Passive damping of vibrations of prestressed thin-walled structures using piezoelectric elements connected to electric circuits

In this paper we estimate the influence of static pressure on the efficiency of passive damping of vibrations of a circular cylindrical shell using an open piezoelectric ring connected to a passive electric circuit of different configurations. The mathematical formulation of the problem of the dynamic behavior of an electroelastic structure is based on the variational principle of virtual displacements, which is written in matrix form taking into account the prestressed non-deformed state caused by applied loads, and as well as the work done by external surface forces, inertial forces and external electric field to transfer the free charge. The resulting equation is supplemented by the equation for the electric circuit, which relates the voltage at the piezoelectric ring with the corresponding electric charge. The problem is solved in the framework of 3D formulation using the finite element method. As examples, we consider the serial RL- and resistive R-circuits. Their parameters providing the most effective damping of free vibrations in terms of their decay rate are determined by solving the optimization problem. The results obtained by numerical simulations are compared with the results calculated by the analytical formulas at different values of static pressure.


Introduction
The key point in the problems of passive damping of thin-walled structures vibrations with the help of piezoelectric elements is the choice of parameters of the shunt electric circuit.The analytical expressions for calculating the optimum values of resistance and inductance of series [1] and parallel [2] RL-circuits, which were proposed at the end of the last century, have proved their efficiency and are still used by many researchers.It is known that their accuracy is significantly reduced in the case of damping higher modes or for a dense frequency spectrum [3].Later, shunt circuits of various designs have been considered: multimodal [4][5][6], with negative capacitance [7,8], adaptive (with automatic impedance tuning during operation at varying resonant frequencies) [9,10] and different type of switching shunts [11,12].One of the factors that can influence the optimal values of shunt circuit parameters is a prestress.In practice, this phenomenon often occurs in structures after different technological operations or as a result of the action of external loads.
In this work, using as an example a circular cylindrical shell, the effect of static internal pressure on the efficiency of vibrations damping with the help of a piezoelectric ring connected to an external electrical circuit of various configurations is evaluated.

General equations
Let us consider the problem of damping of free vibrations of a thin-walled cylindrical shell being under the action of internal pressure P (Figure 1a).The key idea behind the approach is to connect an open piezoelectric ring located on structure surface to a passive electrical circuit of various types.The computation scheme and geometric dimensions of the shell are shown in Figure 1a.A curvilinear surface of the shell is approximated by a set of flat segments (Figure 1b) [13].Each of them is supposed to satisfy the relations of the classical laminated plate theory [14] and the equations of the linear theory of piezoelasticity [15,16], written for the case of the plane stress state [14].When modeling a structure with piezoelectric ring on its outer surface it is assumed that the shell consists of one elastic orthotropic layer and one piezoelectric layer (Figure 1c).Using hypotheses stated in [17], we can write for each n-th layer [18,19]: or Hereinafter, the overbar denotes the quantities written in the coordinate system (x, ȳ, z), which is associated with the lateral surface of the structure (Figure 1b).σ and ε are the vectors containing components of the stress and strain tensors of an orthotropic lamina in a plane stress state; Ē and D are the vectors of electric field intensity and electric displacements; c is the reduced stiffness matrix evaluated at constant electric field; ẽ is the matrix of reduced piezoelectric constants; ε is the matrix of reduced dielectric constants evaluated at constant mechanical strain.If there is no piezoelectric effect in the n-th layer, the term containing matrix ẽ should be excluded.
The elements of the reduced matrices c(n) , ẽ(n) and ε(n) are determined in a well-known manner [18] using the engineering constants [14]: where and are the Young moduli of the material in the x and ȳ directions, ν 12 is the shear module in the xȳ plane (Figure 1b).Using the assumptions that the electric field intensity vector is normal to the electroded surfaces and its intensity in the piezoelectric ring is uniform [17], we have: where ψ is the potential difference between the top and bottom electroded surface of the piezoelectric ring, h p is the thickness of the piezoelectric ring.
The strains in the flat segments are determined using the relations of the nonlinear von Karman thin plate theory (small strains but moderate rotations) [20] T and the matrix of linear factors W has size 6 × 9 with the following nonzero components: The forces and moments in the flat segments can be written in the matrix form using expressions (1), (4): The coefficients entering into the matrices S and G are calculated as follows: The equation ( 2) can also be rewritten taking into account expression (4): The mathematical formulation of the natural vibrations problem for cylindrical shell with piezoelectric ring is based on the variational principle of virtual displacements, which is written in the absence of external loads (both mechanical and electrical) [17]: and is supplemented by the equation for a series RL-circuit connecting the voltage ψ at piezoelectric ring to the corresponding electric charge q: The subscripts "s" and "p" denote the belongingness of the quantity to the structure and piezoelectric element; d = {ū, v, w, θ x, θ ȳ, θ z } T is the generalized vector of the thin-walled structure displacements, including rotation angles θ x, θ ȳ, θ z respect to the corresponding axes of the coordinate system (x, ȳ, z); R and L are the values of resistance and inductance of the RL-circuit connected to the piezoelectric ring; J are the inertia matrices; C p is the capacitance of the piezoelectric ring.Linearization of the relation (5) with respect to the state with a small deviation from the initial equilibrium position caused by the action of a static load, after a series of simplifications, allows us to obtain from the expression (9) the equation (11), which takes into account the prestressed undeformed state of the structure [21]: Here the elements of the matrix σ 0 are found from the condition WSε (0) L = σ 0 w and the vector ε (0) L is the solution of the corresponding static problem.
The formulation of the problem of natural vibrations is based on the representation of the solution in the exponential form: where ũ is the function depending only on coordinates x; i is the imaginary unit; λ = ω + iγ is the characteristic index; ω is the natural frequency of vibrations; γ is the value, characterizing damping of the system.
Substituting expression (12) into the system of equations ( 10) and ( 11) allows us to obtain: The problem solution is found by the finite element method.After implementing the known procedures [13,20], the general matrix equation takes the form: where typical finite element matrices are determined in a well-known manner: Here: N is the shape functions for the generalized vector of the nodal displacements of the thinwalled structure d; B and V are the matrices of relation coupling the strains εL and w with the nodal displacements of the shell finite element, which are obtained using the relations (5).The matrices with overbar are formed in the coordinate system associated with the lateral surface of the structure.Their transformation to the global Cartesian coordinate system (x, y, z) is performed for each element using the matrix of the directional cosines in the known way [13,22].The matrix of geometric stiffness K g is formed using the initial forces and moments caused by the action of internal pressure.They are determined by solving the static problem.More details of the finite element procedures and verification examples are given in articles [21][22][23].Equation ( 14) is reduced to the generalized eigenvalue problem and is solved using the implicitly restarted Arnoldi method [24]: where I is the unit matrix.

IC-MSQUARE-2023
Journal of Physics: Conference Series 2701 (2024) 012101 First, consider a circuit consisting of single resistance of value R. In this case, a change in R leads to a change in the imaginary parts γ of eigenvalues λ, whereas its real parts ω remains practically constant.The optimal value of resistance, at which the damping is maximum, can be calculated by the analytical expression [1]: where ω o/c m and ω s/c m are the m-th natural frequencies of vibrations for the system with an open circuit (hereinafter referred to as "o/c") and closed electrodes (short circuit, hereinafter referred to as "s/c"), C 0 is the static capacity of the piezoelectric ring.
The damping ratios ξ m = γ m /ω m of some vibration modes as a function of the resistance R are given at Figure 2a for a shell being under static pressure P = 50 MPa.As can be seen from the figure, each curve has one pronounced extremum, the value of which is different for a certain mode with number m.The optimal resistance can be approximately estimated by the expression ( 16), but the relative error ∆R m (see Eq. ( 17)) will depend on the applied load.To illustrate the above, the Figure 2b shows the change in the value of ∆R m depending on the pressure P for first three modes with non-zero electromechanical coupling coefficient K m .
Here the resistance value R calc m is obtained based on the solution of a sequence of natural vibrations problems (15) and condition (18): The obtained results showed that expression (16) makes it possible to estimate the optimal resistance value with an accuracy of ±5%.The exception is the third mode (ξ 3 ) at P = 0.

IC-MSQUARE-2023 Journal of Physics: Conference Series 2701 (2024) 012101
The reason for such a significant difference is the dense frequency spectrum in the vicinity of the frequency ω 3 .With an increase in the static load, the spectrum in a given frequency range becomes sparse, and the error of the analytical formula decreases.The next examples consider a resonant electrical RL-circuit, consisting of the series-connected resistor R and the inductance coil L. By varying the values of circuit elements R and L, it is possible to reduce the resonance amplitude, as well as to increase the rate of free vibration damping [1,25,26].Comparison between the numerical and analytical results is presented in Table 1 for the first natural frequency of vibrations.The parameters R opt 1 and L opt 1 are calculated by expressions obtained in [1] using the pole placement method.The values R calc 1 and L calc 1 are determined from the solution of optimization problem using the condition: where λ m and λ mc are the complex eigenvalues corresponding to the damped (m-th) mode and the mode of the electrical circuit connected to the piezoelectric ring.The relative error ∆L 1 is calculated according to expression (17), replacing R with L.

Conclusion
The developed finite element algorithm is used to choose parameters of a shunt electrical circuit for problems of passive vibration damping of prestressed thin-walled structures.The optimal values of circuit elements can be estimated using analytical expression with an accuracy of 5% for various values of static pressure.However, for dense frequency spectrum the relative error significantly increases.The results have been shown that the approach proposed makes it possible to obtain higher damping ratios and it provides the smallest difference between the natural vibration frequencies of the structure and the electric circuit.

Figure 1 .
Figure 1.Circular cylindrical shell equipped with a piezoelectric ring connected to passive electric circuit (a), shell finite element (b), cross-section of the laminated structure (c).

Figure 2 .
Figure 2. Damping ratio ξ m (P = 50 MPa) (a) and relative error ∆R m (b) depending on the resistance R of a passive electric circuit.

Table 1 .
Optimal parameters of the shunt RL-circuit calculated by different methods for the first natural frequency of vibrations.