A piezoelectric type non-linear energy sink optimal target energy transmission threshold acquisition method

Abstract

The present application relates to a kind of piezoelectric type nonlinear energy sink optimal target energy transfer threshold acquisition method, for elastic plate-piezoelectric type nonlinear energy sink system, method includes the following steps: establishing the two-degree-of-freedom model of elastic plate-piezoelectric nonlinear energy sink system, obtains coupled system dynamics equation;Using harmonic balance method to solve the forced vibration response of system, obtains the response surface of main system;According to the nonlinear modal of coupled system dynamics equation obtained by system;The expression of nonlinear modal is mapped with system response, obtains the target frequency response curve of the change of predicted coupled system response surface, and then the lower limit and upper limit of target energy transfer threshold are solved, and the optimal target energy transfer threshold is obtained.Compared with prior art, the calculation process is reduced, and the calculation result is tested, which can ensure accuracy.

Description

Technical Field

[0001] This invention relates to the field of nonlinear energy trap technology, and in particular to a method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap. Background Technology

[0002] Figure 1 The schematic diagram of the system under consideration is shown. It consists of a clamping plate, piezoelectric elements, and a piezoelectric shunt circuit containing an inductor, resistor, negative capacitor, and nonlinear capacitor. Two piezoelectric elements are attached to opposite sides of the clamping plate. The positions of the piezoelectric elements are determined by the mode shape of the fixed plate and are attached at locations with larger mode shape displacements. To suppress plate vibration, a piezoelectric element is attached to the center of the plate according to the first mode shape of the fixed plate. The piezoelectric elements are connected in parallel, and the electrodes of the piezoelectric elements are shunted from the circuit. The shunt circuit acts as a non-electric current source (NES). With the occurrence of the TET phenomenon, vibrational energy is transferred from the mechanical system to the circuit, applying a harmonic excitation F to the plate.

[0003] However, in this system, most existing methods for determining the target energy transfer excitation threshold involve establishing the motion equations of a two-degree-of-freedom system consisting of a linear master structure and a nonlinear energy trap, and then directly solving the model to obtain the optimal target energy transfer threshold. However, this approach involves complex calculations and the results are not very accurate. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art, such as complex calculation process and low accuracy of calculation results, and to provide a method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap.

[0005] The objective of this invention can be achieved through the following technical solutions:

[0006] A method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap, used in an elastic plate-piezoelectric nonlinear energy trap system, the system including a clamping plate, a piezoelectric sheet, and a piezoelectric shunt circuit, the method comprising the following steps:

[0007] A two-degree-of-freedom model of the elastic plate-piezoelectric nonlinear energy trap system is established, and the dynamic equations of the coupled system are obtained.

[0008] The forced vibration response of the system is solved by the harmonic balance method, and the response surface of the main system is obtained; the nonlinear modes of the coupled system are obtained according to the dynamic equations of the coupled system.

[0009] By mapping the expression of the nonlinear mode to the system response, the target frequency response curve for predicting the change of the response surface of the coupled system is obtained. The target frequency response curve at the lower limit of the target energy transfer threshold is defined as the curve at which the system just begins to show the suppression response; the target frequency response curve at the upper limit of the target energy transfer threshold is defined as the curve at which the peak value of the high-amplitude branch of the main system is equal to the plateau amplitude.

[0010] The optimal target energy transfer threshold is obtained by solving for the lower and upper limits of the target energy transfer threshold.

[0011] Furthermore, the expression for the dynamic equations of the coupled system is as follows:

[0012]

[0013]

[0014] In the formula, m is the mass of the plate, r is the generalized displacement, c is the damping value determined by the damping ratio of the plate material, k is the stiffness of the plate, θ is the electromechanical coupling coefficient, and C p For the equivalent capacitance, q a =q / θ, where q is the voltage and charge on the electrode, φ(x,y) is the vibration model, F is the harmonic excitation on the plate, L is the Lagrangian function of the system, R is the resistance value, and k lin k is the negative capacitance coefficient, k3 is the nonlinear capacitance coefficient, and θ 2 L, θ 2 R, θ 2 k lin and θ 4 k3 is equivalent to the mass, damping value, linear stiffness, and cubic stiffness of the grounded NES, respectively.

[0015] Furthermore, the process of using the harmonic balance method to solve the forced vibration response of the system and obtain the response surface of the main system is as follows:

[0016] Using the first harmonic balance method, based on the dynamic equations of the coupled system, an approximate periodic solution for the forced vibration of the coupled system is obtained. The analysis is restricted to 1:1 resonance, yielding:

[0017]

[0018]

[0019] in,

[0020] μ = LC p ω i 2 ,

[0021] λ na=RC p ω i 2 , κ=C p k lin ,

[0022] Let F(t), r(t) and q a 3 Represented by the first harmonic term:

[0023] F(t)=F 1c cos(ωt)+F 1s sin(ωt)

[0024] r(t) = r 1c cos(ωt)+r 1s sin(ωt)

[0025] q a (t)=q 1c cos(ωt)

[0026] Ignoring higher harmonic terms and performing a harmonic balance process, the response surface equation of the principal system is obtained as follows:

[0027]

[0028]

[0029]

[0030] ∈λ na ωq 1c +∈ω i 2 r 1s =0

[0031]

[0032]

[0033] In the formula, F1 is the amplitude of the simple harmonic excitation, and r1 is the amplitude of the displacement of the main system.

[0034] Furthermore, the process of obtaining the nonlinear modes of the coupled system includes:

[0035] Remove the external force and damping terms from the dynamic equations of the coupled system and substitute them into the simple harmonic term r(t) = r 1c cos(ωt) and q a (t)=q 1c cos(ωt), we get q 1c and r 1cThe calculation expression is obtained, and then the absolute value of the amplitude is taken to obtain the nonlinear mode of the coupled system;

[0036] The obtained q 1c and r 1c The calculation expression is:

[0037]

[0038]

[0039] Furthermore, the calculation expression for the lower limit of the target energy transfer threshold is as follows:

[0040]

[0041] In the formula, F b This represents the lower limit of the target energy transfer threshold. To find the frequency corresponding to the maximum point by taking the derivative with respect to ω and setting the derivative to zero,

[0042] Furthermore, in the calculation of the upper limit of the target energy transfer threshold, for the case where μ = 2, let... Using the Shengjin formula, the frequency corresponding to the point where the peak value of the high-amplitude branch of the main system equals the plateau amplitude is obtained as follows:

[0043]

[0044] in,

[0045]

[0046]

[0047]

[0048]

[0049] B = bc - 9ad

[0050] A = b 2 -3ac

[0051] Based on the frequency corresponding to the point where the peak value of the high-amplitude branch of the main system equals the amplitude of the plateau, the upper limit of the target energy transfer threshold on the system response surface is obtained, i.e.:

[0052]

[0053] In the formula, F e The upper limit of the target energy transfer threshold.

[0054] Furthermore, in the calculation of the upper limit of the target energy transfer threshold, for the case where μ = 1, the calculation is performed as follows: The nonlinear mode of the amplitude-frequency plane of the time-harmonic excitation is mapped onto the displacement plane of the system response, and the excitation value corresponding to the minimum point of the corresponding curve is taken as the upper limit of the excitation threshold for target energy transfer.

[0055] Furthermore, the piezoelectric sheet is connected to both sides of the clamping plate, and the piezoelectric sheet is connected to a piezoelectric shunt circuit, which includes an inductor, a resistor, a negative capacitor and a nonlinear capacitor connected in series.

[0056] Furthermore, the platform amplitude is within a certain range of harmonic excitation amplitude. The response of the main system is a tilted platform, the resonance peak of the main system is effectively reduced, and the nonlinear energy trap plays an efficient vibration suppression role. When the harmonic excitation amplitude exceeds this range, the system response shows a new branch, and the branch peak value is higher than the platform amplitude.

[0057] Furthermore, the method also includes controlling the elastic plate-piezoelectric nonlinear energy trap system based on the obtained optimal target energy transfer threshold.

[0058] Compared with the prior art, the present invention has the following advantages:

[0059] (1) After constructing a two-degree-of-freedom model of the elastic plate-piezoelectric nonlinear energy trap system, this invention obtains the response surface of the main system and the nonlinear mode of the coupled system by coupling the system dynamic equations. The two are mapped to each other, and the nonlinear mode is determined to be the target frequency response curve for predicting the change of the response surface of the coupled system. Then, based on the change properties of the curve, the lower limit and upper limit of the target energy transfer threshold are obtained. This calculation process reduces the amount of calculation, and the calculation results have been verified to ensure accuracy.

[0060] (2) When it comes to vibration suppression of elastic plates with different structural forms and parameters, the parameters of the piezoelectric nonlinear energy trap can be quickly matched according to the optimal target energy characteristics to realize the design of the nonlinear energy trap. Attached Figure Description

[0061] Figure 1 This is a schematic diagram of a theoretical model of an elastic plate-piezoelectric nonlinear energy trap system provided in an embodiment of the present invention;

[0062] Figure 2 This is a flowchart illustrating a method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap, as provided in an embodiment of the present invention.

[0063] Figure 3(a) is a schematic diagram of the frequency response of a coupled system of a main system provided in an embodiment of the present invention;

[0064] Figure 3(b) is a schematic diagram of the frequency response of a coupling system of a nonlinear energy trap provided in an embodiment of the present invention.

[0065] Figure 4 This is a schematic diagram comparing numerical simulation results with first harmonic balance results provided in an embodiment of the present invention;

[0066] Figure 5(a) is a schematic diagram of the nonlinear modes of a main system provided in an embodiment of the present invention;

[0067] Figure 5(b) is a schematic diagram of the nonlinear modes of a nonlinear energy trap provided in an embodiment of the present invention;

[0068] Figure 6(a) is a schematic diagram comparing nonlinear modes and system response in a main system provided in an embodiment of the present invention;

[0069] Figure 6(b) is a schematic diagram comparing nonlinear modes and system response in a nonlinear energy trap provided in an embodiment of the present invention;

[0070] Figure 7(a) is a three-dimensional comparison diagram of nonlinear mode and system response when μ=2 provided in an embodiment of the present invention;

[0071] Figure 7(b) is a three-dimensional comparison diagram of nonlinear mode and system response when μ=1 provided in an embodiment of the present invention;

[0072] Figure 8(a) is a schematic diagram of the upper limit verification of the target energy transfer threshold in frequency response provided in an embodiment of the present invention;

[0073] Figure 8(b) is a schematic diagram of the upper limit verification of the target energy transfer threshold on the time domain rate response provided in an embodiment of the present invention;

[0074] Figure 9(a) is a schematic diagram of the frequency response of an upper limit of the target energy transfer threshold provided in an embodiment of the present invention;

[0075] Figure 9(b) is a schematic diagram of the displacement response of a target energy transfer threshold upper limit provided in an embodiment of the present invention;

[0076] Figure 10(a) is a schematic diagram of the upper limit verification of the target energy transfer threshold in frequency response provided in an embodiment of the present invention;

[0077] Figure 10(b) is a schematic diagram of the upper limit verification of the target energy transfer threshold in the time domain response provided in an embodiment of the present invention;

[0078] In the figure, HBM stands for Incremental Harmonic Balance Method, Numericalsimulation represents the numerical simulation results, NNMs represents Nonlinear Normal Modes, and Responses represents the system response. Detailed Implementation

[0079] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0080] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0081] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0082] In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, or the orientation or positional relationship in which the product of this invention is usually placed during use. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0083] It should be noted that the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this application, "multiple" means two or more, unless otherwise explicitly specified.

[0084] Furthermore, terms such as "horizontal" and "vertical" do not imply that components must be absolutely horizontal or suspended, but rather that they can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal than "vertical," not that the structure must be completely horizontal, but can be slightly tilted.

[0085] Example 1

[0086] This embodiment provides a method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap, used in an elastic plate-piezoelectric nonlinear energy trap system. The system includes a clamping plate, piezoelectric sheets, and a piezoelectric shunt circuit. The piezoelectric sheets are respectively connected to both sides of the clamping plate and connected to the piezoelectric shunt circuit. The piezoelectric shunt circuit includes an inductor, a resistor, a negative capacitor, and a nonlinear capacitor connected in series. The method includes the following steps:

[0087] S1: Establish a two-degree-of-freedom model of the elastic plate-piezoelectric nonlinear energy trap system and obtain the dynamic equations of the coupled system;

[0088] S2: The forced vibration response of the system is solved using the harmonic balance method to obtain the response surface of the main system; the nonlinear modes of the coupled system are obtained according to the dynamic equations of the coupled system; the expression of the nonlinear mode is mapped to the system response to obtain the target frequency response curve for predicting the change of the response surface of the coupled system. The target frequency response curve of the lower limit of the target energy transfer threshold is defined as the curve at which the system just begins to show the suppressed response; the target frequency response curve of the upper limit of the target energy transfer threshold is defined as the curve at which the peak value of the high amplitude branch of the main system is equal to the plateau amplitude.

[0089] S3: Solve for the upper and lower limits of the excitation threshold range under two different conditions, and verify with numerical analysis;

[0090] S4: Method for determining the upper limit of the two excitation threshold intervals of the elastic plate-piezoelectric nonlinear energy trap system.

[0091] The following is a detailed description.

[0092] I. Dynamic Model of the Elastic Plate-Piezoelectric Nonlinear Energy Trap System

[0093] Energy-based modeling is based on Hamilton's principle, which is used to establish... Figure 1 The governing equations of the system shown, and the variational operator (VI), can be expressed as:

[0094]

[0095] Among them, L=T-U+W e Let T be the system's Lagrangian function, T be the system's kinetic energy, U be the system's potential energy, and W be the system's potential energy. e It refers to the strain energy and potential energy of the piezoelectric element, W ncIt is a futile effort that is not conservative.

[0096] Assume the deflection w of the plate is the product of the mode shape φ(x, y) and the general coordinate r(t):

[0097] w(x,y,t)=φ(x,y)r(t) (2)

[0098] Where φ represents the mode shape and r represents the generalized displacement.

[0099] Based on boundary conditions, classical laminar flow theory, and the constitutive equations of piezoelectric material systems, T, U, W e W nc They are respectively:

[0100]

[0101]

[0102]

[0103] W nc =r(t)φ(x) f y f )F(t)-V p，1 (t)q1-V p，2 (t)q2 (6)

[0104] In the formula, Vol b For the volume of the plate, Vol p S is the volume of a single piezoelectric element. b ′、T b Let S be the strain and stress vector of the plate. p ′、T p V represents the strain and stress vectors of a single piezoelectric element, E3 represents the electric field, D3 represents the electric displacement, and the subscript 3 indicates that the electric field direction is along the z-axis. p，1 V p，2 q and q are the voltage and charge on the electrode, respectively. By substituting equation (4) into equation (1) and solving the equation, we get:

[0105]

[0106] Where m is mass, k is stiffness, c is the damping value determined by the damping ratio of the plate material, θ is the electromechanical coupling coefficient, and C p This is the equivalent capacitance.

[0107] Consider the effect of the shunt circuit on the dynamic equations of the coupled system. The piezoelectric electrode is shunted by an inductor, resistor, negative capacitor, and nonlinear capacitor. According to Kirchhoff's voltage law, the following voltage-charge relationship is obtained:

[0108]

[0109] Where L is the inductance value, R is the resistance value, and k is the capacitance value. lin is the negative capacitance coefficient, and k3 is the nonlinear capacitance coefficient. Substituting equation (8) into equation (7), the dynamic equation of the coupled system is transformed into the following form:

[0110]

[0111] Let q a =q / θ, multiplying both sides of the second equation (9) by θ, we get:

[0112]

[0113] Equation (8) is structurally equivalent to a mechanically grounded NES system. θ 2 L, θ 2 R, θ 2 k lin and θ 4 k3 is equivalent to the mass, damping value, linear stiffness, and cubic stiffness of the grounded NES, respectively.

[0114] II. Harmonic Balance

[0115] 2.1 Forced Vibration Response

[0116] The first harmonic balance method is used to solve for the approximate periodic solution of the forced vibration of the coupled system, limiting the analysis to 1:1 resonance. Equation (10) is simplified to (11):

[0117]

[0118] in,

[0119]

[0120] Let F(t), r(t) and q a 3 Represented by the following first harmonic term:

[0121]

[0122] Substituting equation (13) into equation (11), ignoring higher harmonic terms, and performing a harmonic balance process, we get:

[0123]

[0124] Equation (14) is the algebraic expression for the displacement amplitude components of the main system, the charge displacement amplitude of the piezoelectric nonlinear energy trap, and the amplitude components of the harmonic excitation. q 1cUsing ω as independent variables, we obtain closed-form expressions for the displacement amplitude components and harmonic excitation amplitude components of the main system, thus obtaining the response surface. Therefore, the amplitudes of the main system displacement and harmonic excitation are:

[0125]

[0126] In the formula, F1 is the amplitude of the simple harmonic excitation, and r1 is the amplitude of the displacement of the main system.

[0127] Figure 3 shows the displacement amplitude-frequency response curves of the coupled system under different harmonic excitation amplitudes.

[0128] Within a certain range of harmonic excitation amplitude, the main system's response exhibits a slightly tilted "plateau" around 82Hz, effectively reducing the resonance peak and allowing the nonlinear energy trap to exert a highly efficient vibration suppression effect. When the harmonic excitation amplitude exceeds this range, the system response exhibits a new branch, with the branch peak value exceeding the plateau amplitude, and the vibration suppression effect of the nonlinear energy trap on the main system deteriorates. The boundaries of this harmonic excitation amplitude range are defined as the lower and upper threshold limits for forced vibration target energy transfer.

[0129] Numerical methods were used to verify the accuracy of the first harmonic balance method in determining the system response. The initial condition was zero. When the system exhibited a modulated response, the maximum value of the response was taken as the amplitude at that frequency point. Figure 4 As shown in the figure. The results indicate that when the initial conditions are zero, the numerical results are consistent with the first harmonic balance results.

[0130] 2.2 Nonlinear Modes

[0131] To find the nonlinear modes of the coupled system, by removing the external force and damping terms from the dynamic equations of the coupled system, we get:

[0132]

[0133] The simple harmonic term r(t) = r 1c cos(ωt) and q a (t)=q 1c Substituting cos(ωt) into equation (16), we get:

[0134]

[0135] Equation (17) can be solved to obtain:

[0136]

[0137] q 1c and r 1c Both are functions of ω. By taking the absolute value of the amplitude, the nonlinear modes of the coupled system can be obtained, as shown in Figure 5.

[0138] The relationship between the system's nonlinear modes and system response is analyzed, and the comparison is shown in Figure 6. The results show that the maximum value and the "plateau" amplitude of the nonlinear mode S11- are approximately the same, and the evolution of the coupled system response can be accurately predicted using the nonlinear mode.

[0139] III. Excitation Threshold Ranges for Two Optimal Target Energy Transfer Characteristics

[0140] From a mathematical perspective, nonlinear modes establish q 1c The relationship between ω and the system response surface is mapped onto the system response surface, i.e., equation (13), to obtain a curve on the response surface. This curve always passes through the extreme points of the response surface and can also accurately predict the changes in the response surface of the coupled system, as shown in Figure 7. By finding the target point on the nonlinear mode, the corresponding harmonic excitation amplitude can be obtained, thereby obtaining the desired frequency response curve, which is called the target frequency response curve. For the system shown in Figure 7(a), the target frequency response curve of the lower limit of the target energy transfer threshold can be defined as the curve at which the system just begins to show the suppression response. The target frequency response curve of the upper limit of the target energy transfer threshold can be defined as the curve where the peak value of the high amplitude branch of the main system is equal to the amplitude of the "platform". Its significance is that the peak value of the high amplitude branch of the control system response does not exceed the amplitude of the "platform".

[0141] Furthermore, for the coupled system when μ=1, the nonlinear modes and frequency response of the system are shown in Figure 7(b). The results show that the high-amplitude branch of the system response appears above the "plateau" amplitude, that is, it is impossible to find a frequency response where the peak value of the high-amplitude branch is approximately the same as the amplitude of the "plateau". The target frequency curve at this time can be defined as the curve when the high-amplitude branch just appears.

[0142] Therefore, the excitation threshold range for the optimal target energy transfer characteristics of the two nonlinear energy traps was determined.

[0143] The amplitude of the harmonic excitation corresponding to the extreme point of the nonlinear mode S11- of the main system is approximately regarded as the lower threshold of the target energy transfer of the coupled system. First, find the target point of the lower threshold of the target energy transfer, that is, the maximum point of the nonlinear mode S11- of the main system. Write r according to equation (17). 1c The expression for ω:

[0144]

[0145] Differentiating equation (18) with respect to ω and setting the derivative to zero, we obtain the frequency corresponding to the maximum point as follows:

[0146]

[0147] Substituting equation (19) into equation (17), we get and This is also the "platform" amplitude. Based on this maximum point, a solution can be obtained on the system response surface. The harmonic excitation amplitude of this solution is the lower threshold of target energy transfer, i.e.:

[0148]

[0149] 3.1 Solving for the upper limit of the threshold in the first case

[0150] Solving for the upper threshold of target energy transfer: For the first case μ = 2, that is, when the frequency response of the coupled system just shows a high-amplitude branch, its peak value is smaller than the "plateau" amplitude, as shown in Figure 6(a). Let:

[0151]

[0152] Based on the curve shape of the nonlinear mode in Figure 5(a), the frequency solution corresponding to the S11-maximum has three real roots, one of which is a double root. Using the Shengjin formula to solve equation (21), the frequency corresponding to the point where the amplitude of S11+ is equal to the S11-maximum is:

[0153]

[0154] in,

[0155]

[0156]

[0157] Similarly, a solution is obtained from this point on the system response surface, and the amplitude of the harmonic excitation of this solution is the upper threshold for target energy transfer. Using F... e This represents the upper limit of the target energy transfer threshold, i.e.:

[0158]

[0159] Substituting the coupled system parameters into equation (25), we obtain the upper limit of the target energy transfer threshold under these parameter conditions as F. e =9.62N. This value was chosen as the amplitude of the harmonic excitation to obtain the system's frequency response curve, and numerical verification was performed to confirm that the harmonic excitation frequency was... The time-domain response of the system under the initial condition r = 0.003m is shown in Figures 8(a) and 8(b). The results show that the peak value of the high-amplitude branch of the main system's response under this harmonic excitation amplitude is approximately equal to the amplitude of the "plateau," and the peak value of the high-amplitude branch is still within an acceptable range.

[0160] 3.2 Solving for the upper limit of the threshold in the second case

[0161] For the second case, μ = 1, meaning the peak value of the high-amplitude branch of the coupled system appears just above the "plateau" amplitude, analytically deriving the extreme points on the nonlinear mode S11+ of the harmonic excitation amplitude-frequency plane is too complex. Therefore, a numerical method is used to solve it, yielding the following result. The nonlinear mode S11+ in the harmonic excitation amplitude-frequency plane is shown in Figure 9(a). Mapping this nonlinear mode S11+ to the harmonic excitation amplitude-principal system displacement plane yields the result shown in Figure 9(b). The excitation value corresponding to the minimum point of this curve is the upper limit of the excitation threshold for target energy transfer.

[0162] Substituting the parameters, the excitation threshold is calculated to be [2.96N, 5.71N], and the maximum value of the nonlinear mode S11 is... When a high-amplitude branch appears, the minimum value of the harmonic excitation amplitude-frequency plane nonlinear mode S11+ corresponds to r. 1c =0.83mm, which is greater than the maximum value of the nonlinear mode S11, proving that the peak value at the beginning of the high-amplitude branch is greater than the "plateau" amplitude. Figures 10(a) and 10(b) show the amplitude F of the harmonic excitation. e The main system frequency response at ω = 5.71 N and the main system time-domain response at the minimum frequency ω = 79.7 Hz corresponding to the minimum point in Figure 9(a), with initial condition r = 0.001 m, are shown. Figure 10(a) shows the newly emerging high-amplitude response branch. The time-domain response in Figure 10(b) indicates that the system response did not converge to the high-amplitude branch at this point, but rather converged to the lower low-amplitude response curve. The target energy transfer threshold remains highly accurate under these conditions.

[0163] When solving for the upper limit of the target energy transfer threshold, it is necessary to first analyze the location of the high-amplitude branch. If it appears below the "plateau" amplitude, the frequency response corresponding to the upper limit of the target energy transfer threshold will show that the peak value of the high-amplitude branch is approximately the same as the "plateau" amplitude. If the high-amplitude branch appears above the "plateau" amplitude, the frequency response corresponding to the upper limit of the target energy transfer threshold will show that the high-amplitude branch just appears. The criterion for judgment is whether the displacement amplitude of the main system corresponding to the minimum value on the nonlinear mode S11+ of the harmonic excitation amplitude-frequency plane is greater than the maximum displacement value of the main system in the nonlinear mode S11-. The former is the peak value when the high-amplitude branch just appears, and the latter is the "plateau" amplitude. If it is less than the former, it means that the high-amplitude branch appears below the "plateau" amplitude.

[0164] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap, used in an elastic plate-piezoelectric nonlinear energy trap system, the system comprising a clamping plate, a piezoelectric sheet, and a piezoelectric shunt circuit, characterized in that, The method includes the following steps: A two-degree-of-freedom model of the elastic plate-piezoelectric nonlinear energy trap system is established, and the dynamic equations of the coupled system are obtained. The forced vibration response of the system is solved by the harmonic balance method, and the response surface of the main system is obtained; the nonlinear modes of the coupled system are obtained according to the dynamic equations of the coupled system. By mapping the expression of the nonlinear mode to the system response, the target frequency response curve for predicting the change of the response surface of the coupled system is obtained. The target frequency response curve at the lower limit of the target energy transfer threshold is defined as the curve at which the system just begins to show the suppression response; the target frequency response curve at the upper limit of the target energy transfer threshold is defined as the curve at which the peak value of the high-amplitude branch of the main system is equal to the plateau amplitude. The optimal target energy transfer threshold is obtained by solving for the lower and upper limits of the target energy transfer threshold. The expression for the dynamic equations of the coupled system is: In the formula, For the quality of the board, For generalized displacement, The damping value is determined by the damping ratio of the plate material. For the stiffness of the plate, The electromechanical coupling coefficient is... This is the equivalent capacitance. , The voltage and charge on the electrodes are the two components of the electrode. For vibration model, For the simple harmonic excitation of the board, Let Lagrange function be the system. This is the resistance value. It has a negative capacitance coefficient. The capacitance coefficient is nonlinear. , , and These are respectively equivalent to the mass, damping value, linear stiffness, and cubic stiffness of a mechanically grounded NES; The process of using the harmonic balance method to solve the forced vibration response of the system and obtain the response surface of the main system is as follows: Using the first harmonic balance method, based on the dynamic equations of the coupled system, an approximate periodic solution for the forced vibration of the coupled system is obtained. The analysis is restricted to 1:1 resonance, yielding: in, Let F(t), r(t) and Represented by the first harmonic term: Ignoring higher harmonic terms and performing a harmonic balance process, the response surface equation of the principal system is obtained as follows: In the formula, For the amplitude of the harmonic excitation, The displacement amplitude of the main system.

2. The method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap according to claim 1, characterized in that, The process of obtaining the nonlinear modes of the coupled system includes: Remove the external force and damping terms from the dynamic equations of the coupled system and substitute in the simple harmonic terms. and ,get and The calculation expression is obtained, and then the absolute value of the amplitude is taken to obtain the nonlinear mode of the coupled system; Received and The calculation expression is: 。 3. The method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap according to claim 2, characterized in that, The calculation expression for the lower limit of the target energy transfer threshold is as follows: In the formula, This represents the lower limit of the target energy transfer threshold. To By taking the derivative and setting it to zero, we obtain the frequency corresponding to the maximum point. , .

4. The method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap according to claim 2, characterized in that, In the calculation of the upper limit of the target energy transfer threshold, for The situation makes Using the Shengjin formula, the frequency corresponding to the point where the peak value of the high-amplitude branch of the main system equals the plateau amplitude is obtained as follows: in, Based on the frequency corresponding to the point where the peak value of the high-amplitude branch of the main system equals the amplitude of the plateau, the upper limit of the target energy transfer threshold on the system response surface is obtained, i.e.: In the formula, The upper limit of the target energy transfer threshold. = , = .

5. The method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap according to claim 2, characterized in that, In the calculation of the upper limit of the target energy transfer threshold, for Calculate the following situation: The nonlinear mode of the amplitude-frequency plane of the time-harmonic excitation is mapped onto the displacement plane of the system response, and the excitation value corresponding to the minimum point of the corresponding curve is taken as the upper limit of the excitation threshold for target energy transfer.

6. The method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap according to claim 1, characterized in that, The piezoelectric elements are respectively connected to both sides of the clamping plate. The piezoelectric elements are connected to a piezoelectric shunt circuit, which includes an inductor, a resistor, a negative capacitor, and a nonlinear capacitor connected in series.

7. The method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap according to claim 1, characterized in that, The platform amplitude is within a certain range of harmonic excitation amplitude. The response of the main system is a tilted platform, the resonance peak of the main system is effectively reduced, and the nonlinear energy trap plays an efficient vibration suppression role. When the harmonic excitation amplitude exceeds this range, the system response shows a new branch, and the branch peak is higher than the platform amplitude.

8. The method for obtaining the optimal target energy transfer threshold of a piezoelectric nonlinear energy trap according to claim 1, characterized in that, The method further includes controlling the elastic plate-piezoelectric nonlinear energy trap system based on the obtained optimal target energy transfer threshold.

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