Circumferential rotation traveling wave regulation and control method in circular plate structure of additional dynamic vibration absorber

Abstract

The invention discloses a method for regulating and controlling circumferential rotation traveling waves in a circular plate structure of an additional dynamic vibration absorber. The method comprises the following steps: firstly, establishing a dynamic model of a coupling system of the dynamic vibration absorber and the circular plate structure; under the external exciting force, an approximate degenerate orthogonal mode is selected according to the kinetic model, the vibration displacement of the circular plate structure is obtained, and then the displacement envelope extreme value is determined; and establishing a circumferential rotation traveling wave regulation and control model of the coupling system, obtaining an amplitude envelope according to a displacement envelope extreme value, further obtaining excitation parameters, and performing circumferential rotation traveling wave regulation and control. According to the method, the dynamic vibration absorbers are reasonably selected, excitation parameters are optimized, efficient regulation and control of the circumferential rotation traveling waves of the circular plate structure are achieved under the condition of cyclic symmetry breaking, the driving efficiency of the traveling waves of the asymmetric circular plate structure is effectively improved, accurate control over the traveling waves of the structure is achieved, and the service life of the structure is prolonged. And a new thought and technical support are provided for piezoelectric actuator design, precise driving and control, structural vibration suppression and energy optimization distribution in practical engineering application.

Description

Technical Field

[0001] The invention relates to a method for regulating a circular plate structure, and in particular to a method for regulating circumferential rotating traveling waves in a circular plate structure with an additional dynamic vibration absorber. Background Art

[0002] Structural traveling waves have important application value in piezoelectric drive, acoustic suspension, transporting objects, and testing material impedance. Maintaining the cyclic symmetry of the structure is crucial for generating structural traveling waves. Traditional traveling wave excitation methods usually assume that plate and shell structures have perfect symmetry, but in practical applications, due to the introduction of dynamic coupling structures such as dynamic vibration absorbers, bolt connections, and rivets, the symmetry assumption is usually difficult to achieve. The cyclic symmetry of the structure allows the degenerate modes of the structure to be excited simultaneously at a single external frequency. By applying a suitable phase difference to the cosine and sine degenerate modes, pure traveling wave propagation can be generated. At present, it is difficult to effectively realize traveling wave propagation in actual asymmetric circular plate structures, and it is difficult to solve the problem of traveling wave parameter detuning caused by modal coupling. Summary of the Invention

[0003] In order to solve the problems existing in the background technology, the present invention provides a method for controlling circumferential rotating traveling waves in a circular plate structure with an additional dynamic vibration absorber. The present invention optimizes the dynamic parameters and excitation frequency of the system by constructing a standing wave ratio function and a traveling wave exponential cost function, thereby solving the problem of detuning of traveling wave parameters caused by modal coupling in existing methods. The method performs modal analysis on the asymmetric circular plate structure of the additional dynamic vibration absorber and finds that the coupling mechanism has degenerate modes similar to those of a uniform plate and shell structure. By optimizing and adjusting the excitation frequency of the system, structural traveling waves are effectively generated in the asymmetric structure. The present invention can reasonably select the weak detuned degenerate modes caused by the additional dynamic vibration absorber to realize traveling wave propagation.

[0004] The technical solution adopted in the present invention is:

[0005] The method for controlling circumferential rotating traveling waves in a circular plate structure of an additional dynamic vibration absorber of the present invention comprises:

[0006] 1) Establish a dynamic model of the coupled system consisting of a circular plate structure with an attached dynamic vibration absorber.

[0007] 2) Under the excitation of the external excitation force, the approximate degenerate orthogonal mode of the coupled system is selected according to the dynamic model of the coupled system, and the vibration displacement of the circular plate structure is obtained, and then the extreme value of the displacement envelope of the vibration displacement of the circular plate structure is determined.

[0008] 3) A circumferential rotating traveling wave control model of the coupled system is established, and the extreme values ​​of the displacement envelope of the vibration displacement of the circular plate structure are input into the circumferential rotating traveling wave control model. After processing, the amplitude envelope of the vibration displacement of the circular plate structure is output, and then the excitation parameters of the external excitation force are obtained, thereby performing circumferential rotating traveling wave control on the circular plate structure with an attached dynamic vibration absorber.

[0009] In the step 1), the circular plate structure is horizontally supported on the top surfaces of a plurality of exciters, each of which provides an external excitation force, and the dynamic vibration absorber is attached to the top surface of the circular plate structure, thereby forming a coupling system.

[0010] In the step 1), the dynamic model of the coupled system includes an antisymmetric modal characteristic equation and a symmetric modal characteristic equation, which are as follows:

[0011] a) Antisymmetric modal characteristic equation Δ 1n as follows:

[0012]

[0013] Where ω is the excitation frequency of the external excitation force applied by the exciter; and are the contributions of the nth-order Bessel functions of the first and second kinds caused by the edge bending moment of the circular plate, respectively; and are the contributions of the nth-order first and second modified Bessel functions caused by the edge bending moment of the circular plate, respectively; and are the contributions of the nth-order Bessel functions of the first and second kinds caused by the shear force at the edge of the circular plate, respectively; and are the contributions of the first and second modified Bessel functions of the nth order caused by the shear force at the edge of the circular plate, respectively; β is a constant coefficient, β = αR, α represents the spatial wave number, and R is the radius of the circular plate structure; J n () and Y n () are the nth order first and second kind Bessel functions, J n,β () and Y n,β () are the calculated values ​​of the nth order first and second kind Bessel functions at the edge of the circular plate respectively; I n () and K n () are the modified Bessel functions of the first and second kinds of the nth order, respectively, I n,β () and K n,β () are the calculated values ​​of the first and second modified Bessel functions of the nth order at the edge of the circular plate; ν is the Poisson's ratio.

[0014] b) The symmetric modal characteristic equation Δ2 is as follows:

[0015]

[0016] Wherein, k0 is the stiffness of the dynamic vibration absorber; D is the bending stiffness of the circular plate structure; ω0 is the natural frequency of the dynamic vibration absorber; is the weight coefficient of the nth-order first-kind Bessel function caused by the coupling between the dynamic vibration absorber and the circular plate; r0 and φ0 are the radial distance and initial angular position of the dynamic vibration absorber on the circular plate structure, respectively; is the weight coefficient of the first-order modified Bessel function caused by the coupling between the dynamic vibration absorber and the circular plate; Φ 1n () and Φ 2n () are the nth order cosine function and sine function respectively; ε n is the nth order constant term.

[0017] 4. The method for controlling circumferential rotating traveling waves in a circular plate structure of an additional dynamic vibration absorber according to claim 2, characterized in that in step 2), the excitation amplitude and phase of each exciter are adjusted according to the dynamic model of the coupled system so that the approximately degenerate cosine and sine modes of the coupled system maintain a phase difference of π / 2, that is, the approximately degenerate orthogonal modes of the coupled system are selected as follows:

[0018]

[0019] Where N is the total number of exciters; is the excitation amplitude of the i-th exciter; Φ 1n () and Φ 2n () are the nth order cosine function and sine function respectively; φ i is the angular position of the i-th exciter under the circular plate structure.

[0020] 5. The method for controlling circumferential rotating traveling waves in a circular plate structure of an additional dynamic vibration absorber according to claim 4, wherein in step 2), the vibration displacement W of the circular plate structure is as follows:

[0021]

[0022] where r and φ are the radial distance and angular position of a point on the circular plate structure, respectively; D is the bending stiffness of the circular plate structure; ε n is the nth order constant term; J is the coupling external force term between the nth-order Bessel function and the cosine function (j=1) or sine function (j=2) caused by the i-th exciter; n () and I n () are the nth order first kind Bessel function and the first kind modified Bessel function respectively; α is the spatial wave number; The coupled external force term of the n-th order modified Bessel function of the i-th exciter and the cosine term (j=1) or sine term (j=2); is the external force contribution of the i-th exciter in the radial direction.

[0023] In step 3), the circumferential rotating traveling wave control model is as follows:

[0024]

[0025] Among them, ω opt is the amplitude envelope of the vibration displacement of the circular plate structure; ω is the excitation frequency of the external excitation force applied by the exciter; SWR is the standing wave ratio function, and the standing wave ratio function SWR = 0 represents that the waveform envelope amplitude is constant, corresponding to a traveling wave, and SWR = 1 represents that the envelope amplitude is fluctuating, corresponding to a standing wave; TWI is the rotating traveling wave index; r0 is the radial distance at which the dynamic vibration absorber is arranged on the circular plate structure; φ is the angular position of a point on the circular plate structure; c max and c min are respectively the maximum and minimum values ​​of the displacement envelope of the vibration displacement of the circular plate structure, that is, the extreme values ​​of the displacement envelope of the vibration displacement of the circular plate structure.

[0026] In the step 4), the excitation parameters of the external excitation force include the excitation force amplitude, excitation frequency, phase difference and angular position of each exciter on the circular plate structure.

[0027] The electronic device of the present invention comprises: a memory and a processor coupled to each other, wherein the memory stores program data, and the processor calls the program data to execute the method described above.

[0028] The computer-readable storage medium of the present invention stores program data thereon, and is characterized in that the program data implements the method described above when executed by a processor.

[0029] The method of the present invention first establishes the system's dynamic control equations based on the dynamic characteristics of an asymmetric circular plate structure with weak damping and an attached dynamic vibration absorber. Combining boundary conditions and the distribution of the external excitation force, the Lagrange daily coefficient variation method is used to solve the circular plate's lateral vibration displacement, as well as the modal shapes and forced vibration response of the coupled system. The degenerate mode separation and eigenvalue trajectory deflection phenomena of the circular plate bending caused by the spring-mass dynamic vibration absorber are then analyzed to obtain approximate degenerate modes of the system. The variation of the circular plate's eigenfrequency with the absorber's spring stiffness and mass parameters is determined to regulate the propagation of traveling waves. Finally, optimization conditions for traveling waves in a circumferentially rotating structure are proposed, and a standing wave ratio function and a traveling wave exponential cost function are constructed. Approximate degenerate orthogonal modes of the circular plate and absorber coupling system are selected, the external excitation frequency is optimized, and the traveling wave dynamic response of the circular plate under driver excitation is calculated to verify the performance of the traveling wave response of the asymmetric circular plate structure.

[0030] The beneficial effects of the present invention are:

[0031] By rationally selecting the spring stiffness and mass coefficient of the dynamic vibration absorber and optimizing the excitation amplitude, frequency, and phase difference of the three exciters, this method achieves efficient control of circumferentially rotating traveling waves in a circular plate structure even when cyclic symmetry is broken. This method effectively improves the driving efficiency of traveling waves in asymmetric circular plate structures and enables precise control of structural traveling waves. This method provides new insights and important technical support for piezoelectric actuator design, precision drive and control technology, structural vibration suppression, and optimized energy distribution in practical engineering applications.

[0032] The generation mechanism and dynamic control method of traveling waves in the plate and shell structure of the present invention can be widely used in the fields of piezoelectric drives, near-field acoustic levitation motors, micro-robot drives, acoustic impedance tubes and engine rotating blade structures, and are of great significance to the application needs of national defense science and technology, aerospace, and intelligent manufacturing. Specifically, the present invention precisely controls the traveling wave response of an asymmetric, damped circular plate under the action of an additional dynamic vibration absorber to meet the needs of high-precision, high-reliability and precision driving technology in complex working environments. By rationally designing the external excitation frequency, phase and position, efficient propagation of traveling waves in an asymmetric circular plate structure is achieved. The present invention has broad application prospects in the fields of precision drive and control technology, structural vibration suppression and energy optimization distribution. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is a schematic diagram of an asymmetric circular plate structure with an additional dynamic vibration absorber;

[0034] Figure 2 (a) is when the spring stiffness ratio is μ k =10, the characteristic frequency of the coupling system between the circular plate and the dynamic vibration absorber changes with the mass ratio μ m Schematic diagram of the changing law of Figure 2 (b) is a schematic diagram showing the influence of the absorber mass ratio on the first nine modes of the circular plate;

[0035] Figure 3 It is a schematic diagram of the variation of standing wave ratio function and traveling wave index with excitation frequency and damping coefficient, where: Figure 3 (a) is a schematic diagram showing the variation of the standing wave ratio function with the damping coefficient C under the action of uniformly distributed drivers. Figure 3 (b) is a schematic diagram showing the variation of the traveling wave index with the damping coefficient C under the action of uniformly distributed drivers. Figure 3 (c) is a schematic diagram showing the variation of the standing wave ratio function with the damping coefficient C under the action of non-uniformly distributed drivers. Figure 3 (d) is a schematic diagram of the variation of the traveling wave index with the damping coefficient C under the action of non-uniformly distributed drivers;

[0036] Figure 4 There is damping C = 0.1N·s / m·m 3 Schematic diagram of the vibration displacement response of a circular plate along the radius r = 0.8R, where: Figure 4 (a) is a schematic diagram of the vibration displacement response of a circular plate along the radius r = 0.8R when the driving parameter ω = 19.341 rad / s under the action of a uniformly distributed driver. Figure 4 (b) is a schematic diagram of the vibration displacement response of the circular plate along the radius r = 0.8R when the driving parameter ω = 19.429 rad / s under the action of the non-uniformly distributed driver. DETAILED DESCRIPTION

[0037] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0038] The present invention proposes a method for exciting a driver with a phase difference, generating circumferential rotating traveling waves in an asymmetric circular plate structure of an additional dynamic vibration absorber. By constructing a cost function, a traveling wave driver for the circular plate structure is realized. The specific implementation steps of the method for controlling circumferential rotating traveling waves in the circular plate structure of the additional dynamic vibration absorber of the present invention are as follows:

[0039] 1) Establish a dynamic model for a coupled system consisting of a circular plate structure with attached dynamic vibration absorbers. The circular plate structure is horizontally supported on the top surfaces of several exciters, each of which provides external excitation force. The dynamic vibration absorbers are attached to the top surface of the circular plate structure, thus forming a coupled system. The dynamic model of the coupled system includes antisymmetric modal characteristic equations and symmetric modal characteristic equations, as follows:

[0040] a) Antisymmetric modal characteristic equation Δ 1n as follows:

[0041]

[0042] Where ω is the excitation frequency of the external excitation force applied by the exciter; and are the contributions of the nth-order Bessel functions of the first and second kinds caused by the edge bending moment of the circular plate, respectively; and are the contributions of the nth-order first and second modified Bessel functions caused by the edge bending moment of the circular plate, respectively; and are the contributions of the nth-order Bessel functions of the first and second kinds caused by the shear force at the edge of the circular plate, respectively; and are the contributions of the first and second modified Bessel functions of the nth order caused by the shear force at the edge of the circular plate, respectively; β is a constant coefficient, β = αR, α represents the spatial wave number, and R is the radius of the circular plate structure; J n () and Y n () are the nth order first and second kind Bessel functions, J n,β () and Y n,β () are the calculated values ​​of the nth order first and second kind Bessel functions at the edge of the circular plate respectively; I n () and K n () are the modified Bessel functions of the first and second kinds of the nth order, respectively, I n,β () and K n,β () are the calculated values ​​of the first and second modified Bessel functions of the nth order at the edge of the circular plate; ν is the Poisson's ratio.

[0043] b) The symmetric modal characteristic equation Δ2 is as follows:

[0044]

[0045] Where k0 is the stiffness of the dynamic vibration absorber; D is the bending stiffness of the circular plate structure D = Eh 3 / [12(1-ν 2 )], E is Young's modulus, h is the thickness of the circular plate structure; ω0 is the natural frequency of the dynamic vibration absorber, m0 is the mass of the dynamic vibration absorber; is the weight coefficient of the nth-order first-kind Bessel function caused by the coupling between the dynamic vibration absorber and the circular plate; r0 and φ0 are the radial distance and initial angular position of the dynamic vibration absorber on the circular plate structure, respectively; is the weight coefficient of the first-order modified Bessel function caused by the coupling between the dynamic vibration absorber and the circular plate; Φ 1n () and Φ 2n () are the nth order cosine function and sine function respectively; ε n is the nth order constant term.

[0046] A dynamic model of an asymmetric circular plate structure with an attached dynamic vibration absorber is established to reveal the vibration characteristics of the coupled system. Considering a uniform, linearly elastic, isotropic circular plate structure with a radius of R, and assuming the lateral vibration displacement of the circular plate structure is w, the system dynamics governing equation after coupling the dynamic vibration absorber is as follows:

[0047]

[0048] Where ρ is the mass density of the circular plate structure; and are the first-order derivative and second-order derivative of the lateral vibration displacement of the circular plate structure, respectively; r and φ are the radial distance and angular position of a point on the circular plate structure, respectively, and t is the time; C is the viscous damping coefficient of the circular plate structure; ▽ is the gradient operator; q() includes the coupling force between the circular plate structure and the dynamic vibration absorber and the external excitation force. The external excitation force is in the form of multi-point excitation and is as follows in the polar coordinate system:

[0049]

[0050] Where z(t) represents the vibration displacement of the dynamic vibration absorber at time t; N is the total number of exciters; δ() is the Dirac delta function; is the excitation amplitude of the i-th exciter; r i and φ i are the radial distance and angular position of the i-th exciter under the circular plate structure; e jωt is the external simple harmonic excitation force applied by the exciter at time t,

[0051] The vibration displacement z(t) of the dynamic vibration absorber satisfies the following governing equation:

[0052]

[0053] in, is the second-order derivative of the vibration displacement of the dynamic vibration absorber.

[0054] The circular plate structure is modeled using free boundary conditions as follows:

[0055]

[0056]

[0057] Among them, w r , M and V are the bending slope, bending moment and shear force at the boundary of the circular plate structure respectively; w φφ is the second-order partial derivative of the circular plate vibration displacement with respect to the coordinate φ, M| r＝R and V| r＝RThey represent the calculated values ​​of the circular plate bending moment and shear force at the edge position r=R respectively.

[0058] When the system reaches a steady-state response, the lateral displacement response of the circular plate structure and the vibration displacement of the dynamic vibration absorber are:

[0059] w(r,φ,t)=W(r,φ)e jωt

[0060]

[0061] Where W() is the vibration displacement of the circular plate structure.

[0062] The Lagrange daily coefficient variation method is used to solve the vibration displacement of the circular plate structure:

[0063]

[0064] α 4 =(ρhω 2 +jCω) / D

[0065] Among them, A jn and B jn are the first and second constant coefficients to be determined respectively; q jn () is the coupling external force term between the nth-order Bessel function and the cosine function (j=1) or the sine function (j=2).

[0066]

[0067] Where H() is a step function.

[0068] Substituting into the equation we can solve for the constant coefficient A jn and B jn as follows:

[0069]

[0070] in, is the coupling external force term between the nth-order Bessel function and the cosine function (j=1) or sine function (j=2) caused by the i-th exciter, The coupled external force term is the n-th order modified Bessel function of the i-th exciter and the cosine term (j=1) or the sine term (j=2).

[0071] Finally, the definition of β=αR is introduced to obtain the antisymmetric modal characteristic equation and symmetric modal characteristic equation of the circular plate and dynamic vibration absorber coupling system.

[0072] The present invention first establishes the system's dynamic control equations based on the dynamic characteristics of an asymmetric circular plate structure with an additional dynamic vibration absorber with weak damping. Combined with the boundary conditions and the distribution form of the external excitation force, the Lagrange daily coefficient variation method is used to solve the lateral vibration displacement of the circular plate, and the modal vibration shape and forced vibration response of the coupled system are solved.

[0073] like Figure 1 As shown, in a specific embodiment of the present invention, it is assumed that the radius of the circular plate structure with the center O is R = 1m, the thickness is h = 0.001m, the Young's modulus is E = 210GPa, the Poisson's ratio is ν = 0.33, and the material density is ρ = 7800kg / m 3 Meanwhile, a discrete spring-mass dynamic absorber was placed on the circular plate structure at a radial distance r0 = 0.45 m, with an initial angular position of φ0 = 0 rad. Given these known parameters, a dynamic model was used to analyze the system's modal characteristics and characteristic frequency curves. This revealed the characteristic frequency variation patterns, modal coupling characteristics, and forced vibration response of the coupled system.

[0074] 2) Under the excitation of an external excitation force, the approximate degenerate orthogonal modes of the coupled system are selected based on the dynamic model of the coupled system, and the vibration displacement of the circular plate structure is obtained, thereby determining the extreme values ​​of the displacement envelope of the vibration displacement of the circular plate structure; the approximate degenerate orthogonal modes of the coupled system of the circular plate and the vibration absorber are selected, and the external excitation frequency is optimized to verify the performance of the traveling wave response of the asymmetric circular plate structure. According to the dynamic model of the coupled system, the excitation amplitude and phase of each exciter are adjusted so that the approximate degenerate cosine and sine modes of the coupled system maintain a phase difference of π / 2. That is, the approximate degenerate orthogonal modes of the coupled system are selected as follows:

[0075]

[0076] Where N is the total number of exciters; is the excitation amplitude of the i-th exciter; Φ 1n () and Φ 2n () are the nth order cosine function and sine function respectively; φ i is the angular position of the i-th exciter under the circular plate structure.

[0077] In the case of three exciters, the three exciters are located on the same circle and at arbitrary angular positions. Separating the real and imaginary parts of the above equation yields:

[0078]

[0079] Where F1, F2, and F3 are the excitation forces of the three exciters, respectively; ψ2 and ψ3 are the phase differences between the first and second exciters and between the first and third exciters, respectively; φ2 and φ3 are the angular positions of the second and third exciters under the circular plate structure, respectively.

[0080] When the phase differences ψ2 and ψ3 between the three exciters are known, the conditions for the generation of circumferential rotating traveling waves in a symmetrical circular plate structure without a dynamic vibration absorber are as follows:

[0081]

[0082]

[0083] The vibration displacement W of the circular plate structure is as follows:

[0084]

[0085] where r and φ are the radial distance and angular position of a point on the circular plate structure, respectively; D is the bending stiffness of the circular plate structure; ε n is the nth order constant term; J is the coupling external force term between the nth-order Bessel function and the cosine function (j=1) or sine function (j=2) caused by the i-th exciter; n () and I n () are the nth order first kind Bessel function and the first kind modified Bessel function respectively; α is the spatial wave number; The coupled external force term of the n-th order modified Bessel function of the i-th exciter and the cosine term (j=1) or sine term (j=2); is the external force contribution of the i-th exciter in the radial direction.

[0086] When the dynamic vibration absorber is not attached, the stiffness k0 and mass m0 of the dynamic vibration absorber are both 0, that is, the vibration displacement of the circular plate structure without the dynamic vibration absorber is obtained.

[0087] This paper analyzes the degenerate mode separation and eigenvalue trajectory deflection of circular plate bending caused by a spring-mass dynamic vibration absorber, obtains the approximate degenerate mode of the coupled system, and determines the variation of the circular plate's eigenfrequency with the absorber's spring stiffness and mass parameters to control the propagation of traveling waves. The first and second dimensionless parameters μ are introduced. k and μ m , the influence of the spring stiffness and mass of the dynamic vibration absorber on the vibration mode of the coupled system is obtained as follows:

[0088] μ k =k0R 2 / D

[0089] μ m =m0 / mp

[0090] m p =πρhR 2

[0091] Among them, m p is the mass of the circular plate.

[0092] In the embodiment of the present invention, Figure 2 As shown in (a), when the spring stiffness coefficient μ of the dynamic vibration absorber is selected k = 10, the characteristic frequency f of the coupled system is given as the mass ratio μ m The frequency interval between the symmetric mode and the antisymmetric mode of the coupled system changes with the mass ratio μ m Increases and gradually decreases. By reasonably selecting the coupling parameters, the degenerate modes of the coupled system can be significantly regulated, laying an important foundation for dynamic regulation of the structural traveling wave propagation. Among them, the dotted line corresponds to the characteristic frequency of the dynamic vibration absorber, the dashed curve corresponds to the antisymmetric modal characteristic frequency of the uniform circular plate structure, and the solid line corresponds to the symmetric modal characteristic frequency of the coupled system of the circular plate and the dynamic vibration absorber. When the spring-mass dynamic vibration absorber is attached to the circular plate structure system, a new coupling characteristic frequency appears between the original plate vibration frequencies near the natural frequency of the dynamic vibration absorber. When the spring stiffness is greater than μ k When the value of is small, it can be observed that the mass ratio μ m When the stiffness ratio μ is increased, the intersection of the first three lowest frequency curves is very strong, and the lowest frequency curve and the natural frequency curve of the vibration absorber tend to merge, indicating that the vibration characteristics of the coupled system are dominated by the vibration absorber. However, for higher-order characteristic frequencies, due to the weak coupling effect between the dynamic vibration absorber and the circular plate, the vibration of the circular plate gradually becomes dominant. When the stiffness ratio μ is further increased, the vibration of the circular plate gradually becomes dominant. k When the dynamic vibration absorber has an influence on the vibration mode of the circular plate, the influence area of ​​the dynamic vibration absorber on the vibration mode of the circular plate gradually expands, and the natural frequency curve of the dynamic vibration absorber intersects with the characteristic frequency curve of the circular plate at more orders. In the area below the frequency curve of the dynamic vibration absorber, the characteristic frequency of the coupling system of the circular plate and the dynamic vibration absorber (symmetrical mode) is lower than the characteristic frequency of the uniform circular plate (antisymmetric mode), while in the area above the frequency curve of the vibration absorber, the characteristic frequency of the coupling system is higher than the characteristic frequency of the uniform circular plate.

[0093] Fix the spring rate to μ k =10, such as Figure 2 As shown in (b), the first nine modes of the circular plate and dynamic vibration absorber coupling system are given as the mass ratio μ m The change law of spring stiffness is μ k and mass ratio μ m will significantly affect the modal characteristics of the circular plate. mAs the mass ratio μ increases, the sixth-order coupled mode approaches the fifth-order antisymmetric uniform circular plate mode, and the sixth-order eigenfrequency is slightly higher than the eigenfrequency of the fifth-order uniform circular plate. m As the dynamic absorber parameters are appropriately selected, the asymmetric circular plate structure with the attached dynamic absorber exhibits degenerate modes similar to those of a uniform circular plate structure, i.e., approximately orthogonal cosine and sine modes at the same frequency, providing a theoretical basis for the subsequent realization of traveling waves.

[0094] 3) Establish a circumferential rotation traveling wave control model for the coupled system. Input the extreme value of the displacement envelope of the circular plate structure's vibration displacement into the circumferential rotation traveling wave control model. After processing, the amplitude envelope of the circular plate structure's vibration displacement is output, and then the excitation parameters of the external excitation force are obtained, thereby performing circumferential rotation traveling wave control on the circular plate structure with the dynamic vibration absorber attached. The circumferential rotation traveling wave control model is as follows:

[0095]

[0096] Among them, ω opt is the amplitude envelope of the vibration displacement of the circular plate structure; ω is the excitation frequency of the external excitation force applied by the exciter; SWR is the standing wave ratio function, and the standing wave ratio function SWR = 0 represents that the waveform envelope amplitude is constant, corresponding to a traveling wave, and SWR = 1 represents that the envelope amplitude is fluctuating, corresponding to a standing wave; TWI is the rotating traveling wave index; r0 is the radial distance at which the dynamic vibration absorber is arranged on the circular plate structure; φ is the angular position of a point on the circular plate structure; c max and c min are respectively the maximum and minimum values ​​of the displacement envelope of the vibration displacement of the circular plate structure, that is, the extreme values ​​of the displacement envelope of the vibration displacement of the circular plate structure.

[0097] Using the standing wave ratio function and the traveling wave index function, the conditions for the occurrence of circumferential traveling waves in the coupled system of the circular plate structure and the dynamic vibration absorber are established between two adjacent modal frequencies. Due to the introduction of the dynamic vibration absorber, modulated standing waves will be generated in the coupled system instead of perfect circumferential traveling waves. Then, by appropriately selecting the absorber mass and spring stiffness, two adjacent modes on two adjacent branches of the eigenvalue trajectory curve can be obtained, both of which have almost the same natural frequency ω (n,m) and Where (n,m) represents the mth root of the transcendental function associated with the nth-order Bessel function.

[0098] A cost function is introduced to evaluate the quality of the circumferential traveling wave in the asymmetric case, and the propagation direction of the traveling wave is predicted by the traveling wave index based on complex orthogonal decomposition. Assume that along a given circular plate radius r = r0, the displacement response of the circular plate is a complex time domain signal Z, and the time domain signal is sampled at time interval Δt throughout the entire period, with the number of sampling points being The signal is discretized in the spatial domain along the circumferential direction to construct the complex correlation matrix R. The circumferential rotation traveling wave index of the asymmetric circular plate structure with an attached dynamic vibration absorber is TWI, which is as follows:

[0099]

[0100] Where cond() is the condition number of the matrix, real() and imag() represent the real and imaginary parts respectively; u1 is the first eigenvector of the complex correlation matrix R; It represents the number of sampling points sampled at time interval Δt in the entire cycle; is the complex conjugate of the complex time domain signal Z; the traveling wave index corresponding to a pure traveling wave is TWI=1, and the traveling wave index corresponding to a pure standing wave is TWI=0.

[0101] The excitation parameters of the external excitation force include the excitation force amplitude, excitation frequency, phase difference and angular position of each exciter on the circular plate structure.

[0102] Finally, the present invention proposes the optimization conditions for the traveling waves of the circumferentially rotating structure, constructs the standing wave ratio function and the traveling wave index cost function, selects the approximate degenerate orthogonal mode of the coupling system of the circular plate and the dynamic vibration absorber, optimizes the external excitation frequency, calculates the traveling wave dynamic response of the circular plate under the excitation of the driver, and verifies the performance of the traveling wave response of the asymmetric circular plate structure.

[0103] After obtaining the amplitude envelope of the circular plate's vibration displacement, we first examine the effect of the excitation frequency on the traveling wave response of an asymmetric circular plate coupled to a dynamic vibration absorber, maintaining a fixed radius of r = 0.45R and varying the driver's excitation frequency near the plate's undamped natural frequency. Three spatially evenly distributed exciters are used, with the following parameters defined:

[0104] F1=F2=F3=5N

[0105] (φ1,φ2,φ3)=(ψ1,ψ2,ψ3)=(0,2π / 3,4π / 3)rad

[0106] like Figure 3 (a) and Figure 3As shown in (b), if the external damping of the coupled system is weak, the standing wave ratio function takes a local minimum at the undamped natural frequency of the structure, and the traveling wave index decreases with increasing frequency. In the case of weak damping, the standing wave ratio curve has one local minimum and two local maxima, while in the case of strong damping, there is only a single local maximum, the standing wave ratio function is close to SWR=0, and the traveling wave index is close to TWI=1, indicating that the circular plate responds to a near-perfect circumferential rotating traveling wave.

[0107] Secondly, three non-uniformly distributed exciters are used, and the parameters are defined as follows:

[0108] (F1,F2,F3)=(5,-1.093,4.553)N

[0109] (φ1,φ2,φ3)=(0,5π / 9,6π / 5)rad

[0110] (ψ1,ψ2,ψ3)=(0,2π / 3,4π / 3)rad

[0111] like Figure 3 (c) and Figure 3 As shown in (d), in the undamped case, the standing wave ratio function and the traveling wave index reach their minimum and maximum values, respectively, at a frequency between two adjacent natural frequencies. The cost function reaches only one local maximum as the damping coefficient increases. Because the nonuniform actuator arrangement destroys the structural cyclic symmetry, the amplitude and propagation direction of the traveling wave response are very sensitive to the excitation frequency. Increasing the external viscous damping coefficient can lead to detuning of the traveling wave waveform.

[0112] In the case of broken cyclic symmetry of the structure, the circumferential traveling wave of the circular plate structure can still be effectively regulated by designing the mechanical properties of the circular plate and the spring-mass absorber. The optimal excitation frequency can be obtained by optimizing the standing wave ratio function and the traveling wave index near the two adjacent modes of the coupled system, such as Figure 4 (a) and Figure 4 (b) shows that the actuator is uniformly and non-uniformly distributed and has a damping of C = 0.1N·s / m·m 3 In this case, it can be seen from the vibration displacement response of the circular plate along the radius r = 0.8R when the driving parameter ω = 19.341 rad / s and the vibration displacement response of the circular plate along the radius r = 0.8R when the driving parameter ω = 19.429 rad / s that the circular plate structure can still generate high-quality modal traveling waves, and the strong asymmetry will cause the circular plate to have modulated standing waves.

[0113] like Figure 3 and Figure 4As shown in the figure, based on the analysis of modal characteristics and the variation of coupling parameters, the coupling parameters are further optimized to achieve spatial separation and constrained control of structural waves. The standing wave ratio function and the traveling wave index show significant regularity near the characteristic frequency of the coupled system. Under the condition of symmetrically distributed driver arrangement, the asymmetric circular plate has a near-perfect circumferential rotation traveling wave response near the undamped natural frequency of the coupled system. Under the condition of asymmetrically distributed driver arrangement, the amplitude and propagation direction of the traveling wave response are very sensitive to the excitation frequency because the non-uniform driver arrangement destroys the cyclic symmetry. The increase in the external viscous damping coefficient will lead to detuning of the traveling wave waveform.

[0114] The method of the present invention first establishes the system's dynamic control equations based on the dynamic characteristics of an asymmetric circular plate structure with weak damping and an attached dynamic vibration absorber. Combining boundary conditions and the distribution of the external excitation force, the Lagrange daily coefficient variation method is used to solve the circular plate's lateral vibration displacement, as well as the modal shapes and forced vibration response of the coupled system. The degenerate mode separation and eigenvalue trajectory deflection phenomena of the circular plate bending caused by the spring-mass dynamic vibration absorber are then analyzed to obtain approximate degenerate modes of the system. The variation of the circular plate's eigenfrequency with the absorber's spring stiffness and mass parameters is determined to regulate the propagation of traveling waves. Finally, optimization conditions for traveling waves in a circumferentially rotating structure are proposed, and a standing wave ratio function and a traveling wave exponential cost function are constructed. Approximate degenerate orthogonal modes of the circular plate and absorber coupling system are selected, the external excitation frequency is optimized, and the traveling wave dynamic response of the circular plate under driver excitation is calculated to verify the performance of the traveling wave response of the asymmetric circular plate structure.

[0115] The embodiments described in this specification are merely examples of implementations of the present invention, and the scope of protection of the present invention should not be construed as limited to the specific forms and parameters described in the embodiments. The scope of protection of the present invention covers, but is not limited to, vibration control technologies and traveling wave drive schemes in the following structural forms: uniform circular plates, circular ring structures, circular plates with complex boundary conditions, multi-layer composite circular plates, and asymmetric structures with localized enhancement areas or multiple coupling modes.

[0116] Those skilled in the art will appreciate that the embodiments of the present application can be provided as methods, systems or computer program products. Therefore, the application can adopt the form of a complete hardware embodiment, a complete software embodiment or an embodiment in combination with software and hardware. Moreover, the application can adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, optical storage, etc.) that contain computer-usable program code. The scheme in the embodiments of the present application can be implemented in various computer languages. The application is described according to the flow chart of the method, system and computer program product of the embodiments of the present application.

[0117] Although the preferred embodiments of the present application have been described, those skilled in the art may make additional changes and modifications to these embodiments once they have learned the basic creative concepts. Therefore, the present invention is intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present application.

[0118] Obviously, those skilled in the art may make various changes and modifications to the present application without departing from the spirit and scope of the present application. Thus, if these modifications and variations of the present application fall within the scope of the equivalent technology of the present invention, the present application is intended to include these modifications and variations.

Claims

1. A method for controlling circumferential rotating traveling waves in a circular plate structure with an additional dynamic vibration absorber, characterized in that: include: 1) Establish a dynamic model of the coupled system consisting of a circular plate structure with a dynamic vibration absorber attached; 2) Under the excitation of the external excitation force, the approximate degenerate orthogonal mode of the coupled system is selected according to the dynamic model of the coupled system, and the vibration displacement of the circular plate structure is obtained, and then the extreme value of the displacement envelope of the vibration displacement of the circular plate structure is determined; 3) A circumferential rotating traveling wave control model of the coupled system is established, and the extreme values ​​of the displacement envelope of the vibration displacement of the circular plate structure are input into the circumferential rotating traveling wave control model. After processing, the amplitude envelope of the vibration displacement of the circular plate structure is output, and then the excitation parameters of the external excitation force are obtained, thereby performing circumferential rotating traveling wave control on the circular plate structure with an attached dynamic vibration absorber.

2. The method for controlling circumferential rotating traveling waves in a circular plate structure of an additional dynamic vibration absorber according to claim 1, characterized in that: In the step 1), the circular plate structure is horizontally supported on the top surfaces of a plurality of exciters, each of which provides an external excitation force, and the dynamic vibration absorber is attached to the top surface of the circular plate structure, thereby forming a coupling system.

3. The method for controlling circumferential rotating traveling waves in a circular plate structure of an additional dynamic vibration absorber according to claim 2, characterized in that: In the step 1), the dynamic model of the coupled system includes an antisymmetric modal characteristic equation and a symmetric modal characteristic equation, which are as follows: a) Antisymmetric modal characteristic equation Δ 1n as follows: Where ω is the excitation frequency of the external excitation force applied by the exciter; and are the contributions of the nth-order Bessel functions of the first and second kinds caused by the edge bending moment of the circular plate, respectively; and are the contributions of the nth-order first and second modified Bessel functions caused by the edge bending moment of the circular plate, respectively; and are the contributions of the nth-order Bessel functions of the first and second kinds caused by the shear force at the edge of the circular plate, respectively; and are the contributions of the first and second modified Bessel functions of the nth order caused by the shear force at the edge of the circular plate, respectively; β is a constant coefficient, β = αR, α represents the spatial wave number, and R is the radius of the circular plate structure; J n () and Y n () are the nth order first and second kind Bessel functions, J n,β () and Y n,β () are the calculated values ​​of the nth order first and second kind Bessel functions at the edge of the circular plate respectively; I n () and K n () are the modified Bessel functions of the first and second kinds of the nth order, respectively, I n,β () and K n,β () are the calculated values ​​of the first and second modified Bessel functions of the nth order at the edge of the circular plate; ν is the Poisson's ratio; b) The symmetric modal characteristic equation Δ2 is as follows: Wherein, k0 is the stiffness of the dynamic vibration absorber; D is the bending stiffness of the circular plate structure; ω0 is the natural frequency of the dynamic vibration absorber; is the weight coefficient of the nth-order first-kind Bessel function caused by the coupling between the dynamic vibration absorber and the circular plate; r0 and φ0 are the radial distance and initial angular position of the dynamic vibration absorber on the circular plate structure, respectively; is the weight coefficient of the first-order modified Bessel function caused by the coupling between the dynamic vibration absorber and the circular plate; Φ 1n () and Φ 2n () are the nth order cosine function and sine function respectively; ε n is the nth order constant term.

4. The method for controlling circumferential rotating traveling waves in a circular plate structure of an additional dynamic vibration absorber according to claim 2, characterized in that: In step 2), the excitation amplitude and phase of each exciter are adjusted according to the dynamic model of the coupled system so that the approximately degenerate cosine and sine modes of the coupled system maintain a phase difference of π / 2. That is, the approximately degenerate orthogonal modes of the coupled system are selected as follows: Where N is the total number of exciters; is the excitation amplitude of the i-th exciter; Φ 1n () and Φ 2n () are the nth order cosine function and sine function respectively; φ i is the angular position of the i-th exciter under the circular plate structure.

5. The method for controlling circumferential rotating traveling waves in a circular plate structure of an additional dynamic vibration absorber according to claim 4, characterized in that: In the step 2), the vibration displacement W of the circular plate structure is as follows: where r and φ are the radial distance and angular position of a point on the circular plate structure, respectively; D is the bending stiffness of the circular plate structure; ε n is the nth order constant term; J is the coupling external force term between the nth-order Bessel function and the cosine function (j=1) or sine function (j=2) caused by the i-th exciter; n () and I n () are the nth order first kind Bessel function and the first kind modified Bessel function respectively; α is the spatial wave number; The coupled external force term of the n-th order modified Bessel function of the i-th exciter and the cosine term (j=1) or sine term (j=2); is the external force contribution of the i-th exciter in the radial direction.

6. The method for controlling circumferential rotating traveling waves in a circular plate structure of an additional dynamic vibration absorber according to claim 1, characterized in that: In step 3), the circumferential rotating traveling wave control model is as follows: Among them, ω opt is the amplitude envelope of the vibration displacement of the circular plate structure; ω is the excitation frequency of the external excitation force applied by the exciter; SWR is the standing wave ratio function; TWI is the rotating traveling wave index; r0 is the radial distance of the dynamic vibration absorber on the circular plate structure; φ is the angular position of a point on the circular plate structure; c max and c min are respectively the maximum and minimum values ​​of the displacement envelope of the vibration displacement of the circular plate structure, that is, the extreme values ​​of the displacement envelope of the vibration displacement of the circular plate structure.

7. The method for controlling circumferential rotating traveling waves in a circular plate structure of an additional dynamic vibration absorber according to claim 1, characterized in that: In the step 4), the excitation parameters of the external excitation force include the excitation force amplitude, excitation frequency, phase difference and angular position of each exciter on the circular plate structure.

8. An electronic device, characterized in that: include: A memory and a processor coupled to each other, wherein the memory stores program data, and the processor calls the program data to execute the method according to any one of claims 1 to 7.

9. A computer-readable storage medium having program data stored thereon, characterized in that: When the program data is executed by a processor, the method according to any one of claims 1 to 7 is implemented.

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