Method for broadband collection of friction-induced stick-slip vibration and implementation device

Abstract

The invention discloses a method for broadband collection of friction-induced stick-slip vibration and an implementation device, and the method can effectively convert the low-frequency friction-induced stick-slip vibration into electric energy through the relative movement between a moving sliding block and a fixed wall surface by using a friction electrification principle; the motion sliding block is located on an elastic clamping beam, clamping force is applied to the two ends of the clamping beam, potential energy in the beam is changed, a monostable nonlinear structure and a bistable nonlinear structure are formed, the problems that a traditional energy vibrator is weak in response and narrow in frequency band under low-frequency vibration are solved through the design, and the energy efficiency is improved. And the self potential energy can be adjusted according to the strength and frequency change of the stick-slip vibration so as to adapt to the change of the vibration environment, and the friction-induced low-frequency stick-slip vibration energy can be effectively converted. According to the invention, the response of the energy vibrator can be enhanced under the condition of low-frequency vibration, and the energy conversion efficiency of friction-induced stick-slip vibration is improved. And a new technical scheme is provided for efficient utilization of low-frequency friction induction energy.

Description

Technical Field

[0001] The present invention relates to the field of friction-induced stick-slip vibration monitoring, and particularly to a method and an implementation device for collecting friction-induced stick-slip vibration with a wide frequency band. Background Art

[0002] Friction-induced stick-slip vibration is a typical self-excited vibration, commonly found in various mechanical systems, such as dry friction systems with low relative sliding speeds like cutting systems, windshield wipers, and vehicle base braking. Especially during the train braking process, when the train speed is close to a complete stop, the vehicle will exhibit obvious flutter behavior, accompanied by low-frequency creep groaning noise, seriously affecting the riding comfort of passengers. Additionally, practical engineering applications have also shown that this flutter behavior can cause problems such as chipping of the train brake pads, seriously endangering traffic safety. Although this vibration is harmful and difficult to completely eliminate, due to its considerable energy, energy harvesting technology can be used to convert it into electrical energy, and the harvested electrical energy can power micro-sensors installed on the braking interface to passively monitor the health status of the braking interface.

[0003] Existing technologies mainly rely on piezoelectric materials or electromagnetic induction devices to convert vibration energy into electrical energy. However, piezoelectric materials have low response efficiency to low-frequency vibrations (such as vibrations in the 0.1 - 100 Hz range during the train braking creep stage) and are prone to performance degradation due to mechanical fatigue; electromagnetic devices require precise alignment and continuous relative motion, making it difficult to adapt to the complex multi-degree-of-freedom vibration environment of the braking interface. As a new technology, triboelectric nanogenerators have good low-frequency response capabilities and are well-suited for low-frequency vibration excitations such as friction-induced stick-slip vibration. There are still certain deficiencies in the method and implementation device for collecting friction-induced stick-slip vibration with a wide frequency band in terms of energy conversion efficiency and adaptability. In traditional methods, the energy oscillator has a weak response under low-frequency vibration conditions, resulting in limited energy harvesting effect for friction-induced stick-slip vibration. At the same time, the working frequency bands of these methods are relatively narrow and cannot cover a wide range of vibration frequencies, making it difficult for them to effectively adapt to different vibration environments. In addition, existing devices cannot dynamically adjust their own potential energy according to the strength and frequency changes of stick-slip vibration, so that under different vibration conditions, their energy conversion efficiency is low and it is difficult to effectively utilize external vibration energy. This fixed potential energy characteristic limits the adaptive ability of the device, resulting in a decrease in energy conversion rate during low-frequency stick-slip vibration and affecting the stability and practicality of energy harvesting. Therefore, how to improve the low-frequency response ability, broaden the working frequency band, and achieve adaptive potential energy adjustment to more efficiently convert friction-induced stick-slip vibration energy remains an urgent problem in this field.

[0004] To this end, the present application proposes a method and an implementation device for collecting friction-induced stick-slip vibrations in a wide frequency band, aiming to effectively collect the vibration energy caused by stick-slip vibrations, and using an efficient nanogenerator as an energy conversion medium to convert the relative motion between friction interfaces into electrical energy. Summary of the Invention

[0005] In order to overcome the shortcomings and deficiencies of the prior art, the present invention provides a method and an implementation device for collecting friction-induced stick-slip vibrations in a wide frequency band.

[0006] In a first aspect, the technical solution adopted by the present invention is a method for collecting friction-induced stick-slip vibrations in a wide frequency band, and the method includes:

[0007] Step S1: Design a variable steady-state vibration structure with adjustable potential energy based on the principle of slender beam buckling under compression. The vibration structure includes a main elastic beam vibration structure, one end of which is fixed and clamped on a base, and the other end is subjected to longitudinal pressure through a nut to adjust the potential energy inside the elastic beam and change its vibration characteristics. There is a pair of sliders in the middle part of the elastic beam, and the end faces of the sliders are in contact with the wall surfaces, forming two pairs of friction pairs, and constructing the initial dynamic motion equation of the elastic beam.

[0008] Step S2: Set a triboelectric nanogenerator on the friction pairs to convert the relative motion between the friction pairs into electrical energy. The triboelectric nanogenerator consists of two copper foil electrodes and a polytetrafluoroethylene (PTFE) film placed between the two layers of copper foil electrodes. This design utilizes the principle of Maxwell displacement current to generate electron transfer, and can achieve the purpose of converting the relative motion between the friction pairs into electrical energy.

[0009] Step S3: According to the structure designed in steps S1 and S2, derive the potential energy function and kinetic energy function of the elastic beam, and construct the dynamic motion equation of the elastic beam based on the Lagrange equation.

[0010] Step S4: Model the frictional force between the friction pairs in step S1 according to Coulomb's friction law, and construct a frictional force equation.

[0011] Step S5: Construct the electrodynamic equation of the triboelectric nanogenerator in step S2 according to Kirchhoff's law.

[0012] Step S6: Synthesize the dynamic equations in steps S3, S4, and S5 to obtain the mechanical-electrical coupling control equation of the entire energy harvesting system.

[0013] Step S7: Use the mechanical-electrical coupling control equation in step S6 to solve the motion and output voltage of the system, optimize the key parameters in this design, compare the energy output performance between the single-steady state and the multi-steady state, and achieve the adaptive adjustment of different steady-state structures according to different vibration environments to reach the optimal output of the system.

[0014] Furthermore, for the initial dynamic equation of motion of the elastic beam, for the compressed elastic beam structure, the internal potential energy function is divided into two parts. One part is the strain energy generated by the elongation of the beam along its length direction, and the other part is the strain energy generated by the bending deformation of the elastic beam. The two parts are superimposed to generate the total elastic potential energy. The elastic strain expressions of the two parts are respectively:

[0015]

[0016]

[0017] Among them, ε1 is the strain energy generated by the elongation of the beam along its length direction, ε2 is the strain energy generated by the bending deformation of the elastic beam, u′ is the first-order derivative of the transverse displacement of a certain point on the elastic beam with respect to the coordinate, w′ is the first-order derivative of the longitudinal displacement of a certain point on the elastic beam with respect to the coordinate, z is the distance from a certain point on the elastic beam to its neutral layer, u" is the second-order derivative of the transverse displacement of a certain point on the elastic beam with respect to the coordinate, and w" is the second-order derivative of the longitudinal displacement of a certain point on the elastic beam with respect to the coordinate.

[0018] The total potential energy function of the system is calculated and the expression is:

[0019]

[0020] Among them, U represents the potential energy function, E is the elastic modulus of the elastic beam, and dV represents the volume element.

[0021] Furthermore, the potential energy function and kinetic energy function of the elastic beam are expressed as:

[0022]

[0023] Among them, T represents the kinetic energy of the elastic beam, ρ is the density of the elastic beam, A is the cross-sectional area of the elastic beam, L is the length of the elastic beam, y is the external excitation displacement, is the first-order derivative of the external excitation with respect to time, is the first-order derivative of the transverse displacement of the elastic beam with respect to time, is the first-order derivative of the longitudinal displacement of the elastic beam with respect to time, δ is the Dirac symbol, m is the mass of the slider, x is the coordinate of the elastic beam along its length, dx is the length element on the elastic beam;

[0024] Due to the inextensibility of the beam, the longitudinal displacement and transverse displacement have the following relationship:

[0025]

[0026] Among them, P is the magnitude of the pre-tightening force, and dζ is the element of a point on the elastic beam.

[0027] The transverse displacement of the clamping beam is updated in the form of a series of orthogonal modal vibration modes multiplied by the time coordinate, and the expression is:

[0028] w(x,t) = q(t)φ(x)

[0029] where q(t) represents the modal coordinate related to time, and φ(x) represents the vibration mode of the elastic beam;

[0030] Define the Lagrangian:

[0031]

[0032] where, represents the Lagrangian, T is the kinetic energy of the system, and U is the potential energy of the system.

[0033] Furthermore, the dynamic motion equation of the elastic beam is constructed based on the Lagrange equation, and the expression is:

[0034]

[0035] where the expressions of each coefficient are:

[0036] M = ρA∫0 L φ 2 dx + mφ(0.5L) 2

[0037] D = ∫0 L cφ 2 dx

[0038] λ = ρA∫0 L [∫0 x φ′ 2 dζ] 2 dx + m[∫0 0.5L φ′ 2 dζ] 2

[0039] α = EI∫ o L φ″ 2 dx - P∫0 L φ′ 2 dx

[0040]

[0041] Γ = ρA∫0 L φdx + mφ(0.5L)

[0042] where, represents the second derivative of the modal coordinate with respect to time, represents the first derivative of the modal coordinate with respect to time, is the second derivative of the external excitation with respect to time, c is the damping coefficient, I is the moment of inertia of the elastic beam, φ′ represents the first derivative of a point on the mode shape function with respect to the coordinate, and φ″ represents the second derivative of a point on the mode shape function with respect to the coordinate.

[0043] Further, the frictional force equation has the following expression:

[0044]

[0045] where F k represents the dynamic frictional force, μ k1 and μ k2 are the dynamic friction coefficients of the slider between the metal film and the dielectric material respectively, N is the normal contact force between the slider and the wall exerted by the internal compression spring of the slider, H is the height of the slider, P1 * and represent the initial positions of the slider, which are the lengths of the slider overlapping with the dielectric material film patch and the lengths of the slider overlapping with the metal electrode at the initial moment when the excitation starts. sgn represents the sign function, and vr represents the relative sliding speed between the slider and the wall;

[0046] Assume P1 * = k2H, then the relationship between k1 and k2 is as follows:

[0047] k1 + k2 = 1

[0048] where k1 and k2 represent position coefficients. Without loss of generality, considering the sticking state during the friction process, the condition for sticking to occur is:

[0049]

[0050] |F s | ≤ F smax

[0051] where F smax is the maximum static frictional force, |F s | is the absolute value of the static frictional force, following the following principle:

[0052]

[0053] where μ s1 and μ s2 are the static friction coefficients of the slider between the metal film and the dielectric material film respectively. In the sticking state, the magnitude of the static frictional force F s is:

[0054]

[0055] Furthermore, for the electrodynamic equation of the triboelectric nanogenerator, the triboelectric nanogenerator is simplified to a first-order lumped parameter circuit with an ideal voltage source in series with a capacitor. According to Kirchhoff's law, the electrodynamic equation can be obtained:

[0056]

[0057] where V is the voltage across the external resistor, Q is the charge transferred between the two electrodes, C is the equivalent capacitance of the triboelectric nanogenerator, and V oc is the open-circuit voltage of the circuit;

[0058] The expressions for the equivalent capacitance and the open-circuit voltage are:

[0059]

[0060] where ψ is the surface triboelectric charge density, t s is the thickness of the dielectric material, ε0 and ε r are the vacuum permittivity and the relative permittivity respectively, and L is the length of the slider;

[0061] According to Ohm's law, the voltage across the external resistor is: The equivalent circuit differential equation of the TEH is:

[0062]

[0063] where R is the value of the external resistor.

[0064] In a second aspect, the technical solution adopted by the present invention is an apparatus for realizing broadband collection of friction-induced stick-slip vibrations. The apparatus includes: a main vibration elastic beam structure, one end of the main vibration elastic beam structure is clamped at a fixed end, and one end of the main vibration elastic beam structure is clamped on a movable support block and clamped with a clamping block.

[0065] There is a pre-tightening nut on the movable end side that can apply a pre-tightening force P to compress the elastic beam from the free state to a mono-stable and bi-stable structure. There is a slider cavity in the middle of the elastic beam. In the internal cross-sectional view of the slider cavity, there is a slider that can move along the cavity on each of the left and right sides of the main vibration elastic beam structure.

[0066] The inside of the slider is a hollow structure with a compression spring inside, which can provide a spring force to press the left and right sliders against the left and right side walls respectively. A PTFE film is pasted on the surface of the slider. Correspondingly, there is a copper foil at the upper end of the wall and a PTFE film at the lower end.

[0067] The PTFE film on the slider, the copper foil on the side wall and the PTFE film together form a triboelectric nanogenerator.

[0068] Beneficial effects:

[0069] The present invention provides a method and an implementation device for collecting broadband friction-induced stick-slip vibration. Through the relative movement between a moving slider and a fixed wall surface, and by utilizing the principle of triboelectrification, the present invention can effectively convert low-frequency friction-induced stick-slip vibration into electrical energy. At the same time, the moving slider is located on an elastic clamping beam. By applying clamping forces at both ends of the clamping beam, the potential energy inside the beam is changed, forming a single-stable and double-stable nonlinear structure. This design not only overcomes the problems of weak response and narrow frequency band of traditional energy oscillators under low-frequency vibration, but also can adjust its own potential energy according to the strength and frequency changes of the stick-slip vibration to adapt to the changes in the vibration environment, and can effectively convert the low-frequency stick-slip vibration energy induced by friction. The present invention can enhance the response of the energy oscillator under low-frequency vibration conditions and improve the energy conversion efficiency of the friction-induced stick-slip vibration. At the same time, this method broadens the working frequency band, enables the device to adapt to vibration environments with different frequency ranges, and improves the stability of energy collection. In addition, the device can dynamically adjust its own potential energy according to the strength and frequency changes of the stick-slip vibration, thereby optimizing the vibration energy conversion process and enabling it to maintain a high energy collection efficiency under different working conditions. This adaptive adjustment mechanism effectively improves the applicability of the device in complex vibration environments and provides a new technical solution for the efficient utilization of low-frequency friction-induced energy. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 is the overall process flow chart of the method of the present invention;

[0071] Figure 2 is the three-dimensional structure schematic diagram of the device of the present invention;

[0072] Figure 3 is the internal sectional view of the slider cavity of the present invention;

[0073] Figure 4 is the change trend diagram of the static equilibrium point and elastic potential energy of the elastic beam of the present invention with the pre-tightening force;

[0074] Figure 5 is the curve diagram of the output voltage and the corresponding slider vibration displacement under the 3Hz excitation of the present invention;

[0075] Figure 6 is the change trend diagram of the slider vibration amplitude and the output voltage amplitude with the vibration frequency in the double-stable mode of the present invention;

[0076] Figure 7 is the curve diagram of the relationship between the slider vibration amplitude and the frequency of the linear system of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0077] It should be noted that, without conflict, the embodiments in the present application and the features in the embodiments can be combined with each other. The present application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0078] The present invention utilizes the Maxwell displacement current effect of the triboelectric nanogenerator on the dry friction interface to convert the frictional-induced stick-slip vibration energy into electrical energy. At the same time, in order to overcome the disadvantages of narrow working frequency band and poor adaptability of traditional energy harvesters, the present invention proposes a method and an implementation device for wide-band collecting frictional-induced stick-slip vibration with adjustable potential energy. This method changes the internal potential energy by adjusting the pre-tightening force applied at both ends of the clamping beam, and according to the magnitude of the pre-tightening force, it can form a single-stable or double-stable vibration structure. The advantage is that it can adapt to the vibration environment through its own adjustment according to the frequency and intensity of the stick-slip vibration, achieving an adaptive energy conversion effect.

[0079] As Figure 1 shown, a method for wide-band collecting frictional-induced stick-slip vibration, the method comprising:

[0080] Step S1, designing a variable-stable vibration structure with adjustable potential energy by using the principle of slender beam buckling under compression. The vibration structure includes a main elastic beam vibration structure, one end of which is clamped and fixed on a base, and the other end is applied with longitudinal pressure through a nut to adjust the internal potential energy of the elastic beam and change its vibration characteristics. There is a pair of sliders in the middle part of the elastic beam, and the end faces of the sliders are in contact with the wall surfaces, forming two pairs of friction pairs, and constructing the initial dynamic motion equation of the elastic beam;

[0081] Specifically, by using the principle of slender beam buckling under compression, a variable-stable vibration structure with adjustable potential energy is designed. The structure includes an elastic beam as the main vibration structure, one end of which is fixed on a base, and the other end is applied with longitudinal pressure through a nut, thereby adjusting the internal potential energy of the elastic beam and changing its vibration characteristics. This design enables the elastic beam to form a single-stable or multi-stable vibration mode under different force conditions, improving the adaptability of energy collection. In addition, a pair of sliders is arranged in the middle part of the elastic beam, and the end faces of the sliders are in contact with the wall surfaces, forming two pairs of friction pairs, so that the vibration of the beam is affected by friction at the end faces of the sliders. This structural design lays the foundation for vibration energy collection and provides a basis for subsequent mathematical modeling by constructing the initial dynamic motion equation of the elastic beam.

[0082] Step S2, setting a triboelectric nanogenerator on the friction pair to convert the relative motion between the friction pairs into electrical energy. The triboelectric nanogenerator consists of two copper foil electrodes and a polytetrafluoroethylene (PTFE) film placed between the two layers of copper foil electrodes; this design utilizes the Maxwell displacement current principle to generate electron transfer, and can achieve the purpose of converting the relative motion between the friction pairs into electrical energy;

[0083] Specifically, a triboelectric nanogenerator is integrated at the position of the friction pair to convert the relative motion between the friction pairs into electrical energy. The triboelectric nanogenerator mainly consists of two copper foil electrodes and a polytetrafluoroethylene (PTFE) film placed between the two layers of copper foil electrodes. When the vibration structure undergoes periodic motion, the friction pair between the slider and the wall surface generates relative displacement of contact and separation, thereby generating charge transfer between the copper foil electrodes and forming a voltage output. This energy conversion mechanism can effectively collect the energy of friction-induced stick-slip vibration and provide a stable electrical energy output, enabling the system to not only have the ability to convert mechanical vibration but also achieve efficient collection of electrical energy.

[0084] Step S3: Based on the structure designed in steps S1 and S2, derive the potential energy function and kinetic energy function of the elastic beam, and construct the dynamic motion equation of the elastic beam according to the Lagrange equation;

[0085] Specifically, based on the structural design of steps S1 and S2, derive the potential energy function and kinetic energy function of the elastic beam, and construct the dynamic motion equation of the elastic beam according to the Lagrange equation. The Lagrange equation can describe the energy balance relationship of the system. By analyzing the motion law of the elastic beam under different initial potential energies and force conditions, a mathematical model is established. This process can not only reflect the steady-state characteristics of the structure but also provide a basis for subsequent electromechanical coupling modeling. By constructing this dynamic equation, the response characteristics of the vibration system can be deeply studied, and its energy collection efficiency can be further optimized.

[0086] Step S4: Model the frictional force between the friction pairs in step S1 according to Coulomb's friction law and construct a frictional force equation;

[0087] Specifically, according to Coulomb's friction law, model the frictional force of the friction pair in step S1 and construct a frictional force equation. Coulomb's friction law shows that the magnitude of the frictional force is related to the normal pressure of the contact surface and the coefficient of friction. In this system, the frictional interaction between the slider and the wall surface directly affects the vibration behavior of the elastic beam. Therefore, it is necessary to accurately describe the variation characteristics of the frictional force. Frictional force modeling can not only reveal the influence of the friction pair on the vibration mode of the system but also provide a mathematical expression of frictional energy consumption for subsequent electromechanical coupling analysis.

[0088] Step S5: Construct the electrodynamic equation of the triboelectric nanogenerator in step S2 according to Kirchhoff's law;

[0089] Specifically, according to Kirchhoff's laws, the electrodynamic equations of the triboelectric nanogenerator are constructed to describe the process of charge accumulation and transmission during the energy conversion process. Kirchhoff's laws include the current law (KCL) and the voltage law (KVL), which are used to analyze the circuit characteristics of the triboelectric nanogenerator. During the movement of the triboelectric nanogenerator, charge transfer occurs, forming a dynamic electric field. Therefore, it is necessary to combine the electrodynamic equations to describe the charge accumulation, dissipation, and voltage output characteristics. This modeling process can predict the voltage change law under different vibration conditions, laying a foundation for optimizing the energy output of the system.

[0090] Step S6: Obtain the mechanical-electrical coupling control equation of the entire energy harvesting system by synthesizing steps S3, S4, and S5.

[0091] Specifically, by synthesizing step S3 (vibration structure dynamics), step S4 (friction force equation), and step S5 (electrodynamic equation), the mechanical-electrical coupling control equation of the entire energy harvesting system is obtained. This equation combines the motion equation of the elastic beam, the friction force characteristics, and the charge accumulation behavior of the triboelectric nanogenerator to form a complete energy conversion model. This mechanical-electrical coupling equation can describe the overall dynamic characteristics of the system and reveal the specific mechanism of the conversion of mechanical energy into electrical energy, providing a theoretical basis for optimizing the design and adjustment strategy of the energy harvesting system.

[0092] Step S7: Solve the motion and output voltage of the system using the mechanical-electrical coupling control equation in step S6, optimize the key parameters in this design, compare the energy output performance between the monostable and multistable states, and achieve adaptive adjustment of different stable-state structures according to different vibration environments to reach the optimal output of the system.

[0093] Specifically, using the mechanical-electrical coupling control equation in step S6, solve the motion behavior and output voltage of the system, and optimize the key parameters. Through numerical simulation, analyze the influence of different stable states (monostable, multistable) on the energy output, and compare the energy harvesting performance under different working conditions. In addition, the system can adaptively adjust the stable-state structure according to different vibration environments to optimize the energy output, enabling it to maintain high efficiency under different conditions. Finally, through experimental verification and parameter optimization, ensure that the system can achieve the best energy harvesting performance in a complex vibration environment.

[0094] Furthermore, for the initial dynamic motion equation of the elastic beam, for the compressed elastic beam structure, the internal potential energy function is divided into two parts. One part is the strain energy generated by the elongation of the beam along its length direction, and the other part is the strain energy generated by the bending deformation of the elastic beam. The two parts are superimposed to generate the total elastic potential energy. The elastic strain expressions of the two parts are respectively:

[0095]

[0096] Among them, ε1 is the strain energy generated by the elongation of the beam along its length direction, ε2 is the strain energy generated by the bending deformation of the elastic beam, u′ is the first derivative of the transverse displacement of a certain point on the elastic beam with respect to the coordinate, w′ is the first derivative of the longitudinal displacement of a certain point on the elastic beam with respect to the coordinate, z is the distance from a certain point on the elastic beam to its neutral layer, u″ is the second derivative of the transverse displacement of a certain point on the elastic beam with respect to the coordinate, and w″ is the second derivative of the longitudinal displacement of a certain point on the elastic beam with respect to the coordinate.

[0097] The total potential energy function of the system is calculated and the expression is:

[0098]

[0099] Among them, U represents the potential energy function, E is the elastic modulus of the elastic beam, and dV represents the volume element.

[0100] Furthermore, the potential energy function and kinetic energy function of the elastic beam are expressed as:

[0101]

[0102] Among them, T represents the kinetic energy of the elastic beam, ρ is the density of the elastic beam, A is the cross-sectional area of the elastic beam, L is the length of the elastic beam, y is the external excitation displacement, is the first derivative of the external excitation with respect to time, is the first derivative of the transverse displacement of the elastic beam with respect to time, is the first derivative of the longitudinal displacement of the elastic beam with respect to time, δ is the Dirac symbol, m is the mass of the slider, x is the coordinate of the elastic beam along its length, dx is the length element on the elastic beam;

[0103] Due to the inextensibility of the beam, the longitudinal displacement and transverse displacement have the following relationship:

[0104]

[0105] Among them, P is the magnitude of the pre-tightening force, and dζ is the element of a point on the elastic beam.

[0106] The transverse displacement of the clamped beam is updated to the form of a series of orthogonal mode shapes multiplied by the time coordinate, and the expression is:

[0107] w(x,t) = q(t)φ(x)

[0108] Among them, q(t) represents the mode coordinate related to time, and φ(x) represents the mode shape of the elastic beam;

[0109] Define the Lagrangian:

[0110]

[0111] wherein, represents the Lagrangian, T is the kinetic energy of the system, and U is the potential energy of the system.

[0112] Furthermore, the dynamic equation of the elastic beam is constructed based on the Lagrange equation, and the expression is:

[0113]

[0114] where the expressions of each coefficient are:

[0115] M = ρA∫0 L φ 2 dx + mφ(0.5L) 2

[0116] D = ∫0 L cφ 2 dx

[0117] λ = ρA∫0 L [∫0 x φ′2dζ] 2 dx + m[∫0 0.5L φ′2dζ] 2

[0118] α = EI∫0 L φ″2dx - P∫0 L φ′ 2 dx

[0119]

[0120] Γ = ρA∫0 L φdx + mφ(0.5L)

[0121] wherein, represents the second derivative of the modal coordinate with respect to time, represents the first derivative of the modal coordinate with respect to time, is the second derivative of the external excitation with respect to time, c is the damping coefficient, I is the moment of inertia of the elastic beam, φ′ represents the first derivative of a point on the vibration mode function with respect to the coordinate, and φ″ represents the second derivative of a point on the vibration mode function with respect to the coordinate.

[0122] Furthermore, the friction force equation has the expression:

[0123]

[0124] wherein, F k represents the dynamic friction force, μ k1 and μ k2is the dynamic friction coefficient between the slider and the dielectric material, N is the normal contact force between the slider and the wall applied by the internal compression spring of the slider, H is the height of the slider, and P1 * and represent the initial position of the slider, which is the length of the slider overlapping with the dielectric material film patch and the length of the slider overlapping with the metal electrode at the initial moment when the excitation starts. sgn represents the sign function, and vr represents the relative sliding speed between the slider and the wall;

[0125] Assume P l * = k2H, then the relationship between k1 and k2 is as follows:

[0126]

[0127] where k1 and k2 represent position coefficients. Without loss of generality, considering the sticking state during friction, the condition for sticking to occur is:

[0128]

[0129] |F s | ≤ F smax

[0130] where F smax is the maximum static friction force, |F s | is the absolute value of the static friction force, following the following principle:

[0131]

[0132] where μ s1 and μ s2 are the static friction coefficients between the slider and the metal film and the dielectric material film respectively. In the sticking state, the magnitude of the static friction force F s is:

[0133]

[0134] Furthermore, for the electrodynamic equation of the triboelectric nanogenerator, the triboelectric nanogenerator is simplified to a first-order lumped parameter circuit with an ideal voltage source in series with a capacitor. According to Kirchhoff's law, the electrodynamic equation can be obtained:

[0135]

[0136] where V is the voltage across the external resistor, Q is the charge transferred between the two electrodes, C is the equivalent capacitance of the triboelectric nanogenerator, and V oc is the open-circuit voltage of the circuit;

[0137] The expressions for the equivalent capacitance and the open-circuit voltage are as follows:

[0138]

[0139]

[0140] where ψ is the surface frictional charge density, t s is the thickness of the dielectric material, ε0 and ε r are the permittivity of free space and the relative permittivity respectively, and L is the length of the slider;

[0141] According to Ohm's law, the voltage across the external resistor is: The differential equation of the equivalent circuit of TEH is:

[0142]

[0143] where R is the value of the external resistor.

[0144] As Figure 2 and Figure 3 shown, a device for realizing broadband collection of friction-induced stick-slip vibration, which is used to implement a method for broadband collection of friction-induced stick-slip vibration. The device includes: a main vibration elastic beam structure 6, with a length of 80 mm, a width of 20 mm, a thickness of 0.2 mm, and the material is stainless steel; one end of the main vibration elastic beam structure 6 is clamped on the fixed end 1, and one end of the main vibration elastic beam structure 6 is clamped on the movable support block 10 and clamped with the clamping block 7.

[0145] On the movable side, there is a pre-tightening nut 8 that can apply a pre-tightening force P to compress the elastic beam from the free state to the single-stable and double-stable structures. There is a slider cavity 3 in the middle of the elastic beam. The internal height of the slider cavity is 20 mm, the width is 20 mm. On the left and right sides of the internal sectional view of the slider cavity of the main vibration elastic beam structure 6, there are two sliders 12 and 13 that can move along the cavity.

[0146] The inside of the slider is a hollow structure, with compression springs 15 and 16 inside. The stiffness of the springs can be replaced according to the actual required frictional force, and its function is to provide spring forces to press the left and right sliders against the left and right side walls 2 and 9 respectively. PTFE films 11 and 14 are pasted on the surface of the slider. Correspondingly, there is a copper foil 4 at the upper end of the wall and a PTFE film 5 at the lower end.

[0147] The PTFE film on the slider, the copper foil on the side wall and the PTFE film together form a triboelectric nanogenerator.

[0148] When the entire structure is subjected to external excitation, relative displacement will occur between the slider and the wall due to the inertial force. Especially when the external excitation frequency is low, the relative movement speed of the slider is low. Due to the difference in the static and dynamic friction coefficients between the slider and the wall, the slider will generate stick-slip vibration relative to the wall. The relative displacement caused by this vibration will generate Maxwell displacement current inside the nanogenerator, thus converting the stick-slip vibration energy into electrical energy.

[0149] As Figure 4 shown is the variation trend of the static equilibrium point and elastic potential energy of the elastic beam with the pre-tightening force in this embodiment. It can be found from the figure that when the pre-tightening force is small, the elastic beam has only one static equilibrium point, indicating that the system is monostable at this time; when the pre-tightening force exceeds a certain critical point, the critical point of the system suddenly becomes two, indicating that the system has become a bistable system. From the potential energy functions under different pre-tightening forces, it can be seen that when the pre-tightening force is less than the critical value, the potential energy decreases with the increase of the pre-tightening force; when the pre-tightening force exceeds the critical value, the shape of the potential energy function changes, and the origin in the middle changes from the lowest potential energy point to a local highest point, forming symmetric potential wells on both sides. Moreover, with the further increase of the pre-tightening force, the depth of the potential well becomes deeper. By adjusting the magnitude of the pre-tightening force, the vibration behavior of the elastic beam can be changed.

[0150] Figure 5 shown are the voltage signal generated by the triboelectric nanogenerator and the vibration displacement signal of the slider under an external excitation of 3 Hz. It can be seen from the figure that the slider has obvious stick-slip vibration phenomenon. When the slider is in the sticking state, its vibration displacement remains constant, and correspondingly its output voltage value is zero. Only when the slider is in the sliding state, its output voltage value is non-zero, which proves that the device can effectively collect stick-slip vibration energy.

[0151] Figure 6 shown is the variation of the vibration amplitude of the slider and the output voltage amplitude with the vibration frequency. Figure 7 shown is the amplitude-frequency curve of the linear vibration energy harvester. It can be found that the voltage amplitude and vibration amplitude of this energy harvester first increase with the increase of the frequency. When the frequency is greater than a certain critical value, the vibration and voltage suddenly disappear. Its effective frequency bandwidth can reach 17 Hz, which is much larger than the effective frequency bandwidth of the linear system, about 2 Hz.

[0152] In the description of the present invention, it should be noted that, unless otherwise clearly specified and defined, the terms "set", "installed", "connected", "connected", "fixed" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be directly connected or indirectly connected through an intermediate medium, and it can be the communication inside two components. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood through specific situations.

[0153] Although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various equivalent changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalent scope.

Claims

1. A method for collecting friction-induced stick-slip vibrations over a wide frequency band, characterized in that , the method comprises: Step S1, using the principle of compression instability of a slender beam to design a variable steady-state vibration structure with adjustable potential energy, the vibration structure includes an elastic beam vibration main structure, one end of which is fixed on a base and clamped, and the other end is subjected to longitudinal pressure by a nut to adjust the internal potential energy of the elastic beam to change its vibration characteristics, a pair of sliders are provided in the middle part of the elastic beam, the end faces of the sliders are in contact with the wall surface to form two pairs of friction pairs, and the initial dynamic motion equation of the elastic beam is constructed; Step S2, arranging a friction nanogenerator on the friction pair to convert the mutual motion between the friction pairs into electrical energy, wherein the friction nanogenerator comprises two copper foil electrodes and a polytetrafluoroethylene film placed between the two copper foil electrodes; Step S3, deriving the potential energy function and kinetic energy function of the elastic beam according to the structures designed in steps S1 and S2, and constructing the dynamic motion equation of the elastic beam according to the Lagrange equation; Step S4, modeling the friction force between the friction pairs in step S1 according to Coulomb's friction law, and constructing a friction force equation; Step S5, constructing the electrodynamic equation of the friction nanogenerator in step S2 according to Kirchhoff's law; Step S6, combining steps S3, S4 and S5 to obtain the electromechanical coupling control equation of the entire energy harvesting system; Step S7, using the electromechanical coupling control equation in step S6 to solve the motion and output voltage of the system, and optimizing the key parameters in the design, comparing the energy output performance between the monostable state and the multistable state, and realizing adaptive adjustment of different steady-state structures according to different vibration environments to achieve the optimal output of the system.

2. A method for collecting friction-induced stick-slip vibrations over a wide frequency band as claimed in claim 1, characterized in that: The initial dynamic motion equation of the elastic beam, for the compressed elastic beam structure, the internal potential energy function is divided into two parts, one part is the strain energy generated by the elongation of the beam along its length direction, and the other part is the strain energy generated by the bending deformation of the elastic beam. The two parts are superimposed to generate the total elastic potential energy. The elastic strain expressions of the two parts are: Among them, ε1 is the strain energy generated by the elongation of the beam along its length direction, ε2 is the strain energy generated by the bending deformation of the elastic beam, u′ is the first-order derivative of the lateral displacement of a point on the elastic beam with respect to the coordinate, w′ is the first-order derivative of the longitudinal displacement of a point on the elastic beam with respect to the coordinate, z is the distance from a point on the elastic beam to its neutral layer, u″ is the second-order derivative of the lateral displacement of a point on the elastic beam with respect to the coordinate, and w″ is the second-order derivative of the longitudinal displacement of a point on the elastic beam with respect to the coordinate; The total potential energy function of the system is calculated and expressed as: Among them, U represents the potential energy function, E is the elastic modulus of the elastic beam, and dV represents the volume element.

3. A method for collecting friction-induced stick-slip vibrations over a wide frequency band as claimed in claim 1, characterized in that: The potential energy function and kinetic energy function of the elastic beam are expressed as follows: Where T represents the kinetic energy of the elastic beam, ρ is the density of the elastic beam, A is the cross-sectional area of ​​the elastic beam, L is the length of the elastic beam, and y is the external excitation displacement. is the first-order derivative of the external excitation with respect to time, is the first-order derivative of the lateral displacement of the elastic beam with respect to time, is the first-order derivative of the longitudinal displacement of the elastic beam with respect to time, δ is the Dirac symbol, m is the mass of the slider, x is the coordinate of the elastic beam along its length, and dx is the length element of the elastic beam; Due to the inextensibility of the beam, the longitudinal displacement and the lateral displacement have the following relationship: Among them, P is the preload force, dζ is the infinitesimal element of a point on the elastic beam; The lateral displacement of the clamped beam is updated as a series of orthogonal mode shapes multiplied by the time coordinate, expressed as: w(x,t)=q(t)φ(x) Where q(t) represents the time-dependent modal coordinates, φ(x) represents the vibration mode of the elastic beam; Define the Lagrangian: in, represents the Lagrangian, T is the kinetic energy of the system, and U is the potential energy of the system.

4. A method for collecting friction-induced stick-slip vibrations over a wide frequency band as claimed in claim 1, characterized in that: The dynamic motion equation of the elastic beam is constructed based on the Lagrange equation, and the expression is: The expressions of each coefficient are: in, represents the second derivative of the modal coordinates with respect to time, represents the first derivative of the modal coordinates with respect to time, is the second derivative of the external excitation with respect to time, c is the damping coefficient, I is the moment of inertia of the elastic beam, φ′ represents the first derivative of a point on the vibration mode function with respect to the coordinate, and φ″ represents the second derivative of a point on the vibration mode function with respect to the coordinate.

5. A method for collecting friction-induced stick-slip vibrations over a wide frequency band as claimed in claim 1, characterized in that: The friction force equation is expressed as: Among them, F k represents the dynamic friction, μ k1 and μ k2 are the dynamic friction coefficients of the slider between the metal film and the dielectric material, N is the normal contact force between the slider and the wall surface applied by the compression spring inside the slider, H is the height of the slider, and represents the initial position of the slider, which is the slider length overlapping with the dielectric material film patch and the slider length overlapping with the metal electrode at the initial moment of the start of excitation, sgn represents the sign function, and vr represents the relative sliding speed between the slider and the wall; Assumptions Then there is the following relationship between k1 and k2: k1+k2=1 Where k1 and k2 represent position coefficients. Without loss of generality, considering the adhesion state during the friction process, the condition for adhesion to occur is: |F s |≤F smax Among them, F smax is the maximum static friction, |F s | is the absolute value of static friction, and the following principles apply: Among them, μ s1 and μ s2 are the static friction coefficients of the slider between the metal film and the dielectric material film, respectively. In the adhesive state, the magnitude of the static friction force F s for:

6. A method for collecting friction-induced stick-slip vibrations over a wide frequency band as claimed in claim 1, characterized in that: The electrodynamic equation of the triboelectric nanogenerator simplifies the triboelectric nanogenerator into a first-order lumped parameter circuit in which an ideal voltage source and a capacitor are connected in series. According to Kirchhoff's law, the electrodynamic equation can be obtained: Where V is the voltage across the external resistor, Q is the charge transferred between the two electrodes, C is the equivalent capacitance of the triboelectric nanogenerator, and V oc is the open circuit voltage of the circuit; The expression of equivalent capacitance and open circuit voltage is: Where ψ is the surface friction charge density, t s is the thickness of the dielectric material, ε0 and ε e are the vacuum dielectric constant and relative dielectric constant respectively, L is the length of the slider; According to Ohm's law, the voltage across the external resistor is: The equivalent circuit differential equation of TEH is: Among them, R is the size of the external resistance.

7. The device for realizing wide-band collection of friction-induced stick-slip vibration according to claim 1, characterized in that: The device comprises: a main vibration elastic beam structure (6), one end of the main vibration elastic beam structure (6) is clamped on a fixed end (1), and one end of the main vibration elastic beam structure (6) is clamped on a movable support block (10) and clamped by a clamping block (7); A pre-tightening nut (8) is provided on the movable end side to load a pre-tightening force P, thereby compressing the elastic beam from a free state to a monostable and bistable structure. A slider cavity (3) is provided in the middle of the elastic beam. The internal cross-sectional view of the slider cavity shows that there are sliders (12) and (13) on the left and right sides of the main vibration elastic beam structure (6) for moving along the cavity. The interior of the slider is a hollow structure, and there are compression springs (15) and (16) inside, which are used to provide spring force to press the left and right sliders onto the left and right side walls (2) and (9) respectively. PTFE films (11) and (14) are attached to the surfaces of the sliders. Correspondingly, there is a copper foil (4) at the upper end of the wall surface and a PTFE film (5) at the lower end. The PTFE film on the slider and the copper foil and PTFE film on the side wall together form a friction nanogenerator.

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