Optimal Placement of Piezoelectric Plates to Control Multimode Vibrations of a Beam

Abstract

Damping of vibrations is often required to improve both the performance and the integrity of engineering structures, for example, gas turbine blades. In this paper, we explore the possibility of using piezoelectric plates to control the multimode vibrations of a cantilever beam. To develop an effective control strategy and optimize the placement of the active piezoelectric elements in terms of vibrations amplitude reduction, a procedure has been developed and a new analytical solution has been proposed. The results obtained have been corroborated by comparison with the results from a multiphysics finite elements package (COMSOL), results available in the literature, and experimental investigations carried out by the authors.

1. Introduction

The vibration control is a problem of great interest in many engineering fields since it allows avoiding problems connected with the vibrations. Blade vibrations in aircraft engines, for example, are often induced by interactions between blades and fluid and the associated fatigue phenomena can give rise to catastrophic failures [1–3]. Typically, passive damping systems, such as friction damping, are used to increase the blade life. These systems are very effective, but, in contrast to active damping elements, they are not able to change their characteristics depending on the system response. In the last two decades, the adoption of piezoelectric elements has received considerable attention by many researchers for its potential applicability to different areas of mechanical, aerospace, aeronautical, and civil engineering. These elements have an interesting coupling between electrical and mechanical properties: a deformation appears when an electric field is applied and vice versa [4, 5]. Their effectiveness to damp a particular excited mode or a multimode combination strongly depends on their position; in fact the study of their optimal position has received increasing attention. Typically, the aim of these studies is to find the position that minimizes an objective function or maximizes the degree of modal controllability (see [6, 7] for a review). The first study concerned with the optimal position to damp a specified mode has been that of Crawley and de Luis [4]. They found that the actuators should be in regions of higher average strain. Analogous results have been found by other researchers [8, 9]. For a cantilever beam Sunar and Rao [10], Demetriou [11], and Bruant et al. [12] found that the closer the piezoelectric actuators are to the fixed end, the more efficient they are. For a simply supported beam Yang et al. [13] found that, to control one specific mode, the optimal position for the piezoelectric plates is within the regions separated by the vibration nodal lines. In the paper by Barboni et al. [14] the possibility of exciting the flexural dynamics of an Euler-Bernoulli beam, according to a single mode, is examined. The results show that, to excite a desired mode, the actuator must be placed between two consecutive points at which the curvature becomes zero. Aldraihem et al. [15] studied the optimal length and location for different boundary conditions. They investigated beams with one pair and two pairs of piezoelectric actuators. Their optimization criterion was based on beam modal cost and controllability index; moreover, they added a penalty term to consider the actuator length (and the cost, weight, and space factors associated to this). Baz and Poh [16] studied the effects of varying the thickness and material of the bonding layer as well as the position of the piezoelectric actuators. They consider three beam elements to model a cantilever beam and the results show that it is preferable to place the actuator in a region of large strain. Subsequently Yang and Lee presented an analytical model for simultaneous optimization of noncollocated [17] and collocated [18] piezoelectric sensor/actuator placement and feedback control gain. The results show that this procedure can avoid the instability of the structural control system. Q. Wang and C. M. Wang [19] studied modal and multimodal vibrations. They propose a new controllability index and illustrated various beam examples with a pair of collocated piezoelectric actuators. More recently, studies about their use in blades of turbomachinery have been carried out; however, only few of these concern active damping [20–23]. Unfortunately in many real cases the loads applied to the structure excite more than one mode, with different amplitudes, and the excited modes can change during service. Therefore, the implementation of an active system that is capable of changing the work configuration of the piezoelectric plates can increase considerably their efficiency in the damping of the vibrations. In this paper, a new function to find the optimal placement of piezoelectric plates to control the multimode vibrations of the cantilever beam, with different amplitudes of the single modes, is defined. An analytical solution is proposed and the results are also compared with the FEM simulations and results from the literature with very good agreement.

2. Governing Equations for Piezoelectric Coupled Beam

In Figure 1 an Euler-Bernoulli cantilever beam with attached piezoelectric patches is schematically shown; the two PZT plates are applied in a symmetrical position with respect to the mid plane. Indicating by 𝛿𝐿_𝑒,𝛿𝐿_in, and 𝛿𝐿_𝑎 the virtual work of the elastic, inertial, and piezoelectric forces, respectively, the principle of the virtual work can be written as (in the following, the PZT plates will be considered perfectly bonded to the structure, their mass and inertia negligible with respect to the mass and inertia of the beam, and their thickness very lower than the thickness of the beam) 𝛿𝐿_𝑒+𝛿𝐿_in+𝛿𝐿_𝑎=0. (1)

Figure 1 

Reference configuration and action of the PZT plates.

Using the pin force model [4] the action of the PZT plates can be modeled by two flexural moments concentrated at the end of the plates (Figure 1) with 𝑀(𝑡)=𝜓𝐸6+𝜓^𝑎𝑐𝑇_𝑎𝑇_𝑏Λ(𝑡),Λ(𝑡)=𝑑₃₁𝑇_𝑎𝑉(𝑡),𝐸𝜓=^𝑏𝑇_𝑏𝐸_𝑎𝑇_𝑎. (2)

So by indicating 𝑤 the vertical displacement, the virtual work of the PZT plates can be written as (the virtual quantities are oversigned by a tilde) 𝛿𝐿_𝑎𝜕𝑤=𝑀𝜕𝑥_{𝑥=𝑎+(ℎ/2)}−𝜕𝑤𝜕𝑥_{𝑥=𝑎−(ℎ/2)}. (3)

The variables 𝑎 and ℎ can vary within the domain: ℎ0≤𝑎−2≤𝐿_𝑏,ℎ0≤𝑎+2≤𝐿_𝑏,0≤𝑎≤𝐿_𝑏,0≤ℎ≤𝐿_𝑏 (4) and their value depends on the modes that must be damped [14].

The virtual work of the elastic and inertial forces can be written, respectively, as 𝛿𝐿_𝑒=𝐸_𝑏𝐼_𝑏^𝐿𝑏₀𝜕²𝑤𝜕𝑥²𝜕²𝑤𝜕𝑥²𝑑𝑥,𝛿𝐿_in=−𝜌𝑆^𝐿𝑏₀𝜕²𝑤𝜕𝑡²𝑤𝑑𝑥. (5)

Using the modal analysis technique, indicating by 𝜙_𝑖(𝑥) the𝑖th flexural modal displacement of the cantilever beam and by 𝑋_𝑖(𝑡) its amplitude, the vertical displacement will be approximated by 𝑤(𝑥,𝑡)=^𝑁_𝑖=1𝑋_𝑖(𝑡)𝜙_𝑖(𝑥), (6) where𝑁is the number of the considered modes.

Substituting the last in (3), (5), considering 𝑤=𝜙_𝑗(𝑥), and successively in (1), the following governing equation is obtained: 𝐌𝐗^..(𝑡)+𝐊𝐗(𝑡)=𝐁(𝑎,ℎ)𝑉(𝑡), (7) where𝐗represents the vector of the amplitudes of the modes,𝐌and𝐊are, respectively, the mass and the stiffness matrix, and𝐁(𝑎,ℎ)is the vector control (the prime denotes the first spatial derivative): 𝐁()=𝑀̆𝜙𝑎,ℎ_′1ℎ𝑎+2−𝜙^′₁ℎ𝑎−2,𝜙^′₂ℎ𝑎+2−𝜙^′₂ℎ𝑎−2𝜙,…,_′𝑁ℎ𝑎+2−𝜙^′_𝑁ℎ𝑎−2 (8) with𝑀̆=(𝜓/(6+𝜓))𝐸_𝑎𝑐𝑇_𝑎𝑇_𝑏(𝑑₃₁/𝑇_𝑎).

If the Rayleigh damping is considered, (7) becomes 𝐌𝐗^..(𝑡)+𝐂𝐗̇(𝑡)+𝐊𝐗(𝑡)=𝐁(𝑎,ℎ)𝑉(𝑡) (9) with 𝐂=𝛼𝐌+𝛽𝐊. (10)

Indicating by𝝎the diagonal matrix of the natural frequencies of the beam, using the normal modes (𝐌=𝐈,𝐊=𝝎²), and assuming𝛽=0, (9) becomes 𝐗^..(𝑡)+𝛼𝐗̇(𝑡)+𝝎²𝐗(𝑡)=𝐁(𝑎,ℎ)𝑉(𝑡). (11)

If a single mode excitation is considered, Barboni et al. [14] found the optimal placement of PZTs. However, in some practical applications, for example, gas turbine blades, the load spectrum can include various modes, each of which is characterized by different amplitudes, and a new strategy needs to be developed. In this paper, the multimode damping will be obtained by applying a counterphase load, by PZTs plates, to the external excitation. The effectiveness of the piezoelectric elements will be measured by the amplitude of the vertical displacement of the free end, so that the most effective (optimal) position, identified in terms of𝑎andℎ(Figure 1), will be the location which maximizes this amplitude. Therefore, assigning a general spectrum to the PZT load (which corresponds to the spectrum of𝑉(𝑡); see (2)) 𝑉(𝑡)=^𝑁𝑠_𝑗=1𝑉_𝑗cos𝜔_𝑗𝑡, (12) where 𝑁𝑠 is the number of excited modes. The solution of (11) becomes 𝑋_𝑖(𝑡)=^𝑁𝑠_𝑗=1−𝑒^{−𝑡𝛼/2}𝐵_𝑖𝑉_𝑗cosh(1/2)𝑡𝛼²−4𝜔²_𝑖𝛼²−4𝜔²_𝑖𝜔²_𝑖−𝜔²_𝑗𝛼²−4𝜔²_𝑖𝛼²𝜔²_𝑗+𝜔²_𝑖−𝜔²_𝑗²−𝑒^{−𝑡𝛼/2}𝐵_𝑖𝑉_𝑗𝛼sinh(1/2)𝑡𝛼²−4𝜔²_𝑖𝜔²_𝑖+𝜔²_𝑗𝛼²−4𝜔²_𝑖𝛼²𝜔²_𝑗+𝜔²_𝑖−𝜔²_𝑗²+𝐵_𝑖𝑉_𝑗𝛼sin𝜔_𝑗𝑡𝜔_𝑗𝛼²𝜔²_𝑗+𝜔²_𝑖−𝜔²_𝑗²+𝐵_𝑖𝑉_𝑗cos𝑡𝜔_𝑗𝜔²_𝑖−𝜔²_𝑗𝛼²𝜔²_𝑗+𝜔²_𝑖−𝜔²_𝑗². (13)

It is possible to obtain an approximate form of (13) considering that the𝑖th term is the dominant one (the 𝑖th term of 𝑉(𝑡) is exciting 𝑋_𝑖(𝑡) near its resonance). After some algebraic manipulation, (13) can be simplified to 𝑋_𝑖(𝑡)≅𝐵_𝑖𝑉_𝑖sin𝜔_𝑖𝑡𝛼𝜔_𝑖−𝑒^{−(1/2)𝑡(𝛼+𝛼2−4𝜔2𝑖)}−1+𝑒^{𝑡𝛼2−4𝜔2𝑖}𝐵_𝑖𝑉_𝑖𝛼𝛼²−4𝜔²_𝑖. (14)

The second term of the right hand side of (14) represents the transient part so that, if this is neglected, the amplitude of the displacement of the free end can be written as 𝑤^⌢(𝑎,ℎ)=^𝑁_𝑖=1𝐵_𝑖(𝑎,ℎ)𝑉_𝑖𝜙_𝑖𝐿_𝑏𝛼𝜔_𝑖. (15)

In the following a bimodal excitation is considered, indicating by 𝑖₁,𝑖₂the indexes of the first and second considered modes respectively, and𝑟≤1the relative contribution to the excitation induced by the second mode with respect to the first one (i.e.,𝑟=1corresponds to the second mode governing the excitation without any contribution from the first mode). From (12): 𝑉(𝑡)=(1−𝑟)cos𝜔_𝑖1𝑡+𝑟cos𝜔_𝑖2𝑡. (16)

Substituting (16) into (15) and introducing 𝑔_𝑖1()=𝑀̆𝑟,𝜉𝛼1−𝑟𝜔_𝑖1𝜙_𝑖1𝐿_𝑏𝜙^′_𝑖1(𝜉),𝑔_𝑖2()=𝑀̆𝑟,𝜉𝛼𝑟𝜔_𝑖2𝜙_𝑖2𝐿_𝑏𝜙^′_𝑖2(𝜉), (17) it is obtained that 𝑤^⌢(𝑟,𝑎,ℎ)=𝑔_𝑖1ℎ𝑟,𝑎+2−𝑔_𝑖1ℎ𝑟,𝑎−2+𝑔_𝑖2ℎ𝑟,𝑎+2−𝑔_𝑖2ℎ𝑟,𝑎−2. (18)

Indicating by(𝜉₁=𝑎+(ℎ/2),𝜉₂=𝑎−(ℎ/2)), respectively, the position of the right and the left ends of the piezoelectric plates it will be 𝑤^⌢𝑟,𝜉₁,𝜉₂=𝑔_𝑖1𝑟,𝜉₁−𝑔_𝑖1𝑟,𝜉₂+𝑔_𝑖2𝑟,𝜉₁−𝑔_𝑖2𝑟,𝜉₂. (19)

If(𝜉̂₁,𝜉̂₂) are the coordinates of the absolute maximum of|𝑤^⌢(𝑟,𝜉₁,𝜉₂)|, it is possible to write 𝑤^⌢𝑟,𝜉₁,𝜉_2max=𝑤^⌢𝑟,𝜉̂₁,𝜉̂₂=𝑔_𝑖1𝑟,𝜉̂₁−𝑔_𝑖1𝑟,𝜉̂₂+𝑔_𝑖2𝑟,𝜉̂₁−𝑔_𝑖2𝑟,𝜉̂₂. (20)

Considering that𝑔_𝑘(𝑟,𝜉)has, for all the considered values of 𝑘, its positive absolute maximum at𝜉̂₁=𝐿_𝑏, this value will also be the abscissa of the positive absolute maximum for |𝑔_𝑖1(𝑟,𝜉₁)−𝑔_𝑖1(𝑟,𝜉₂)|+|𝑔_𝑖2(𝑟,𝜉₁)−𝑔_𝑖2(𝑟,𝜉₂)|. This implies that the PZT plates should always be placed with their right edges terminating at the tip of the beam independently of the excited modes and value of 𝑟, and hence 𝑤^⌢𝑟,𝜉₁,𝜉_2max=𝑤^⌢𝑟,𝐿_𝑏,𝜉̂_2max=𝑔_𝑖1𝑟,𝐿_𝑏−𝑔_𝑖1𝑟,𝜉̂₂+𝑔_𝑖2𝑟,𝐿_𝑏−𝑔_𝑖2𝑟,𝜉̂₂. (21)

Moreover, since 0<𝑔_𝑗𝑟,𝐿_𝑏>𝑔_𝑗(𝑟,𝜉)∀𝜉∈0,𝐿_𝑏,∀𝑗. (22) (21) becomes 𝑤^⌢𝑟,𝜉₁,𝜉_2max=𝑤^⌢𝑟,𝐿_𝑏,𝜉̂_2max=𝑔_𝑖1𝑟,𝐿_𝑏+𝑔_𝑖2𝑟,𝐿_𝑏−𝑓_𝑖1,𝑖2𝑟,𝜉̂₂, (23) where 𝑓_𝑖1,𝑖2𝑟,𝜉̂₂=𝑔_𝑖1𝑟,𝜉̂₂+𝑔_𝑖2𝑟,𝜉̂₂=𝑀̆𝛼)𝜙(1−𝑟^𝑖1𝐿_𝑏𝜔_𝑖1𝜙^′_𝑖1𝜉̂₂𝜙+𝑟^𝑖2𝐿_𝑏𝜔_𝑖2𝜙^′_𝑖2𝜉̂₂. (24)

The ordinate 𝜉̂₂ of the absolute maximum of |𝑤^⌢(𝑟,𝜉₁,𝜉₂)| will coincide with the absolute minimum of the 𝑓_𝑖1,𝑖2(𝑟,𝜉₂). This position changes with the considered modes and the modes ratio𝑟so that the optimal location of the left edge of the PZT elements can be found by solving the following system of equations: 𝜕𝑓_𝑖1,𝑖2𝑟,𝜉₂𝜕𝜉₂=^⌣𝑀𝛼)𝜙(1−𝑟^𝑖1𝐿_𝑏𝜔_𝑖1𝜙_𝑖^′′₁𝜉₂𝜙+𝑟^𝑖2𝐿_𝑏𝜔_𝑖2𝜙_𝑖^′′₂𝜉₂𝜕=0,₂𝑓_𝑖1,𝑖2𝑟,𝜉₂𝜕𝜉²₂=^⌣𝑀𝛼)𝜙(1−𝑟^𝑖1𝐿_𝑏𝜔_𝑖1𝜙_𝑖^′′′₁𝜉₂𝜙+𝑟^𝑖2𝐿_𝑏𝜔_𝑖2𝜙_𝑖^′′′₂𝜉₂>0. (25)

The solution will provide all the local minima and the absolute minimum will then be selected among these.

3. Results and Discussions

This section reports analytical results for different bimodal excitations and their comparison with FEM simulations. These simulations have been performed by using the frequency module of the structural mechanics module of COMSOL Multiphysics software [24]. Specifically, a steel beam of 30 cm of length has been taken into account; with bimodal control in mind and focusing on combinations of the first five modes, the wavelength of the highest mode has been divided into 50 subintervals of length 3 mm, so that Δ𝑎=Δℎ=3 mm and 5000 different combinations for𝑎andℎhave been considered. For each of these, the amplitude of the response of the tip, to a periodic load, with the frequency corresponding to one of the first five eigenfrequencies has been calculated. Moreover the amplitude response to a linear combination of the previous loads has been obtained by superposition of the response to two of the eigenfrequencies. The optimal position has been chosen to be the one which corresponds to the maximum amplitude. In Figure 2;  𝑓_1,2(𝑟,𝜉₂), the position of its absolute minimum, maximum and the position of the center of the piezoelectric plates, for different𝑟ratios, are reported. It is possible to observe that for𝑟=0and𝑟=1(that correspond, resp., to the excitation of the first and second modes) the optimal configuration of the PZT coincides with that obtained in [14]. For𝑟=0,𝑓_1,2(0,𝜉₂)attains its absolute minimum at the boundary points𝜉̂₂=0, so that the optimal configuration is with the piezoelectric plates distributed along the entire length of the beam. For increasing values of𝑟the position of the absolute minimum does not change until the derivative of𝑓_𝑖1,𝑖2(𝑟,𝜉₂)becomes zero in𝜉₂=0(25.1): ̂𝜙𝑟=^1′′(0)𝜙₁𝐿_𝑏𝜙₁^′′(0)𝜙₁𝐿_𝑏−𝜙₂^′′(0)𝜙₂𝐿_𝑏𝜔₁/𝜔₂≅0.5. (26)

Figure 2 

𝑓1,2(𝑟,𝜉2); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right edge (analytical); ×: FEM simulations.

For values of𝑟above𝑟̂, the optimal configuration gradually changes to reach the optimal one for the second mode at𝑟=1. The FEM results show that there is a very good agreement between the numerical simulations and the analytical results. Different mode combinations are represented in Figures 3, 4, 5, 6, and 7: the optimal configuration varies from the optimal value for the individual𝑖₁-excited mode (𝑟=0) to that for the mode𝑖₂(𝑟=1). For all the considered coupled modes, the optimal position changes continuously except for the couple ((1)–(3)), where a different type of behaviour is observed (Figure 5), with a sharp transition at𝑟≅0.8. This is due to the fact that, when𝑟is low, the absolute minimum of 𝑓_1,3(𝑟,𝜉₂) coincides with that obtained for the optimal configuration of the first mode (𝑟=0). When the ratio reaches the value of𝑟≅0.46, a relative minimum appears, but this develops into an absolute minimum and, therefore, coincides with the left end of the piezoelectric plates only when𝑟>0.8; for such values the PZT elements should be active only over about half of the length of the beam. The comparisons with the FEM are in agreement with all the cases analyzed. The proposed analytical method has been also compared with the experimental investigations reported in [25–27]. To test different load spectra and configurations an experimental apparatus has been designed and built; moreover, a MATLAB code has been developed to control the actuation and the modal contributions. Four piezoelectric plates have been used in three different configurations. Experiments have been done with one mode and multimodal vibrations. A good agreement was shown in all the considered cases.

Figure 3 

𝑓2,3(𝑟,𝜉2); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right edge (analytical); ×: FEM simulations.

Figure 4 

𝑓3,4(𝑟,𝜉2); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right edge (analytical); ×: FEM simulations.

Figure 5 

𝑓1,3(𝑟,𝜉2); red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right edge (analytical); ×: FEM simulations.

Figure 6 

𝑓2,5(𝑟,𝜉2);red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right edge (analytical); ×: FEM simulations.

Figure 7 

𝑓3,5(𝑟,𝜉2);red line: PZT plate’s left edge (analytical); green line: PZT position of the centre (analytical); orange line: PZT plate’s right edge (analytical); ×: FEM simulations.

4. Conclusions

In this work a new theoretical model for the optimal placement of piezoelectric plates to control the multimode vibrations of a cantilever beam is proposed. After a detailed description of the theoretical model, bi-modal excitation is considered. The designs of the optimal configurations, for different coupled modes and different relative contributions, are reported. The comparison between numerical and theoretical results is shown to confirm the validity of the optimization technique. This model can be very useful in many real situations when the spectrum of the load excites more eigenmodes and has, therefore, implications for the development of strategies to control multimode vibrations. This methodology has been extended to rotating systems [28, 29]. Its application to real systems subjected to the centrifugal loads, for example, gas turbine blades [30], and impulse loads [31] will constitute the object of further contributions.

Nomenclature

𝑎∶	Position of the center of the piezoelectric plates
𝐁:	Control vector
𝑐:	Beam width
𝐂:	Damping matrix
𝑑31:	Piezoelectric coefficient
𝐸𝑎:	Young’s modulus of the piezoelectric material
𝐸𝑏:	Young’s modulus of the beam material
ℎ:	Piezoelectric plates length
𝐼𝑏:	Inertia moment of the beam
𝐊:	Stiffness matrix
𝐿𝑏:	Beam length
𝑀:	Piezoelectric bending moment
𝐌:	Mass matrix
𝑟:	Percentage coupling coefficient
𝑆:	Cross-section area of the beam
𝑇𝑎:	Piezoelectric thickness
𝑇𝑏:	Beam thickness
𝑉:	Voltage applied to the piezoelectric plates
𝑤:	Vertical displacement
𝑤:	Virtual vertical displacement
𝛼:	Damping coefficient
𝜙𝑖(𝑥):	𝑖th flexural mode of the cantilever beam
𝜌:	Mass density
𝑋𝑖(𝑡):	Amplitude of the𝑖th mode
𝜔𝑖:	Natural frequency.