Method for eliminating inherent frequency splitting of circular periodic structure based on grouping characteristics

Abstract

The present invention discloses a method for eliminating the natural frequency splitting of a circular periodic structure based on grouping characteristics. The method includes: constructing an additional mass block distribution function based on grouping characteristics in the inertial polar coordinate system, including the in-group uniform distribution and the in-group progressive distribution methods; applying Hamilton's principle in the inertial polar coordinate system to establish a dimensionless partial differential vibration equation for the out-of-plane bending vibration of the circular periodic structure; using Galerkin discretization and classical vibration theory to solve and obtain an analytical expression of the dimensionless natural frequency corresponding to the vibration wave number n, and determining the frequency splitting elimination condition; based on the frequency splitting elimination condition, determining specific elimination rules for the two in-group distribution methods. The present invention intuitively gives the natural frequency in an analytical form of the circular periodic structure, which can be substituted with actual parameters to directly solve and verify whether the natural frequency splits, and has characteristics such as universality, economy, and high efficiency.

Description

Technical Field

[0001] The present invention relates to the field of mechanical system vibration, and particularly relates to a method for eliminating the natural frequency splitting of a circular periodic structure based on grouping characteristics. Background Art

[0002] Circular periodic structures are important structures widely used in fields such as transportation, new energy, and aerospace. For example, the annular gears in gear transmission, the stator / rotor in motors, and the micro-rings in gyroscopes. The specific structure is composed of a standard circular structure and periodic units. The natural frequencies corresponding to the two orthogonal vibration modes of each order of elastic vibration of the standard circular structure are the same. However, for circular periodic structures, due to the periodic units (such as periodic mass, stiffness, or external load, etc.) changing the symmetry of the structure, it is possible that the natural frequencies corresponding to the two orthogonal vibration modes are different, that is, the phenomenon of natural frequency splitting occurs. The splitting of natural frequencies will increase the resonance range of the structure. For equipment with multi-frequency complex excitation sources, frequency splitting may lead to more unstable regions of the parametric excitation system, and further lead to problems of large-amplitude vibration and high-intensity noise.

[0003] Accordingly, many scholars have studied the natural frequencies and their splitting behaviors of circular periodic structures, and the results all show that the specific topological distribution of periodic units has a positive effect on eliminating frequency splitting [1-5] , in particular, for periodically spaced periodic units, the method for eliminating natural frequency splitting is to make the vibration wave number n of the circular periodic structure and half of the number of periodic units N / 2 satisfy a non-integer relationship.

[0004] It should be noted that the design or installation of most circular periodic structures is no longer limited to arranging periodic units in an equally spaced manner. Currently, there is little research on methods for eliminating the natural frequency splitting of circular periodic structures with non-equally spaced periodic units.

[0005] References

[0006] [1]R.Perrin, Selection rules for the splitting of the degenerate pairs of natural frequencies of thin circular rings, Acta Acust. United Ac. 25(1971)69–72.

[0007] [2]R.C.Yu,Jr.C.D.Mote,Vibration and parametric excitation in asymmetric circular plates under moving loads,J.Sound Vib.119(1987)409–427.

[0008] [3]X.Wu,R.G.Parker,Vibration of rings on a general elastic foundation,J.Sound Vib.295(2006)194–213.

[0009] [4]Y.Wang,S.Wang,D.Zhu,Dual-mode frequency splitting elimination of ring periodic structures via feature shifting,Proc.Inst.Mech.Eng.Part C J.Mech.Eng.Sci.230(2016)3347–3357.

[0010] [5]N.Gao,S.Wang,J.Wang,Free and parametric vibrations of an elastic ring structure induced by rotating internal and external time-varying excitations,Nonlinear Dyn.110(2022)2079–2099. Summary of the Invention

[0011] The present invention provides a method for eliminating the natural frequency splitting of a ring-shaped periodic structure based on grouping characteristics. The present invention proposes two distribution methods of additional mass units within a group based on grouping characteristics, namely, uniform distribution within the group and progressive distribution within the group. Based on the thin ring hypothesis, a dynamic model of the out-of-plane bending vibration of the ring-shaped periodic structure is established, and the analytical expression of the natural frequency is derived by using the Galerkin discretization method and classical vibration theory, and the rule for eliminating frequency splitting is determined, as described in detail below:

[0012] A method for eliminating the natural frequency splitting of a ring-shaped periodic structure based on grouping characteristics, the method comprising the following steps:

[0013] Based on the inertial polar coordinate system, construct an additional mass block distribution function based on the grouping characteristics, including the within-group uniform distribution and the within-group progressive distribution methods;

[0014] Based on the inertial polar coordinate system, apply Hamilton's principle to establish a dimensionless partial differential vibration equation for the out-of-plane bending vibration of the ring-shaped periodic structure;

[0015] Use Galerkin discretization and classical vibration theory to solve for the analytical expression of the dimensionless natural frequency corresponding to the vibration wave number n, and determine the condition for eliminating frequency splitting;

[0016] Based on the condition for eliminating frequency splitting, determine the specific elimination rules for the two within-group distribution methods.

[0017] The beneficial effects of the technical solution provided by the present invention are as follows:

[0018] 1. For a ring-shaped periodic structure that is difficult to disassemble or has been machined and formed, the lack of symmetry may lead to frequency splitting of the natural frequency, increasing the potential resonance range; according to the theoretical modeling, derivation process, and specific elimination rules of the present invention, the frequency splitting of the natural frequency can be quickly and effectively eliminated;

[0019] 2. For the resonance problems that may exist in related equipment containing ring-shaped periodic structures and are difficult to explain, the present invention can be used to predict and adjust the resonance range of actual rotating machinery, and can also estimate the unstable regions that may be generated in mechanical equipment containing such structures under the influence of time-varying complex excitations, providing theoretical guidance for determining the operating condition range;

[0020] 3. The present invention intuitively gives the natural frequency in an analytical form of the ring-shaped periodic structure, which can substitute actual parameters to directly solve and verify whether the natural frequency splits, and has the characteristics of universality, economy, and high efficiency;

[0021] 4. The grouping mode of the additional mass blocks proposed by the present invention can ensure a certain symmetry of the ring-shaped periodic structure, can be used to expand other methods for eliminating frequency splitting of the natural frequency, and can also provide new ideas and inspirations for the flexible design of similar structures. Description of the Drawings

[0022] Figure 1 It is a schematic diagram of the ring-shaped periodic structure with additional mass blocks based on grouping characteristics provided by the present invention;

[0023] Figure 2a It is a schematic diagram of the natural frequency when N1 = 2 and N2 = 3 are uniformly distributed within the group at the vibration wave number n = 2 according to the present invention;

[0024] Figure 2b It is a schematic diagram of the natural frequency when N1 = 2 and N2 = 3 are uniformly distributed within the group at the vibration wave number n = 3 according to the present invention;

[0025] Figure 3aSchematic diagram of the natural frequencies when N1 = 3 and N2 = 3 are progressively distributed within the groups at the vibration wave number n = 2 according to the present invention;

[0026] Figure 3b Schematic diagram of the natural frequencies when N1 = 3 and N2 = 3 are progressively distributed within the groups at the vibration wave number n = 3 according to the present invention;

[0027] Figure 4a Schematic diagram of the natural frequencies when N1 = 3 and N2 = 4 are progressively distributed within the groups at the vibration wave number n = 2 according to the present invention;

[0028] Figure 4b Schematic diagram of the natural frequencies when N1 = 3 and N2 = 4 are progressively distributed within the groups at the vibration wave number n = 3 according to the present invention. Detailed implementation manners

[0029] To make the objectives, technical solutions and advantages of the present invention clearer, the following further describes the implementation manners of the present invention in detail.

[0030] The embodiment of the present invention provides a method for eliminating the natural frequency splitting of a circular periodic structure based on grouping characteristics. According to the distribution function of additional mass blocks based on grouping characteristics, the dimensionless partial differential vibration equation of the circular periodic structure is established by using the Hamilton principle, and then the analytical expression of the natural frequency is derived by using the Galerkin discretization method and the classical vibration theory, and further the specific rules for eliminating frequency splitting related to the distribution function are determined. This method is simple, fast and efficient, and can be used to eliminate the natural frequency splitting of similar periodic structures, and can also be used to guide the preliminary design or later installation of similar periodic structures.

[0031] The embodiment of the present invention is carried out according to the following steps:

[0032] Step 1: Based on the inertial polar coordinate system, construct the mass distribution function of additional mass blocks (a term well-known to those skilled in the art) based on grouping characteristics, including two within-group distribution methods.

[0033] Among them, the origin of the coordinate system o-rθz is located at the geometric center of the circular periodic structure, and the polar axis points to the lower edge of the first mass block in the first group. The mass distribution function m(θ) of the additional mass blocks based on grouping characteristics is:

[0034]

[0035] In the formula, m * is the mass of the additional mass block, N1 is the number of groups, N2 is the number within the group, θ is an arbitrary angle, α i,j is the position angle of the jth mass block in the ith group, and δ() is the Dirac function.

[0036] The two in-group distribution methods based on the grouping characteristics are as follows:

[0037] (1) Uniform distribution within the group:

[0038] The arbitrary position angle α of the additional mass block i,j satisfies: α i,j = 2π(i - 1) / N1 + (j - 1)α, where α is the adjacent interval angle within the group.

[0039] (2) Progressive distribution within the group:

[0040] The arbitrary position angle α of the additional mass block i,j satisfies: α i,j = 2π(i - 1) / N1 + (j - 1)[γ + (j - 2)θ ia , where γ is the initial angle (the included angle between the first and second additional mass blocks within the group), and θ ia is the progressive angle.

[0041] Step 2: Based on the inertial polar coordinate system, apply Hamilton's principle to establish the dimensionless partial differential vibration equation for the out-of-plane bending vibration of the ring-shaped periodic structure.

[0042] The schematic diagram of the ring-shaped periodic structure with additional mass blocks based on the grouping characteristics is as shown in Figure 1 . o-rθz is the inertial polar coordinate system. The ring-shaped periodic structure includes: outer ring 1, additional mass blocks 2, thin web 3, and inner ring 4. R1, R2, and h1 represent the inner circle radius, outer circle radius, and axial thickness of the thin web, respectively. R, h2, and c represent the neutral circle radius, axial thickness, and radial width of the outer ring. α 1,j , α i,j and α N1,j represent the position angles of the jth mass block in the first group, the jth mass block in the ith group, and the jth mass block in the N1th group, respectively. The A-A section shows the neutral plane of the outer ring.

[0043] Use the energy method to establish the dynamic model of the ring-shaped periodic structure. The specific energy expression is:

[0044] (1) Kinetic energy:

[0045]

[0046] In the formula, m0 (m0 = ρA) is the mass per unit arc length of the outer ring, ρ and A (A = ch2) are the material density and cross-sectional area, respectively, is the equivalent mass per unit arc length of the thin web, and w is the out-of-plane vibration displacement.

[0047] (2) Potential energy:

[0048] The tangential strain energy of the ring-shaped periodic structure at the neutral plane can be expressed as:

[0049]

[0050] Wherein, E and are the Young's modulus and the moment of inertia of the cross-section, respectively.

[0051] The axial support potential energy of the outer ring by the thin web can be expressed as:

[0052]

[0053] Wherein, is the equivalent support stiffness per unit arc length.

[0054] (3) Partial differential vibration equation:

[0055] Based on the Hamilton principle

[0056]

[0057] The partial differential vibration equation of the ring-shaped periodic structure can be derived as follows:

[0058]

[0059] Introducing the dimensionless parameters and The dimensionless vibration equation can be obtained. For the sake of convenience of representation, replace and with t and w respectively, and there is:

[0060]

[0061] Step 3: Adopt the Galerkin discretization method and the classical vibration theory to derive the analytical expression of the dimensionless natural frequency corresponding to the vibration wave number n, and then determine the method for eliminating frequency splitting.

[0062] Assume that the out-of-plane vibration response of the ring-shaped periodic structure is:

[0063]

[0064] Wherein, W(t) is a complex function of time t, i is the imaginary unit, and "~" represents the conjugate of the function.

[0065] Substitute the above formula into Equation (6) and take the inner product with e inθ to obtain:

[0066]

[0067] Wherein, the second derivative of the function with respect to time; The second derivative of the conjugate function with respect to time; m t is the total weight of the additional mass block, which can be specifically expressed as:

[0068]

[0069] where m * is the actual weight of the additional mass block.

[0070] Substitute W(t) = w Re (t) + iw Im (t) into Equation (8), then separate the real and imaginary parts and further organize it into matrix form, we get:

[0071]

[0072] where

[0073]

[0074] where w Re (t) and w Im (t) are the real and imaginary part vibration amplitudes of the complex function W(t) respectively; M is the unit mass matrix; q(t) is the vibration amplitude matrix; K is the stiffness matrix; C Δ and S Δ represent the total mass of the additional mass block respectively, which can be expressed as:

[0075]

[0076] where m * is the mass of the additional mass block, N1 is the number of groups, N2 is the number within each group, α i,j is the position angle of the j-th mass block in the i-th group.

[0077] According to the classical vibration theory, substituting into Equation (9) gives the eigenvalue problem:

[0078]

[0079] where ω n is the dimensionless natural frequency; v is the vibration amplitude vector.

[0080] Let the determinant of the above equation be zero, we get the characteristic equation:

[0081]

[0082] where

[0083]

[0084] where and are the coefficients of the characteristic equation.

[0085] The solution of the above equation is the dimensionless natural frequency corresponding to the vibration wave number n, and the analytical expression is:

[0086]

[0087] According to the above equation, to eliminate the natural frequency splitting, it is necessary to ensure that C Δ = S Δ = 0.

[0088] Step 4: Based on the frequency splitting elimination method, determine the specific elimination rules for the two intra-group distribution methods.

[0089] Substitute the specific position angles of the intra-group uniform distribution and the intra-group progressive distribution into C Δ and S Δ so that they satisfy C Δ = S Δ = 0, and then the specific elimination rules can be obtained.

[0090] (1) Intra-group uniform distribution:

[0091] The intra-group uniform distribution means that the position angle of the additional mass unit satisfies α i,j = 2π(i - 1) / N1 + (j - 1)α. Under this distribution, C Δ and S Δ can be further expressed as:

[0092]

[0093] In the formula,

[0094]

[0095]

[0096] According to the above equation, when any of the following two conditions is satisfied, that is, condition (a): Condition (b): When both C △α = 0 and S △α = 0, the natural frequency splitting corresponding to the vibration wave number n can be eliminated. The specific elimination rules are shown in Table 1.

[0097] Condition (a) indicates that when the vibration wave number and half of the number of groups satisfy a non-integer multiple relationship, the natural frequency splitting is eliminated; condition (b) indicates that when condition (a) is not satisfied, adjusting the intra-group interval angle can also achieve the elimination of frequency splitting, that is, the interval angle α needs to satisfy an integer multiple relationship with π / (nN2) and a non-integer multiple relationship with π / n at the same time.

[0098] Table 1 Inherent frequency splitting elimination rules for uniform distribution within a group

[0099]

[0100] (1) Progressive distribution within a group

[0101] The progressive topological distribution within a group means that the position angle of the additional mass unit satisfies α i,j = 2π(i - 1) / N1 + (j - 1)[γ + (j - 2)θ ia , where θ ia is the progressive angle, which can be positive or negative. In particular, when θ ia = 0, the progressive distribution within a group degenerates into a uniform distribution within a group.

[0102] To ensure the progressive relationship, the position angles of adjacent additional mass blocks satisfy α i,j > α i,j-1 , that is, θ ia > -γ / (N2 - 2), and at the same time, the position angle of the last additional mass within the group satisfies α i,N2 < i2π / N1, that is, (N2 - 1)[γ + (N2 - 2)θ ia / 2] < 2π / N1. It should be noted that this distribution needs to satisfy that the number of units within the group is greater than or equal to three. Under this type of distribution, C Δ and S Δ can be expressed as:

[0103]

[0104] In the formula,

[0105]

[0106] According to the above formula, when any one of the following two conditions is satisfied, that is, condition (a): Condition (b): Both and the inherent frequency splitting corresponding to the vibration wave number n can be eliminated.

[0107] It should be noted that condition (b) will change according to N2. Formulas (14) and (15) respectively give the elimination rules when N2 = 3 and N2 = 4:

[0108]

[0109] In the formula,

[0110]

[0111] The specific elimination rules are listed in Table 2.

[0112] Table 2 Inherent frequency splitting elimination rules for the intra-group asymptotic distribution when N2 = 3 (k1 and k2 are integers)

[0113]

[0114]

[0115] Wherein,

[0116]

[0117] The specific elimination rules are listed in Table 3.

[0118] Table 3 Inherent frequency splitting elimination rules for the intra-group asymptotic distribution when N2 = 4

[0119]

[0120] To reveal the relationships between parameters such as the interval angle, vibration wave number, additional mass block weight, and number of groups and the inherent frequency splitting behavior, the inherent frequencies under the influence of different parameters are solved using the basic parameters of the ring-shaped periodic structure in Table 4, as shown in Figures 2 to 4.

[0121] Figure 2a and Figure 2b are the inherent frequencies of the uniform intra-group distribution with N1 = 2 and N2 = 3 varying with the interval angle at vibration wave numbers n = 2 and 3, respectively. When n = 2, the inherent frequency splitting is eliminated at α = π / 6 and π / 3 (Rule 2 in Table 1); when n = 3, it is eliminated at α = π / 9, 2π / 9, and 4π / 9 (Rule 2 in Table 1). Further observation shows that the weight of the additional mass block does not affect the existing inherent frequency splitting behavior, but only affects the degree of splitting, and the greater the weight, the greater the degree of splitting. In addition, the interval angle also affects the degree of splitting.

[0122] Figure 3a and Figure 3b are the inherent frequencies of the intra-group asymptotic topological distribution with N1 = N2 = 3 varying with the initial and asymptotic angles at vibration wave numbers n = 2 and 3, respectively. When n = 2, it satisfies Rule 1 in Table 3, and the inherent frequencies are always degenerate, as shown in Figure 3a ; when n = 3, the frequency splitting is eliminated at γ = π / 9, θ ia = π / 3 (Rule 2 in Table 2: k1 = 1 and k2 = 0) and γ = 4π / 9, θ ia = -π / 3 (Rule 4 in Table 2: k1 = 1 and k2 = 0), as shown in Figure 3b . Figure 4a and Figure 4b are the inherent frequencies of the intra-group asymptotic topological distribution with N1 = 3 and N2 = 4 varying with the initial and asymptotic angles at vibration wave numbers n = 2 and 3, respectively. When n = 2, it satisfies Rule 1 in Table 3, and the inherent frequencies are always degenerate, as shown in Figure 4aAs shown; when n = 3, the positions where the natural frequency splitting is eliminated are no longer discrete, but the initial angle and the asymptotic angle satisfy the relationship γ + θ ia = π / 6 to eliminate the frequency splitting (Rule 2 in Table 3).

[0123] Table 4 Basic parameters of the circular periodic structure

[0124]

[0125]

[0126] In summary, the embodiments of the present invention provide a method for eliminating the natural frequency splitting of a circular periodic structure based on grouping characteristics. According to the distribution function of the additional mass blocks based on grouping characteristics, the in-group uniform distribution and in-group asymptotic distribution methods can be applied to effectively eliminate the natural frequency splitting of the circular periodic structure. This method is simple, fast, and efficient, and can be used to eliminate the natural frequency splitting of similar periodic structures, and can also be used to guide the preliminary design or later installation of similar periodic structures.

[0127] Those skilled in the art can understand that the drawings are only schematic diagrams of a preferred embodiment, and the serial numbers of the above embodiments of the present invention are only for description and do not represent the advantages or disadvantages of the embodiments.

[0128] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for eliminating the natural frequency splitting of a cyclic periodic structure based on grouping characteristics, characterized in that The method includes the following steps: Based on the inertial polar coordinate system, construct an additional mass block distribution function based on the grouping characteristics, including the within-group uniform distribution and the within-group progressive distribution methods; Based on the inertial polar coordinate system, apply Hamilton's principle to establish a dimensionless partial differential vibration equation for the out-of-plane bending vibration of the circular periodic structure; Use Galerkin discretization and classical vibration theory to solve and obtain the analytical expression of the dimensionless natural frequency corresponding to the vibration wave number n, and determine the condition for eliminating frequency splitting; Based on the condition for eliminating frequency splitting, determine the specific elimination rules for the two within-group distribution methods.

2. A method for eliminating the inherent frequency splitting of a cyclic periodic structure based on grouping characteristics according to claim 1, characterized in that The additional mass block distribution function m(θ) based on the grouping characteristics is expressed in the inertial polar coordinate system o-rθz as: where m * is the mass of the additional mass block, N1 is the number of groups, N2 is the number of elements within a group, θ is an arbitrary angle, α i,j is the position angle of the j-th mass block within the i-th group, and δ() is the Dirac function; Position angle α i,j Expressed as: where α is the adjacent interval angle within the group, γ is the initial angle, and θ ia is the asymptotic angle.

3. A method for eliminating the inherent frequency splitting of a cyclic periodic structure based on grouping characteristics according to claim 1, characterized in that The dimensionless partial differential vibration equation for the out-of-plane bending vibration of the circular periodic structure is: where, m0 and m e are respectively the mass per unit arc length of the outer ring and the equivalent mass per unit arc length of the thin web, R is the radius of the neutral circle of the ring-shaped periodic structure, w is the out-of-plane vibration displacement, is the second derivative of the out-of-plane vibration displacement with respect to time t, E is the Young's modulus, I is the moment of inertia of the cross section, is the equivalent support stiffness per unit arc length of the thin web.

4. A method for eliminating the inherent frequency splitting of a cyclic periodic structure based on grouping characteristics according to claim 2, characterized in that, The analytical expression of the dimensionless natural frequency corresponding to the vibration wave number n is: In the formula, 5. A method for eliminating the inherent frequency splitting of a cyclic periodic structure based on grouping characteristics according to claim 4, characterized in that The method uses C Δ and S Δ as the judgment basis: when either C Δ or S Δ is non-zero, the natural frequency splits; when both C Δ and S Δ are zero, the splitting of the natural frequency is eliminated.

6. A method for eliminating the inherent frequency splitting of a cyclic periodic structure based on grouping characteristics according to claim 1, characterized in that, The natural frequency splitting elimination rule for the within-group uniform distribution is: Rule 1, the initial condition is 2n / N1≠integer; Rule 2, the initial condition is 2n / N1 = integer, and the subsidiary conditions are α / [π / (nN2)] = integer and α / (π / n)≠integer.

7. A method for eliminating the natural frequency splitting of the inherent cyclic periodic structure based on the grouping characteristics according to claim 1, characterized in that The natural frequency splitting elimination rule for the within-group progressive distribution is: 1) The natural frequency splitting elimination rule for the within-group progressive distribution when N2 = 3, where k1 and k2 are integers: Rule 1, the initial condition is 2n / N1≠integer; Rule 2, the initial condition is 2n / N1 = integer, The subsidiary conditions are satisfied when γ = (3k1 - 3k2 - 2)π / (3n) and θ ia = (3k2 - k1 + 2)π / n; Rule 3, the initial condition is 2n / N1 = integer, The subsidiary conditions are satisfied when γ = (3k1 - 3k2 - 1)π / (3n) and θ ia = (3k2 - k1 + 1)π / n; Rule 4, the initial condition is 2n / N1 = integer, The subsidiary conditions are satisfied when γ = (3k1 - 3k2 + 1)π / (3n) and θ ia = (3k2 - k1)π / n; Rule 5, the initial condition is 2n / N1 = integer, The subsidiary conditions are satisfied when γ = (3k1 - 3k2 - 1)π / (3n) and θ ia = (3k2 - k1 + 2)π / n; 2) The natural frequency splitting elimination rule for the within-group progressive distribution when N2 = 4: Rule 1, the initial condition is 2n / N1≠integer; Rule 2, the initial condition is 2n / N1 = integer, and the subsidiary condition is satisfied that (γ + θ ia ) / [π / (2n)] = odd number.

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