A method for optimizing optimal target energy delivery characteristics for a non-linear energy sink system

Abstract

This invention relates to a method for optimizing the target energy transfer characteristics of a nonlinear energy trap system, specifically an elastic plate-piezoelectric nonlinear energy trap system. The method includes: establishing a two-degree-of-freedom model of the elastic plate-piezoelectric nonlinear energy trap system to obtain the dynamic equations of the coupled system; solving for the slow-invariant manifold of the coupled system using the complex variable-average method, and obtaining the parameterized design conditions for the coupled system to exhibit a forced vibration response; solving for the forced vibration response of the coupled system using the harmonic balance method to obtain the system response surface; and optimizing the target energy transfer characteristic threshold range of the system by adjusting the system mass ratio, negative linear stiffness, damping, and nonlinear stiffness within the parameterized design conditions, based on the changes in the system response surface. Compared with existing technologies, this invention provides systematic guidance for the parameter design of the elastic plate-piezoelectric nonlinear energy trap system, improving vibration suppression effects and broadening the target energy transfer characteristic threshold range.

Description

Technical Field

[0001] This invention relates to the field of nonlinear energy trap technology, and in particular to a method for optimizing the target energy transfer characteristics of a nonlinear energy trap system. Background Technology

[0002] Figure 1 The schematic diagram of an elastic plate-piezoelectric nonlinear energy trap system is shown. It consists of a clamping plate, piezoelectric sheets, and a piezoelectric shunt circuit containing an inductor, resistor, negative capacitor, and nonlinear capacitor. Two piezoelectric sheets are attached to opposite sides of the clamping plate. The positions of the piezoelectric sheets are determined by the mode shape of the fixed plate and are attached to the positions with larger mode shape displacements. A piezoelectric sheet is attached to the center of the plate, and the piezoelectric sheets are connected in parallel. The electrodes of the piezoelectric sheets are shunted from the circuit, where the shunt circuit acts as the NES (nonlinear energy trap). With the occurrence of the TET (transient energy transfer) phenomenon, vibrational energy is transferred from the mechanical system to the circuit, applying a simple harmonic excitation force F to the plate.

[0003] In this elastic plate-piezoelectric nonlinear energy trap system, existing methods do not provide clear guidance for optimizing the threshold range of the target energy transfer characteristics of the elastic plate-piezoelectric nonlinear energy trap system. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the existing technology by providing an optimal target energy transfer characteristic optimization method for a nonlinear energy trap system. This method provides systematic guidance for the parameter design of an elastic plate-piezoelectric nonlinear energy trap system, which can improve vibration suppression and broaden the threshold range of target energy transfer characteristics.

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

[0006] A method for optimizing the target energy transfer characteristics of a nonlinear energy trap system, wherein the nonlinear energy trap system is an elastic plate-piezoelectric nonlinear energy trap system, 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 slow invariant manifold of the coupled system is solved using the complex variable-average method, and the parameterized design conditions for the coupled system to exhibit a controlled response are obtained.

[0009] The forced vibration response of the coupled system is solved using the harmonic balance method, and the system response surface is obtained.

[0010] By adjusting the system mass ratio, negative linear stiffness, damping, and nonlinear stiffness within parametric design conditions, and based on the changes in the system response surface, the target energy transfer threshold range of the elastic plate-piezoelectric nonlinear energy trap system is optimized.

[0011] Furthermore, the process of optimizing the target energy transfer characteristic threshold range of the elastic plate-piezoelectric nonlinear energy trap system is as follows:

[0012] First, by adjusting the threshold range of the nonlinear stiffness target energy transfer characteristics and the excitation amplitude level, and then appropriately reducing the mass ratio, increasing the negative linear stiffness, or appropriately increasing the damping, the vibration suppression effect is optimized, thereby widening the threshold range of the target energy transfer characteristics.

[0013] Furthermore, by adjusting the threshold range of the nonlinear stiffness target energy transfer characteristics and the excitation amplitude level, the amplitude of the target energy transfer characteristic platform, the threshold, and the excitation amplitude level are reduced by increasing the nonlinear stiffness.

[0014] Furthermore, as the mass ratio decreases, the upper and lower limits of the target energy transfer characteristic threshold of the system gradually decrease, and the threshold range of the target energy transfer characteristic gradually narrows.

[0015] Furthermore, as the negative linear stiffness increases, the upper and lower limits of the target energy transfer characteristic threshold decrease accordingly, and the threshold range gradually widens.

[0016] Furthermore, as the damping increases, the upper and lower limits of the target energy transfer characteristic threshold also gradually increase, and the target energy transfer characteristic threshold range gradually widens.

[0017] Furthermore, the expression for the dynamic equations of the coupled system is as follows:

[0018]

[0019]

[0020] In the formula, m is the mass of the plate, r represents the generalized displacement, c is the damping value determined by the damping ratio of the plate material, k is the stiffness, 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, F is the simple harmonic excitation, φ(x, y) is the mode shape, L is the inductance, R is the resistance, and k is the resistance. 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.

[0021] Furthermore, the process of using the harmonic balance method to solve the forced vibration response of the coupled system and obtaining the system response surface is as follows:

[0022] 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:

[0023]

[0024]

[0025] in,

[0026] μ = LC p ω i 2 ,

[0027] λ na =RC p ω i 2 , κ=C p k lin ,

[0028] Let F(t), r(t) and q a 3 Represented by the first harmonic term:

[0029] F(t)=F 1c cos(ωt)+F 1s sin(ωt)

[0030] r(t) = r 1c cos(ωt)+r 1s sin(ωt)

[0031] q a (t)=q 1c cos(ωt)

[0032] Ignoring higher harmonic terms and performing a harmonic balance process, the response surface equation of the principal system is obtained as follows:

[0033]

[0034]

[0035]

[0036] ∈λ na ωq 1c +∈ω i 2 r 1s =0

[0037]

[0038]

[0039] In the formula, F1 is the amplitude of the simple harmonic excitation, and r1 is the amplitude of the displacement of the main system.

[0040] Furthermore, the computational expression for the slow-invariant manifold is:

[0041]

[0042] In the formula, Z represents the slow invariant manifold, κ represents the stiffness, and μ = LC. p ω i 2 , where is the mass ratio of the coupled system, L is the inductance value, and C is the mass ratio of the coupled system p This is the equivalent capacitance. m is the mass of the plate, ξ na For piezoelectric nonlinear energy trap damping and ω i ratio Z na It is the dimensionless energy of the nonlinear energy trap.

[0043] Furthermore, the parameterized design conditions for the occurrence of a modulated response in a coupled system, derived from the slow invariant manifold, are as follows:

[0044]

[0045] Compared with the prior art, the present invention has the following advantages:

[0046] (1) This invention first determines the range of parameter variations of mass ratio, negative linear stiffness, damping and nonlinear stiffness by analyzing the response of the main system, and clarifies their influence on the energy transfer characteristics of the target. By analyzing the slow invariant manifold of the coupled system, the parameterized design conditions for the coupled system to exhibit a strong damping response are obtained.

[0047] In summary, the mass ratio and negative linear stiffness exhibit the same influence, while the influence of nonlinear stiffness shows proportional characteristics. The general principle for parameter design is as follows: first, adjust the nonlinear stiffness to match the target energy transfer threshold range and excitation amplitude level; then, appropriately reduce the mass ratio, increase the negative linear stiffness, or appropriately increase the damping to optimize vibration suppression and broaden the target energy transfer characteristic threshold range.

[0048] This scheme provides systematic guidance for the parameter design of the elastic plate-piezoelectric nonlinear energy trap system, which can improve the vibration suppression effect and broaden the threshold range of target energy transfer characteristics. Attached Figure Description

[0049] Figure 1 This is a schematic diagram of a model of an elastic plate-piezoelectric nonlinear energy trap system provided in an embodiment of the present invention;

[0050] Figure 2 This is a flowchart illustrating a method for optimizing the target energy transfer characteristics of a nonlinear energy trap system according to an embodiment of the present invention.

[0051] Figure 3 This is a schematic diagram of a hopping mechanism for a slow, invariant manifold provided in an embodiment of the present invention;

[0052] Figure 4(a) is a schematic diagram of the evolution of the main system response when μ = 0.7 provided in an embodiment of the present invention;

[0053] Figure 4(b) is a numerical simulation comparison diagram of the response when μ = 0.7 and F1 = 3.5N provided in the embodiment of the present invention, with stable periodic solution "-" and unstable periodic solution "--".

[0054] Figure 4(c) is a schematic diagram of the 83.1Hz time-domain response when μ = 0.7 provided in an embodiment of the present invention;

[0055] Figure 4(d) is a schematic diagram of numerical comparison of slow invariant manifolds when μ = 0.7 provided in an embodiment of the present invention;

[0056] Figure 4(e) is a schematic diagram of the 83.1Hz time-domain response phase when μ = 0.7 provided in an embodiment of the present invention;

[0057] Figure 5(a) is a schematic diagram of the evolution of the main system response when μ=5 provided in an embodiment of the present invention;

[0058] Figure 5(b) is a numerical simulation comparison diagram of the response when μ=5 and F1=34N provided in the embodiment of the present invention, with stable periodic solution "-" and unstable periodic solution "--".

[0059] Figure 6(a) is a schematic diagram of the main system variation of the influence of mass ratio μ on nonlinear modes provided in an embodiment of the present invention;

[0060] Figure 6(b) is a schematic diagram of the nonlinear energy trap variation of the effect of mass ratio μ on nonlinear modes provided in an embodiment of the present invention;

[0061] Figure 7(a) is a schematic diagram of the displacement effect of a mass ratio μ on the target energy transfer characteristics provided in an embodiment of the present invention;

[0062] Figure 7(b) is a schematic diagram showing the effect of mass ratio μ on the amplitude of harmonic excitation of target energy transfer characteristics in an embodiment of the present invention.

[0063] Figure 8(a) is a schematic diagram of the main system variation of the influence of negative linear stiffness κ on nonlinear modes provided in an embodiment of the present invention;

[0064] Figure 8(b) is a schematic diagram of the nonlinear energy trap variation of the influence of negative linear stiffness κ on nonlinear modes provided in an embodiment of the present invention;

[0065] Figure 9(a) is a schematic diagram of the displacement effect of a negative linear stiffness κ on the energy transfer characteristics of a target provided in an embodiment of the present invention;

[0066] Figure 9(b) is a schematic diagram of the influence of a negative linear stiffness κ on the amplitude of harmonic excitation of the target energy transfer characteristics provided in an embodiment of the present invention.

[0067] Figure 10(a) shows a nonlinear energy trap damping ξ provided in an embodiment of the present invention. na A schematic diagram illustrating the effect of displacement on the target energy transfer characteristics;

[0068] Figure 10(b) shows a nonlinear energy trap damping ξ provided in an embodiment of the present invention. na A schematic diagram illustrating the influence of harmonic excitation amplitude on the target energy transfer characteristics;

[0069] Figure 11(a) is a schematic diagram of the first effect of nonlinear stiffness Ω3 on a slow invariant manifold provided in an embodiment of the present invention;

[0070] Figure 11(b) is a schematic diagram of the second effect of nonlinear stiffness Ω3 on a slow invariant manifold provided in an embodiment of the present invention;

[0071] Figure 12(a) is a schematic diagram of the main system variation of the influence of nonlinear stiffness Ω3 on nonlinear modes provided in an embodiment of the present invention;

[0072] Figure 12(b) is a schematic diagram of the nonlinear energy trap variation of the effect of nonlinear stiffness Ω3 on nonlinear modes provided in an embodiment of the present invention;

[0073] Figure 13(a) is a schematic diagram of the displacement effect of nonlinear stiffness Ω3 on the energy transfer characteristics of the target provided in an embodiment of the present invention;

[0074] Figure 13(b) is a schematic diagram showing the influence of a nonlinear stiffness Ω3 on the harmonic excitation amplitude of the target energy transfer characteristics provided in an embodiment of the present invention. Detailed Implementation

[0075] 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.

[0076] 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.

[0077] 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.

[0078] 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.

[0079] 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.

[0080] 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.

[0081] Example 1

[0082] This embodiment provides a method for optimizing the target energy transfer characteristics of a nonlinear energy trap system, such as... Figure 1 As shown, the nonlinear energy trap system is an elastic plate-piezoelectric nonlinear energy trap system. This system includes a clamping plate, piezoelectric sheets, and a piezoelectric shunt circuit. The piezoelectric sheets are 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, as shown below. Figure 2 As shown, the method includes the following steps:

[0083] 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;

[0084] S2: Solve the slow invariant manifold of the coupled system using the complex variable-average method, and obtain the parameterized design conditions for the coupled system to exhibit a controlled response;

[0085] S3: The forced vibration response of the coupled system is solved by the harmonic balance method, and the system response surface is obtained, which can be used to analyze the target energy transfer characteristics of the system.

[0086] S4: Perform system parameterization design and analyze the effects of system mass ratio, negative linear stiffness, damping and nonlinear stiffness on the system target energy characteristics;

[0087] S5: Obtain the general criteria for parameter design of the elastic plate-piezoelectric nonlinear energy trap system. By adjusting the system mass ratio, negative linear stiffness, damping and nonlinear stiffness within the parameterized design conditions, and based on the changes in the system response surface, optimize the target energy transfer characteristic threshold range of the elastic plate-piezoelectric nonlinear energy trap system.

[0088] The following is a detailed description.

[0089] I. Dynamic Model of the Elastic Plate-Piezoelectric Nonlinear Energy Trap System

[0090] Energy-based modeling, based on Hamilton's principle, allows the system's governing equations to be expressed as:

[0091]

[0092] 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 nc It is a futile effort that is not conservative.

[0093] Assume the deflection w of the plate is the product of the mode shape φ(x, y) and the general coordinate r(t):

[0094] w(x,y,t)=φ(x,y)r(t) (2)

[0095] Where φ represents the mode shape and r represents the generalized displacement.

[0096] 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:

[0097]

[0098]

[0099]

[0100] W nc =r(t)φ(x) f y f )F(t)-V p，1 (t)q1-V p，2 (t)q2 (6)

[0101] 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 S represents the strain and stress vectors 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 Let q be the voltage and charge on the electrode, respectively. Substituting equations (3)-(4) into equation (1) and solving the equation, we get:

[0102]

[0103] 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.

[0104] 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:

[0105]

[0106] 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:

[0107]

[0108] Let q a =q / θ, multiplying both sides of the second equation (9) by θ, we get:

[0109]

[0110] Equation (8) is structurally equivalent to a mechanically grounded NES system. θ 2 L, θ 2 R, θ 2 k linand θ 4 k3 is equivalent to the mass, damping value, linear stiffness, and cubic stiffness of the grounded NES, respectively.

[0111] II. Forced Vibration Response

[0112] 2.1 Solving Slowly Invariant Manifolds Using the Complex Variable-Average Method

[0113] Applying a simple harmonic excitation F(t) to the system, the simple harmonic excitation term is written as ∈F0cos(ωt)=φ(x F y F F(t) / m, equation (10) is transformed into:

[0114]

[0115] in,

[0116] μ = LC p ω i 2 ,

[0117] λ na =RC p ω i 2 , κ=C p k lin ,

[0118] The slow-invariant manifold of the coupled system is analyzed using the complex variable averaging method, with derivation omitted. Assume the oscillation frequency of the coupled system is the same as the excitation frequency, and let:

[0119] ω=ω i +∈σ (12)

[0120] Frequency modulation is achieved through σ, and the resulting equation after averaging is:

[0121]

[0122] in,

[0123]

[0124] Multiscale analysis of equation (13) yields:

[0125]

[0126] The dynamic equations of the system on a slow time scale are obtained as follows:

[0127]

[0128] We obtain the slow-invariant manifold of the system, i.e.:

[0129]

[0130] The system's slow invariant manifold is as follows Figure 3 As shown.

[0131] Taking the derivative of (17) such that it is zero allows us to find the extreme points on the slow-invariant manifold:

[0132]

[0133] Equation (18) is about Z na The condition for a quadratic equation in one variable to have two real roots is:

[0134]

[0135] Z and Z na These are dimensionless energies, and their values ​​are all positive. Therefore, we can conclude that:

[0136]

[0137] Based on equations (19) and (20), the existence conditions of the two extreme points can be obtained as follows:

[0138]

[0139] 2.2 Solving for plateau values ​​and threshold intervals using the harmonic balance method

[0140] Let F(t), r(t) and q a (t) can be represented by the following first harmonic term:

[0141]

[0142] Substituting equation (22) into equation (11), ignoring higher harmonic terms, and performing a harmonic balance process, we get:

[0143]

[0144] Equation (23) is an 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 1c Using 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:

[0145]

[0146] By utilizing nonlinear modes, the amplitude of the system response plateau and the upper limit of the excitation threshold interval F under different parameter conditions can be obtained. e Lower limit F b wait.

[0147] III. Analysis of the Influence of Parameters

[0148] 3.1 Mass ratio of coupled systems

[0149] According to the necessary condition for a coupled system to exhibit a modulated response, μ-κ-1 > 0. When the mass ratio is small, μ = 0.7, and the main system response, as shown in Figures 4(a)-(e), is a weakly modulated response. When the mass ratio is large, μ = 5, and the main system response, as shown in Figures 5(a) and 5(b), exhibits a periodic response. Therefore, when μ is too large or too small, the nonlinear energy trap cannot fully utilize its efficient target energy transfer characteristics. To ensure efficient target energy transfer characteristics, the range of the mass ratio μ of the coupled system is set to 1–4.

[0150] The influence of the mass ratio μ of the coupled system on the nonlinear modes of the system is studied, as shown in Figures 6(a) and 6(b). With the increase of the mass ratio μ, the amplitude of the "plateau" gradually increases, and the "plateau" gradually shifts to lower frequencies. For the nonlinear energy trap, with the increase of the mass ratio μ, the nonlinear mode shifts upward, indicating that the displacement amplitude of the multi-period solution region and the high-amplitude branch of the nonlinear energy trap gradually increases.

[0151] The effects of the mass ratio μ on the upper and lower thresholds of the target energy transfer characteristics, as well as its influence on the amplitude of the main system's "plateau" and the peak value at the moment the high-amplitude branch first appears, are analyzed, as shown in Figures 7(a) and 7(b). It is found that the amplitude of the system response "plateau" and the peak value at the moment the high-amplitude branch first appears are positively correlated with the mass ratio μ. Furthermore, as the mass ratio μ increases, the upper and lower thresholds of the system's target energy transfer characteristics gradually increase, while the threshold range of the target energy transfer characteristics gradually narrows.

[0152] 3.2 Negative linear stiffness

[0153] First, the range of negative linear stiffness κ is defined as -0.9 to -0.1. The effect of negative linear stiffness κ on nonlinear modes is shown in Figures 8(a) and 8(b). The effect of negative linear stiffness κ on nonlinear modes is similar to that of mass ratio μ on nonlinear modes. As the negative linear stiffness κ decreases, the amplitude of the "plateau" gradually increases, and the "plateau" gradually shifts to lower frequencies.

[0154] The effects of negative linear stiffness κ on the upper and lower limits of the target energy transfer characteristic threshold, the amplitude of the "plateau," and the peak value at the initial appearance of the high-amplitude branch are considered, as shown in Figures 9(a) and 9(b). Increasing the negative linear stiffness κ reduces both the amplitude of the "plateau" and the peak value at the initial appearance of the high-amplitude branch; under this parameter condition, the high-amplitude branch always appears below the plateau. The upper and lower limits of the target energy transfer characteristic threshold also decrease with increasing negative linear stiffness κ, and the threshold range gradually widens.

[0155] 3.3 Damping

[0156] According to the necessary conditions for the occurrence of a modulated response in a coupled system The nonlinear energy trap damping ξ can be preliminarily determined. na The range of μ-κ-1 is used to observe the nonlinear energy trap damping ξ by taking smaller and larger values ​​of μ-κ-1 respectively. na The evolution and time-domain performance of the main system response near the upper and lower limits. It was found that when the nonlinear energy trap damping ξ... na When the target energy transfer characteristic threshold is too large, the coupled system mainly exhibits a weakly modulated response within this threshold range, resulting in low efficiency of the target energy transfer characteristic. Furthermore, the nonlinear modes cannot accurately predict the evolution of the main system response at this point. This is detrimental to achieving good vibration suppression and also hinders response prediction and analysis.

[0157] The analysis of nonlinear modes neglects damping, therefore the nonlinear energy trap damping ξ na No effect on nonlinear modes. A nonlinear energy trap damping ξ is set. na The variation range is 0.05 to 0.3, ξ na The influence of the peak value at the moment the high-amplitude branch first appears on the upper and lower limits of the target energy transfer characteristic threshold for the "plateau" amplitude is shown in Figures 10(a) and 10(b). Where r 1b The true "platform" amplitude of the main system response, r b Similar to the previous analysis, this represents the nonlinear mode S11-maxima, i.e., the predicted "plateau" amplitude. The results show that the actual "plateau" amplitude does indeed increase with the nonlinear energy trap damping ξ. na The amplitude increases with the increase of ξ, and gradually exceeds the predicted "plateau" amplitude. The peak value at the beginning of the high-amplitude branch also gradually increases. (b) indicates that as the nonlinear energy trap damping ξ increases, the amplitude increases with the increase of ξ. na As the value increases, the upper and lower limits of the target energy transfer characteristic threshold also gradually increase, and the target energy transfer characteristic threshold range becomes wider.

[0158] 3.4 Nonlinear Stiffness

[0159] When analyzing a slow, invariant manifold, changing the nonlinear stiffness Ω3 does not alter the shape of the manifold; it only causes it to scale up or down proportionally to the change in nonlinear stiffness Ω3. Therefore, it does not change the response characteristics of the coupled system, as shown in Figures 11(a) and 11(b), Z = R 2 Z na =R na 2The change in the maximum value of the slow invariant manifold can qualitatively describe the change in the amplitude of the "plateau". Changing the nonlinear stiffness Ω3 will first change the amplitude of the "plateau", and the initial excitation energy on which the target energy transfer characteristics depend will also increase or decrease, thus changing the threshold of the target energy transfer characteristics.

[0160] The influence of nonlinear stiffness Ω3 on nonlinear modes is analyzed. According to the nonlinear mode equation, similar to the slow invariant manifold, the nonlinear modes are amplified or reduced proportionally to the change of nonlinear stiffness Ω3. Taking the square root of the change of nonlinear stiffness Ω3 gives the change of nonlinear modes. Figures 12(a) and 12(b) verify the above analysis.

[0161] The effects of a nonlinear stiffness Ω3 varying within the range of 0.2 to 2 on the "plateau" amplitude, the peak value at the initial appearance of the high-amplitude branch, and the upper and lower limits of the target energy transfer characteristic threshold are analyzed, as shown in Figures 13(a) and 13(b). The results show that within the analyzed range, the "plateau" amplitude, the peak value at the initial appearance of the high-amplitude branch, and the upper and lower limits of the target energy transfer characteristic threshold all decrease proportionally to the square root of the increase in nonlinear stiffness Ω3, with the high-amplitude branch always appearing below the "plateau." The target energy transfer characteristic threshold range gradually narrows as the nonlinear stiffness Ω3 increases. Therefore, increasing the nonlinear stiffness can reduce the plateau amplitude, lower the threshold, and reduce the excitation amplitude level of the target energy transfer characteristic.

[0162] Therefore, by analyzing the response of the main system, the range of variation of parameters such as mass ratio, negative linear stiffness, damping, and nonlinear stiffness was determined, clarifying their influence on the target energy transfer characteristics. The mass ratio and negative linear stiffness exhibit the same influence law, while the influence of nonlinear stiffness shows proportional characteristics. The general principle for parameter design is derived: first, adjust the nonlinear stiffness to match the target energy transfer characteristic threshold range and excitation amplitude level; then, appropriately reduce the mass ratio, increase the negative linear stiffness, or appropriately increase the damping to optimize the vibration suppression effect and broaden the target energy transfer characteristic threshold range.

[0163] 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 optimizing the target energy transfer characteristics of a nonlinear energy trap system, wherein the nonlinear energy trap system is an elastic plate-piezoelectric nonlinear energy trap system, 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 slow invariant manifold of the coupled system is solved using the complex variable-average method, and the parameterized design conditions for the coupled system to exhibit a controlled response are obtained. The forced vibration response of the coupled system is solved using the harmonic balance method, and the system response surface is obtained. By adjusting the system mass ratio, negative linear stiffness, damping, and nonlinear stiffness within parametric design conditions, and based on the changes in the system response surface, the threshold range of the target energy transfer characteristics of the elastic plate-piezoelectric nonlinear energy trap system is optimized. The expression for the dynamic equations of the coupled system is: In the formula, For the quality of the board, Represents generalized displacement. The damping value is determined by the damping ratio of the plate material. For stiffness, 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 simple harmonic excitation, For vibration modes, This is the inductance value. 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 the grounded NES; The specific process of solving the forced vibration response of a coupled system using the harmonic balance method to obtain the system response surface 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, Displacement of the main system The amplitude of the harmonic excitation.

2. The method for optimizing the target energy transfer characteristics of a nonlinear energy trap system according to claim 1, characterized in that, The specific process for optimizing the target energy transfer characteristic threshold range of the elastic plate-piezoelectric nonlinear energy trap system is as follows: First, by adjusting the threshold range of the nonlinear stiffness target energy transfer characteristics and the excitation amplitude level, and then appropriately reducing the mass ratio, increasing the negative linear stiffness, or appropriately increasing the damping, the vibration suppression effect is optimized, thereby widening the threshold range of the target energy transfer characteristics.

3. The method for optimizing the target energy transfer characteristics of a nonlinear energy trap system according to claim 2, characterized in that, By adjusting the threshold range of the nonlinear stiffness target energy transfer characteristics and the excitation amplitude level, the amplitude of the target energy transfer characteristic platform, the threshold, and the excitation amplitude level are reduced by increasing the nonlinear stiffness.

4. The method for optimizing the target energy transfer characteristics of a nonlinear energy trap system according to claim 2, characterized in that, As the mass ratio decreases, the upper and lower limits of the target energy transfer characteristic threshold of the system gradually decrease, and the threshold range of the target energy transfer characteristic gradually narrows.

5. The method for optimizing the optimal target energy transfer characteristics of a nonlinear energy trap system according to claim 2, characterized in that, As the negative linear stiffness increases, the upper and lower limits of the target energy transfer characteristic threshold decrease accordingly, and the threshold range gradually widens.

6. The method for optimizing the target energy transfer characteristics of a nonlinear energy trap system according to claim 2, characterized in that, As damping increases, the upper and lower limits of the target energy transfer threshold also gradually increase, and the threshold range of the target energy transfer characteristics gradually widens.

7. The method for optimizing the target energy transfer characteristics of a nonlinear energy trap system according to claim 1, characterized in that, The computational expression for the slow invariant manifold is: In the formula, It is a slow-invariant manifold. For stiffness, , which is the mass ratio of the coupled system. This is the inductance value. This is the equivalent capacitance. , For the quality of the board, For piezoelectric nonlinear energy trap damping and ratio , It is the dimensionless energy of the nonlinear energy trap.

8. The method for optimizing the optimal target energy transfer characteristics of a nonlinear energy trap system according to claim 7, characterized in that, The parameterized design conditions for the occurrence of a modulated response in a coupled system, derived from the slow invariant manifold, are as follows: 。

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