plates. This matter is extremely important in terms of practical applications. For this reason it is necessary to conduct research whose main objective is to understand the phenomena associated with vibrations of complex piezoelectric systems.
Work is a continuation and development of decades researches at the Gliwice Center, consist in making the analysis of both the classical methods and nonclassical. Take advantage of nonclassical methods is a more general proposes from modeling in classical meaning. Resolves simple and complex systems irrespective of the type and number of elements included in the test system.
Applied method of structural numbers method was presented and used previously in modeling mechanical systems.
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