Calculation Method for Equivalent Damping and Equivalent Natural Frequency of Internal Resonance Piezoelectric Energy Harvesting System

Abstract

The present invention discloses a calculation method for equivalent resistance damping and equivalent natural frequency of an internal resonance piezoelectric energy harvesting system. An L-shaped piezoelectric vibration energy harvesting system and its corresponding coordinate system are built. The L-shaped piezoelectric vibration energy harvesting system includes an L-shaped vibration energy harvester and a piezoelectric energy harvesting circuit provided with a load resistor R. A piezoelectric energy harvesting equivalent circuit is constructed according to the piezoelectric energy harvesting circuit; all parameters in the L-shaped piezoelectric vibration energy harvesting system are obtained, and through adjustment, the ratio of the first-order and second-order natural frequencies of the system is made equal to 1:2; in combination with the coordinate system, a force-electric coupling control equation of the two-degree-of-freedom L-shaped piezoelectric vibration energy harvesting system is constructed; the multi-scale method is used to solve the first-order and second-order equivalent damping ratios and the first-order and second-order equivalent natural frequencies in the force-electric coupling control equation. Effect: Quantitatively measure the influence degree of the load resistor on the natural frequency and damping ratio in the nonlinear internal resonance theoretical model.

Description

Technical Field

[0001] The present invention belongs to the technical field of vibration energy harvesting, and particularly relates to a calculation method for equivalent resistance damping and equivalent natural frequency of an internal resonance piezoelectric energy harvesting system. Background Art

[0002] For an L-shaped piezoelectric vibration energy harvester, the ratio of the first and second order frequencies of its structure can be approximately designed to satisfy the relationship of 1:2 by designing the geometric dimensions, so as to achieve the 1:2 internal resonance effect under the action of an external excitation to broaden the energy capture frequency band of the structure.

[0003] Related research shows that under the force-electric coupling of a piezoelectric vibration energy harvesting system, the load resistance R in the external circuit will affect the natural frequency and modal damping ratio of the structure, and thus affect the output response of the system. See the literature: (1) Tan T, Yan Z, Hajj M R. Electromechanical decoupled model for cantilever-beam piezoelectric energy harvesters. Appl Phys Lett 2016; 109: 101908. (2) Nie X, Tan T, Yan Z, Yan Z, Hajj M R. Broadband and high-efficient L-shaped piezoelectric energy harvester based on internal resonance. Int J Mech Sci 2019; 159: 287305.

[0004] However, it is difficult to quantitatively measure the influence degree of the load resistance on the natural frequency and damping ratio in the nonlinear internal resonance theoretical model. In the prior art, this technical problem has not been specifically solved, so it is necessary to propose an algorithm to calculate the influence degree of the load resistance on the natural frequency and damping ratio. Summary of the Invention

[0005] Aiming at the above deficiencies of the prior art, the present invention proposes a calculation method for equivalent resistance damping and equivalent natural frequency of an internal resonance piezoelectric energy harvesting system. Based on the internal resonance principle, for the piezoelectric energy harvesting system in an L-shaped piezoelectric vibration energy harvesting system, a targeted design is carried out to measure the influence degree of the load resistance on the natural frequency and damping ratio.

[0006] In order to solve the above technical problems, the present invention adopts the following technical solutions:

[0007] A calculation method for equivalent resistance damping and equivalent natural frequency of an internal resonance piezoelectric energy harvesting system, characterized in that:

[0008] S1: Build an L-shaped piezoelectric vibration energy harvesting system and its corresponding coordinate system. The L-shaped piezoelectric vibration energy harvesting system includes at least an L-shaped vibration energy harvester and a piezoelectric energy harvesting circuit arranged on the L-shaped vibration energy harvester. An external load resistor R is provided in the piezoelectric energy harvesting circuit.

[0009] S2: Construct a piezoelectric energy harvesting equivalent circuit according to the piezoelectric energy harvesting circuit.

[0010] S3: Obtain all the parameters in the L-shaped piezoelectric vibration energy harvesting system, and adjust the lumped mass block in the L-shaped piezoelectric vibration energy harvesting system so that the ratio of the first-order and second-order natural frequencies of the L-shaped piezoelectric vibration energy harvesting system is equal to 1:2.

[0011] S4: Combine the coordinate system of the L-shaped piezoelectric vibration energy harvesting system to construct a force-electric coupling control equation for the two-degree-of-freedom L-shaped piezoelectric vibration energy harvesting system.

[0012] S5: Use the multi-scale method to solve the first- and second-order equivalent damping ratios and the first- and second-order equivalent natural frequencies in the force-electric coupling control equation in step S4.

[0013] Furthermore, the L-shaped vibration energy harvester includes a substrate, a horizontal beam, a vertical beam, a first lumped mass block M1, and a second lumped mass block M2.

[0014] The substrate is connected to one end of the horizontal beam, and the first lumped mass block is connected to the other end of the horizontal beam; the vertical beam is arranged perpendicular to the horizontal beam, and one end of the vertical beam is connected to the first lumped mass block, and the second lumped mass block is arranged on the vertical beam.

[0015] Two piezoelectric wafers are covered along the extending direction of the horizontal beam, and the two piezoelectric wafers are connected to the load resistor R.

[0016] A further technical solution is that the force-electric coupling control equation for the two-degree-of-freedom L-shaped piezoelectric vibration energy harvesting system is:

[0017]

[0018] where q1 and q2 respectively represent the first-order and second-order modal coordinates of the L-shaped piezoelectric vibration energy harvesting system.

[0019] μ1 and μ2 represent the first and second damping coefficients of the L-shaped piezoelectric vibration energy harvesting system, and their expressions are: μ1 = ξ1ω1 and μ2 = ξ2ω2; ξ1 represents the first-order modal damping ratio of the L-shaped piezoelectric vibration energy harvesting system; ξ2 represents the second-order modal damping ratio of the L-shaped piezoelectric vibration energy harvesting system; ω1 represents the first natural frequency of the L-shaped piezoelectric vibration energy harvesting system; ω2 represents the second natural frequency of the L-shaped piezoelectric vibration energy harvesting system;

[0020] m 1...27 and n 1...27 represent the coefficients of the force-electric coupling control equation;

[0021] represents the acceleration excitation; ω b is the excitation frequency; F is the acceleration excitation amplitude;

[0022] V represents the output voltage of the L-shaped piezoelectric vibration energy harvesting system;

[0023] where C p represents the capacitance, and R represents the load resistance;

[0024] where, represents the piezoelectric coupling coefficient; φ represents the vibration mode of the L-shaped vibration energy harvester; φ 11 ′(l1) represents the derivative of the first-order vibration mode of the horizontal beam; φ 12 ′(l1) represents the derivative of the second-order vibration mode of the horizontal beam; l1 represents the horizontal beam;

[0025] A further technical solution is: in step S5, the specific steps for solving using the first and second-order equivalent damping ratios and the first and second-order equivalent natural frequencies in the multi-scale force-electric coupling control equation are as follows:

[0026] S51: Define the perturbation small parameter ε as: εμ′ j = μ j , εθ′ j1 = θ j1 εm′ i = m i , εn′ i = n i ; where j = 1, 2; i = 1···27;

[0027] Introduce the time variable T representing different scales n :

[0028] T n = ε n t, n = 0, 1 (4)

[0029] The derivative with respect to time t can be expressed as:

[0030]

[0031] where

[0032] O(ε 2 ) represents higher-order terms of the second order and above, which are not considered in the subsequent derivation. Here, only the first-order approximation is considered. Then, the solutions of the force-electric coupling control equations (1)-(3) can be written as functions of time variables at different scales:

[0033]

[0034] S52: Substitute formulas (5) and (6) into the force-electric coupling control equations (1)-(3). By equating the coefficients of the zero-th power of the perturbation small parameter ε, the control equations are obtained:

[0035]

[0036] S53: The solutions of the first two differential equations of equation (7) are:

[0037]

[0038] where A1 and A2 are complex functions to be determined, and are the conjugate complex numbers of A1 and A2 respectively,

[0039] Substitute equation (8) into the third equation of equation (7), and the solution of the third differential equation of equation (7) is obtained as:

[0040]

[0041] where cc represents the conjugate complex number of each term on its left;

[0042] S54: Neglect the nonlinear terms and external excitation terms in the force-electric coupling control equations (1)-(3) to obtain the linearized control equation set of the L-type piezoelectric vibration energy harvesting system:

[0043]

[0044] Assume q1≈q 10 , q2≈q 20 , V≈q 30 , then equation (9) is expressed as:

[0045]

[0046] Substitute Equation (11) into the first two equalities of Equation (10) respectively, and the decoupled linearized control equations are obtained:

[0047]

[0048] Among them, the solutions of the first and second-order equivalent natural frequencies of the L-shaped piezoelectric vibration energy harvesting system are respectively expressed as:

[0049]

[0050] The first and second-order equivalent damping ratios of the L-shaped piezoelectric vibration energy harvesting system are respectively expressed as:

[0051]

[0052] Advantages of the present invention:

[0053] The present invention studies the influence of the load resistance on the output performance of the L-shaped piezoelectric vibration energy harvesting system with internal resonance pressure. The force-electric coupling control equation of the L-shaped piezoelectric vibration harvester based on 1:2 internal resonance is given, and the approximate analytical solution of the output response of the harvester is derived by using the multi-scale method.

[0054] The eigenvalues of the Jacobian matrix of the two main resonance modulation equations are used to determine the equilibrium stability of the output response. The approximate analytical solution is verified by the numerical solution, and the verification results are in good agreement. The load resistance has a strong influence on the modal damping ratio of the L-shaped piezoelectric vibration energy harvesting system and can significantly affect the harvesting power, while its influence on the natural frequency of the system can be ignored. Description of the drawings

[0055] Figure 1 It is a schematic diagram of the L-shaped piezoelectric vibration energy harvester model;

[0056] Figure 2 It is a schematic diagram of the coordinate system of the L-shaped piezoelectric vibration energy harvesting system;

[0057] Figure 3 It is a calculation flow chart of the present invention;

[0058] Figure 4 It is a verification schematic diagram of the influence of the load resistance on the first and second-order equivalent natural frequencies;

[0059] Figure 5 It is a verification schematic diagram of the influence of the load resistance on the first and second-order equivalent damping ratios. Detailed implementation manners

[0060] The present invention will be further described in detail below with reference to the accompanying drawings.

[0061] A calculation method for equivalent damping and equivalent natural frequency of an internal resonance piezoelectric energy harvesting system, seeFigure 3 , the specific steps are as follows:

[0062] S1: Build an L-shaped piezoelectric vibration energy harvesting system and its corresponding coordinate system. For details of the coordinate system, see Figure 2 . The L-shaped piezoelectric vibration energy harvesting system includes at least an L-shaped vibration energy harvester and a piezoelectric energy harvesting circuit arranged on the L-shaped vibration energy harvester. An external load resistor R is provided in the piezoelectric energy harvesting circuit;

[0063] See Figure 1 . The L-shaped vibration energy harvester includes a substrate, a horizontal beam, a vertical beam, a first lumped mass M1, and a second lumped mass M2;

[0064] The substrate is connected to one end of the horizontal beam, and the first lumped mass is connected to the other end of the horizontal beam; the vertical beam is arranged perpendicular to the horizontal beam, and one end of the vertical beam is connected to the first lumped mass, and the second lumped mass is arranged on the vertical beam;

[0065] Cover the last two piezoelectric wafers along the extension direction of the horizontal beam, and the two piezoelectric wafers are connected to the load resistor R.

[0066] In this embodiment, the physical and geometric characteristic data of the L-shaped vibration energy harvester structure are shown in Table 1:

[0067] Table 1 Physical and Geometric Characteristic Data Table of L-shaped Vibration Energy Harvester Structure

[0068]

[0069] S2: Construct a piezoelectric energy harvesting equivalent circuit according to the piezoelectric energy harvesting circuit; for details of the specific circuit, see Figure 1 The right circuit diagram.

[0070] S3: Obtain all parameters in the L-shaped piezoelectric vibration energy harvesting system, and adjust the lumped mass in the L-shaped piezoelectric vibration energy harvesting system so that the ratio of the first-order and second-order natural frequencies of the L-shaped piezoelectric vibration energy harvesting system is equal to 1:2;

[0071] S4: Combine the coordinate system of the L-shaped piezoelectric vibration energy harvesting system to construct a force-electric coupling control equation for the two-degree-of-freedom L-shaped piezoelectric vibration energy harvesting system;

[0072] In this embodiment, the force-electric coupling control equation for the two-degree-of-freedom L-shaped piezoelectric vibration energy harvesting system is:

[0073]

[0074] where q1 and q2 represent the first and second modal coordinates of the L-shaped piezoelectric vibration energy harvesting system, respectively;

[0075] μ1 and μ2 represent the first and second damping coefficients of the L-shaped piezoelectric vibration energy harvesting system, and their expressions are: μ1 = ξ1ω1 and μ2 = ξ2ω2; ξ1 represents the first modal damping ratio of the L-shaped piezoelectric vibration energy harvesting system; ξ2 represents the second modal damping ratio of the L-shaped piezoelectric vibration energy harvesting system; ω1 represents the first natural frequency of the L-shaped piezoelectric vibration energy harvesting system; ω2 represents the second natural frequency of the L-shaped piezoelectric vibration energy harvesting system;

[0076] m 1...27 and n 1...27 represent the coefficients of the force-electric coupling control equation;

[0077] where m 1...25 and n 1...25 and the specific values of other parameters can be found in the existing literature: Xiaochun Nie, Ting Tan, Zhimiao Yan, Zhitao Yan, Muhammad R Hajj. Broadband and high-efficient L-shaped piezoelectric energy harvester based on internal resonance[J]. International Journal of Mechanical Sciences 159(2019)287–305.

[0078] where the values of the coefficients of the nonlinear terms in the control equations (1)-(3) refer to pages 301–305 of the above-mentioned literature.

[0079]

[0080]

[0081] represents the acceleration excitation; ω b is the excitation frequency; F is the amplitude of the acceleration excitation;

[0082] V represents the output voltage of the L-shaped piezoelectric vibration energy harvesting system;

[0083] where C p represents the capacitance, and R represents the load resistance;

[0084] where, represents the piezoelectric coupling coefficient; φ represents the vibration mode of the L-shaped vibration energy harvester; φ 11 ′(l1) represents the derivative of the first-order vibration mode of the horizontal beam; φ 12 ′(l1) represents the derivative of the second-order vibration mode of the horizontal beam; l1 represents the horizontal beam.

[0085] S5: Use the multi-scale method to solve the first- and second-order equivalent damping ratios and the first- and second-order equivalent natural frequencies in the force-electric coupling control equation of step S4.

[0086] In step S5, the specific steps for using the multi-scale method to solve the first- and second-order equivalent damping ratios and the first- and second-order equivalent natural frequencies in the force-electric coupling control equation are as follows:

[0087] S51: Define the perturbation small parameter ε as: εμ′ j = μ j ,εθ′ j1 = θ j1 εm i ′ = m i ,εn i ′ = n i ; where j = 1, 2; i = 1···27;

[0088] Introduce the time variable T representing different scales n :

[0089] T n = ε n t, n = 0, 1 (4)

[0090] The derivative with respect to time t can be expressed as:

[0091]

[0092] Among them,

[0093] Combined with the solutions of the force-electric coupling control equations (1)-(3), it can be written as a function of different scale time variables:

[0094]

[0095] S52: Substitute formulas (5) and (6) into the force-electric coupling control equations (1)-(3), and from the equality of the zero-order power coefficients of the perturbation small parameter ε, obtain the control equations:

[0096]

[0097] S53: The solutions of the first two differential equations of equation (7) are:

[0098]

[0099] Among them, A1 and A2 are complex functions to be determined, and are the conjugate complex numbers of A1 and A2 respectively,

[0100] Substituting Equation (8) into the third equation of Equation (7), the solution of the third differential equation of Equation (7) is obtained as:

[0101]

[0102] where cc represents the conjugate complex number of each term on its left;

[0103] S54: Now, ignore the non-linear terms and external excitation terms in the force-electric coupling control equations (1)-(3) to obtain the linearized control equation set of the L-type piezoelectric vibration energy harvesting system:

[0104]

[0105] Assume q1≈q 10 , q2≈q 20 , V≈q 30 , then Equation (9) is expressed as:

[0106]

[0107] Substitute Equation (11) into the first two equations of Equation (10) respectively, then the decoupled linearized control equations are obtained:

[0108]

[0109] Among them, the solutions of the first and second order equivalent natural frequencies of the L-type piezoelectric vibration energy harvesting system are respectively expressed as:

[0110]

[0111] The first and second order equivalent damping ratios of the L-type piezoelectric vibration energy harvesting system are respectively expressed as:

[0112]

[0113] The content of equivalent natural frequency verification is:

[0114] To check the correctness of the approximate solution of the multi-scale method, it is necessary to compare the result of the approximate solution of the multi-scale method with the numerical solution result. The comparison between the approximate solution and the numerical solution of the equivalent natural frequency is shown in detail in Figure 4. In the figure, "MMS" represents the equivalent natural frequency calculated by approximately solving equation (14) using the multi-scale method. "NS" represents the numerical solution of the equivalent natural frequency obtained by solving the governing equations (1)-(3) using the Runge-Kutta method. As can be seen from Figure 4 , the approximate solution obtained by the multi-scale method is in good agreement with the numerical solution within the entire load resistance range, indicating that the calculation method of the approximate solution of the equivalent natural frequency is feasible.

[0115] The content of verifying the equivalent damping ratio is as follows:

[0116] Figure 5 For the comparison between the approximate solution and the numerical value of the equivalent damping ratio, it can be seen from the figure that the variation trend of the approximate solution of the equivalent damping ratio with respect to the resistance is consistent with the variation trend of the corresponding damping ratio of the numerical solution with respect to the resistance, and the two are in good agreement, indicating that the calculation method of the approximate solution of the equivalent damping ratio is feasible.

[0117] The above is only the preferred embodiment of the present invention. It should be noted that for those skilled in the art, without departing from the technical solution of the present invention, several modified and improved technical solutions should also be regarded as falling within the scope protected by the claims of this patent.

Claims

1. A calculation method for equivalent resistance damping and equivalent natural frequency of an internal resonance piezoelectric energy harvesting system, characterized in that The specific steps are as follows: S1: Build an L-shaped piezoelectric vibration energy harvesting system and its corresponding coordinate system. The L-shaped piezoelectric vibration energy harvesting system includes at least an L-shaped vibration energy harvester and a piezoelectric energy harvesting circuit arranged on the L-shaped vibration energy harvester. An external load resistor R is provided in the piezoelectric energy harvesting circuit. S2: Construct a piezoelectric energy harvesting equivalent circuit according to the piezoelectric energy harvesting circuit. S3: Obtain all parameters in the L-shaped piezoelectric vibration energy harvesting system, and adjust the lumped mass block in the L-shaped piezoelectric vibration energy harvesting system so that the ratio of the first-order and second-order natural frequencies of the L-shaped piezoelectric vibration energy harvesting system is equal to 1:

2. S4: Combine the coordinate system of the L-shaped piezoelectric vibration energy harvesting system to construct a force-electric coupling control equation for the two-degree-of-freedom L-shaped piezoelectric vibration energy harvesting system. S5: Use the multi-scale method to solve the first- and second-order equivalent damping ratios and the first- and second-order equivalent natural frequencies in the force-electric coupling control equation in step S4. The L-shaped vibration energy harvester includes a base, a horizontal beam, a vertical beam, a first lumped mass block M1, and a second lumped mass block M2. The base is connected to one end of the horizontal beam, and the first lumped mass block is connected to the other end of the horizontal beam; the vertical beam is arranged perpendicular to the horizontal beam, and one end of the vertical beam is connected to the first lumped mass block, and the second lumped mass block is arranged on the vertical beam. Cover the last two piezoelectric sheets along the extension direction of the horizontal beam, and connect the two piezoelectric sheets to the load resistor R. The force-electric coupling control equation for the two-degree-of-freedom L-shaped piezoelectric vibration energy harvesting system is: Among them, q1 and q2 respectively represent the first-order and second-order modal coordinates of the L-shaped piezoelectric vibration energy harvesting system. μ1 and μ2 respectively represent the first- and second-order damping coefficients of the L-shaped piezoelectric vibration energy harvesting system, and their expressions are: μ1 = ξ1ω1 and μ2 = ξ2ω2; ξ1 represents the first-order modal damping ratio of the L-shaped piezoelectric vibration energy harvesting system; ξ2 represents the second-order modal damping ratio of the L-shaped piezoelectric vibration energy harvesting system; ω1 represents the first-order natural frequency of the L-shaped piezoelectric vibration energy harvesting system; ω2 represents the second-order natural frequency of the L-shaped piezoelectric vibration energy harvesting system. m 1...27 and n 1...27 represent the coefficients of the force-electricity coupling control equation; represents the acceleration excitation; ω b is the excitation frequency; F is the amplitude of the acceleration excitation; V represents the output voltage of the L-shaped piezoelectric vibration energy harvesting system. Among them, C p represents a capacitor, and R represents a load resistor; Among them, represents the piezoelectric coupling coefficient; φ represents the vibration mode of the L-shaped vibration energy harvester; φ 11 ′(l1) represents the derivative of the first-order vibration mode of the horizontal beam; φ 12 ′(l1) represents the derivative of the second-order vibration mode of the horizontal beam.

2. The calculation method of the equivalent damping and equivalent natural frequency of the internal resonance piezoelectric energy harvesting system according to claim 1, characterized in that: In step S5, the specific steps for using the multi-scale method to solve the first- and second-order equivalent damping ratios and the first- and second-order equivalent natural frequencies in the force-electric coupling control equation are: S51: Define the perturbation small parameter ε as: εμ′ j = μ j , εθ′ j1 = θ j1 εm i ′ = m i , εn i ′ = n i ; where j = 1, 2; i = 1 ··· 27; Introduce a time variable T representing different scales n : T n = ε n t,n = 0, 1 (4) The derivative with respect to time t can be expressed as: Among them, Combined with the solutions of the force-electric coupling control equations (1)-(3), they can be written as functions of different-scale time variables: S52: Substitute formulas (5) and (6) into the force-electric coupling control equations (1)-(3). By equating the zero-order power coefficients of the perturbation small parameter ε, the control equations are obtained: S53: The solutions of the first two differential equations of equation (7) are: where A1 and A2 are complex functions to be determined, and are the conjugate complex numbers of A1 and A2 respectively, Substitute equation (8) into the third equation of equation (7) to obtain the solution of the third differential equation of equation (7) as: wherein, cc represents the conjugate complex number of each item on its left; S54: Now, neglect the nonlinear terms and external excitation terms in the force-electric coupling control equations (1)-(3) to obtain the linearized control equation set of the L-type piezoelectric vibration energy harvesting system: Assume q1≈q 10 and q2≈q 20 and V≈q 30 Then Equation (9) can be expressed as: Substitute Equation (11) into the first two equalities of Equation (10) respectively, and then the decoupled linearized control equations can be obtained: Among them, the solutions of the first and second order equivalent natural frequencies of the L-type piezoelectric vibration energy harvesting system are respectively expressed as: The first and second order equivalent damping ratios of the L-type piezoelectric vibration energy harvesting system are respectively expressed as:

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