CN112257213A - Method for describing large-deflection vibration of rubber cylindrical shell - Google Patents

Abstract

A method for describing large-deflection vibration of a rubber cylindrical shell belongs to the field of mechanics and aims to solve the problem that a mathematical model is established to research nonlinear vibration.

Description

Method for describing large-deflection vibration of rubber cylindrical shell

Technical Field

The invention belongs to the field of mechanics, and relates to a method for describing large-deflection vibration of a rubber cylindrical shell.

Background

Rubber is a typical type of super-elastic material, and some important properties (such as high elasticity, large deformation and the like) of the rubber are indispensable parts in engineering. Accordingly, there is a wide range of products composed of these materials, such as tires, gaskets, oil pipes, and the like.Due to the excellent mechanical property, the cylindrical shell is widely applied to the fields of spacecrafts, conveying pipes, engine drums and the like. For thin shell structures and flexible structures, they typically exhibit a non-linear response to a perturbation stimulus.^[1]. In addition, the super-elastic cylindrical shell has the characteristics of geometric nonlinearity and physical nonlinearity, and the study on the nonlinearity, particularly chaotic vibration characteristics, has important significance.

In the field of mechanics, the plate-shell structure is the most economical structure in practical application, and the research on the chaotic vibration of the plate-shell has attracted wide attention. Wang et al^[2]The natural frequency, complex mode and critical velocity of the axially moving rectangular plate were studied. An equation of motion for the plate vibration is established using classical thin plate theory, discussing the effects of the plate distance ratio, motion speed, immersion depth ratio, boundary conditions, stiffness ratio, aspect ratio, and fluid plate density ratio on the free vibration of the moving plate-fluid system. Hao et al^[3]Nonlinear dynamic behaviors of the simply supported functional gradient material rectangular plate in a thermal environment under transverse and planar excitation are analyzed, and periodic solution, quasi-periodic solution and chaotic motion are given. Yang et al^[4]And simplifying by using the Flu gge shell theory and mode orthogonalization, and providing a unified solution for the vibration analysis of the cylindrical shell under the condition of general stress distribution. Li et al^[5]The global bifurcation and multi-pulse chaotic dynamics behaviors of the simply-supported rectangular sheet are researched by utilizing an expanded Melnikov method. Bich et al^[6]Based on an improved Donnell shell theory, nonlinear vibration of the functionally graded cylindrical shell under the action of axial and transverse loads is analyzed, and influences of pre-loading axial compression, functionally graded material characteristics and size ratio on shell behaviors are discussed. Sofiyev et al, using first order shear deformation theory and perturbation^[7]The nonlinear free vibration of the interaction of the functional gradient orthotropic cylindrical shell and the two-parameter elastic foundation is researched. Zhu et al^[8]The nonlinear free vibration behavior of the orthotropic piezoelectric cylindrical shell is researched, and the influence of surface parameters and geometric characteristics is analyzed. Amabili et al^[9]The nonlinear vibration of the water-filled cylindrical shell under the radial simple harmonic excitation is researched by an experimental and numerical method, and the water-filled cylindrical shell has the advantages of simple structure, high efficiency, low cost and low costChaotic motion is found in the frequency domain of the wave response. Yamaguchi et al^[10]Chaotic vibration of the flat cylindrical shell plate under simple harmonic transverse excitation is researched, and the influence of in-plane elastic constraint on shell chaos is discussed. Han et al^[11]The chaotic motion of the elastic cylindrical shell is researched by utilizing a kinetic equation containing quadratic and cubic nonlinear terms, and the effectiveness of analyzing the chaotic motion by using a monomodal model is briefly explained. Kryskoa et al^[12]Chaotic vibration of a closed cylindrical shell in a temperature field was studied. Zhang et al^[13]The resonance response and the chaotic dynamics of the composite material laminated cylindrical shell are researched, and a twinning phenomenon exists between the Pomeau-Manneville type intermittent chaos and the bifurcating of the multiple period. Li et al^[14]On the basis of a Reddy three-order shear theory, the nonlinear transient dynamic response of a functionally gradient material sandwich isotropic homogeneous material and a functionally gradient panel double-curved shell is analyzed by using a new displacement field for the first time. Amabili et al^[15]The geometric nonlinear response of the water-filled simple-supported cylindrical shell is researched, and the result shows that the response undergoes cycle doubling bifurcation, subharmonic response, quasi-periodic response and chaotic behavior along with the increase of the excitation amplitude.

In general, chaotic motion is a part of nonlinear vibration, and research on nonlinear behavior is the basis for analyzing chaotic phenomena. The cause of the non-linear response of the superelastic cylindrical shell is mainly two-fold: first, large deformation of the housing; the second is the nonlinear constitutive relation of the super-elastic material. Large deformations of the structure often introduce geometric non-linearity, which is a major difficulty in studying cylindrical shell non-linear vibrations. Amabili^[16]The original consistent first-order shear deformation theory of all nonlinear terms in displacement and rotation in a plane is retained, and the numerical application of the nonlinear forced vibration of the simple composite material laminated cylindrical shell is provided. In the framework of classical non-linear theory, Krysko et al^[17]The problem of complex vibration of the closed infinite-length cylindrical shell under the action of transverse local load is researched. Breslavsky et al^[18]The nonlinear response of the water-filled simple-supported cylindrical thin shell under multi-harmonic excitation is researched, and the nonlinear dynamic behavior is discussed. Du et al^[19]Using the Lagrange theory andthe multi-scale method researches the nonlinear forced vibration of the infinite-length FGM cylindrical shell and reveals a chaotic path of the system. In addition to geometric non-linearity caused by large deformations, material non-linearity is also an important factor in causing non-linear response. Because the super-elastic material has physical nonlinearity, the corresponding structure is easy to deform greatly, and geometric nonlinearity can be generated after deformation. Thus, studies on superelastic structures typically involve geometric and physical nonlinearities. Thus, a wide range of concerns have been raised regarding the nonlinear response of superelastic structures. Iglesias et al^[20]The large-amplitude axisymmetric free vibration of the incompressible superelasticity orthotropic cylindrical structure is researched, and the fact that the motion of the structure can evolve from periodic motion into quasi-periodic and chaotic motion is proved.

Wait for^[21]The nonlinear vibration behavior of the pre-stretched superelastic circular membrane under the action of limited deformation and varying transverse pressure is analyzed in detail. Breslavsky et al^[22]Free and forced nonlinear vibrations of the superelastic rectangular thin plate were studied and the structure found that the frequency shift between the low amplitude and large amplitude vibrations of the plate decreased with increasing initial deflection.

In the field of dynamics, much research has been done on cylindrical shells or superelastic structures, but few studies have been done on the dynamic behavior of superelastic cylindrical shells. Shahinpool et al^[23]Based on the finite deformation theory, the large-amplitude radial vibration of the super-elastic thin-walled tube is analyzed, and an accurate solution of the simplification problem is obtained. Wang et al^[24]A super-elastic cylindrical tube made of axially transverse isotropic compressible neo-Hookean materials is researched to move symmetrically and nonlinearly in the radial direction and the axial direction. Breslavsky et al^[25]The static and dynamic response of a circular cylindrical shell made of superelastic arterial material was studied, and complex nonlinear dynamic behavior in the driven mode and accompanying modal resonance states was discovered.

Disclosure of Invention

In order to solve the problem of establishing a mathematical model to be capable of researching nonlinear vibration, the invention provides the following technical scheme: a method for describing large-deflection vibration of a rubber cylindrical shell utilizes an energy principle to provide a nonlinear differential equation set for describing the motion of the cylindrical shell.

Has the advantages that:

the invention studies the problem of nonlinear vibration of a thin-walled rubber cylindrical shell made of a classical incompressible material under radial harmonic excitation. The nonlinear differential equation for describing the large-deflection vibration of the rubber cylindrical shell is obtained based on the Donnell nonlinear thin-shell theory, the Lagrange equation and the small strain hypothesis.

Drawings

FIG. 1: a schematic view of a cylindrical shell; (a) sign definition of relative size and displacement; (b) the cross-section in the x-direction is schematic.

FIG. 2: the natural frequency α and β of the casing corresponding to different hoop wave numbers n are 0.02 and 0.5, respectively.

FIG. 3: when the parameters alpha and beta take different values, the radial natural frequency changes, and (a) m is 1 and (b) m is 3.

FIG. 4: relationship between natural frequency and structural parameter, (a) α to ω, β is 0.5, m is 1, (b) α to ω, β is 2, m is 1, (c) β to ω, α is 0.005, m is 1, (d) β to ω, α is 0.02, m is 1.

FIG. 5: radial motion and excitation amplitude F of cylindrical shell_zThe bifurcation diagram of (2).

FIG. 6: poincare sections corresponding to different excitation amplitudes when the cylindrical shell generates chaotic motion, (a) F_z＝5.05N， (b)F_z＝7.3N，(c)F_z＝8.55N，(d)F_z＝9.8N。

FIG. 7: a bifurcation diagram depicting radial motion of the housing without excitation frequencies.

FIG. 8: poincare cross section for describing radial chaotic motion of shell under different excitation frequencies, (a) omega is 0.909 omega₁₄，(b) Ω＝0.998ω₁₄，(c)Ω＝1.001ω₁₄，(d)Ω＝1.0375ω₁₄。

FIG. 9: and when the thickness-diameter ratio takes different values, describing a bifurcation diagram of the radial motion of the cylindrical shell.

FIG. 10: the ratio of thickness to diameter, when different values are taken, describes the poincare section of the radial chaotic motion of the shell, (a) α ═ 0.0101001, (b) α ═ 0.0134950, (c) α ═ 0.0134951, (d) α ═ 0.0135199, (e) α ═ 0.0135220, and (f) α ═ 0.017095.

FIG. 11: a bifurcation diagram describing the radial movement of the shell when the material parameters take different values, (a) mu₁Influence of (b) mu₂The influence of (c).

FIG. 12: no coupling condition: given the excitation amplitude, F_zThe temporal response and Poincar cross-section (a) mode (m-1, N-4), (b) mode (m-1, N-0), (c) mode (m-3, N-0).

FIG. 13: coupling condition: given the excitation amplitude, F_zThe temporal response and Poincar cross-section (a) mode (m-1, N-4), (b) mode (m-1, N-0), (c) mode (m-3, N-0).

Detailed Description

Summary of the invention 1

At present, the nonlinear vibration of the cylindrical shell is researched more, but most of the nonlinear vibration is based on a linear constitutive relation. In particular, there are few reports in the literature of non-linear motion of cylindrical shells based on superelastic constitutive relations. The invention mainly focuses on the super elasticity of the rubber material but not other properties (such as viscoelasticity and the like), and researches some interesting motions, such as period, quasi-period and chaos, of the rubber cylindrical shell which is essentially characterized by the incompressible Mooney-Rivlin material under the action of radial simple harmonic excitation. Section 2 gives the relevant tensor knowledge, and in addition, based on the Donnell nonlinear thin shell theory, gives a control differential equation describing the motion of the rubber cylindrical shell; in section 3, the nonlinear differential equation set is solved by using a Runge-Kutta method, and the influence of periodic excitation, structural parameters and material parameters on the radial vibration of the shell is analyzed through a bifurcation diagram and a Poincare section respectively. Finally, several conclusions reached by the present invention are given in section 4.

2 formula

2.1 tensor basis

Let X be an initial configuration X₀One mass point in the system is marked with the coordinate mapping relation of X ═ χ%₀(X). Accordingly, the current configuration is χ_tThe current position coordinate of the particle is x, and the relationship is shown below

For analysis of χ from initial configuration₀To the current configuration χ_tIs derived from equation (1) with respect to X:

wherein, F ═ dX/dX is deformation gradient tensor, which is convenient for further analysis, and a standard mark is defined^[26]

J＝detF (3)

The Green-Lagrange strain tensor is

Based on the polar decomposition theorem, the deformation gradient tensor is decomposed into the product of an orthogonal rotation tensor R and a symmetric tensor (the cartesian tensor U or the cartesian tensor V), and then the general deformation is decomposed into pure stretching and rotation. The polar decomposition theorem states that there is a unique form of left and right decomposition of the deformation gradient tensor F.

F＝R·U＝V·R (5)

Based on the right-pole decomposition of the deformation gradient tensor F, the right Cauchy-Green deformation tensor is given as follows

C＝F^T·F＝U² (6)

By substituting formula (6) for formula (4)

The invariant of the right Cauchy-Green deformation tensor can be recorded as

2.2 Strain energy function of superelastic Material

As a typical type of superelastic material model, the Mooney-Rivlin model is often used to characterize the nonlinear elastic behavior of rubber materials, and its strain energy function is expressed as follows

Wherein mu₁,μ₂For the material constant, the specific expressions of the principal invariants in combination of formula (7) and formula (8) are as follows

Considering the incompressible property of rubber materials, i.e. having an incompressible condition J-1^[26]. The combination formula (10) can be obtained

Considering the assumption of small strain, epsilon can be obtained_zzThe polynomial expression of (a) is as follows:

substituting equations (12) and (10) for equation (9) yields a specific expression of the incompressible Mooney-Rivlin strain energy function. Considering the complexity of the calculation, only the inclusion of small strains epsilon is considered herein_xx，ε_θθ，ε_zz，ε_xθ，ε_xzAnd ε_θzThe fourth order expansion of (1) then has:

2.3 Shell theory and Displacement Dispersion

A cylindrical coordinate system (x, θ, z) is established at the mid-plane of the thin-walled rubber cylindrical shell, as shown in fig. 1. x, theta, z are axial, circumferential and radial, respectively. Fig. 1(b) shows a cross section of the cylindrical shell perpendicular to the axial direction x, and u, v and w represent displacements of the midplane in three directions x, θ and z, respectively. u. of₁,u₂And u₃Representing the displacement of any particle in three directions. l, h and R represent the initial length, thickness and mid-plane radius of the cylindrical shell, respectively. FIG. 1 (a) is a symbolic definition of relative size and displacement; fig. 1(b) is a schematic cross-sectional view in the x direction.

According to the Kirchhoff-Love hypothesis^[27]Arbitrary particle displacement (u)₁,u₂,u₃) The displacement (u, v, w) from a point on the middle plane satisfies the following relationship

Based on the Donnell nonlinear shallow shell theory, the corresponding strain-displacement relationship can be obtained as follows^[28]

In general, there is ε for a thin shell_zz≈0,ε_xz≈0,ε _θz0. Then the expressions for kinetic and elastic strain energy can be derived as follows:

where ρ and Φ are the material density and strain energy functions, respectively.

For the simple boundary condition of the shell, when x is 0, l, there is

v＝w＝0 (18)

To simplify the problem, an infinite-degree-of-freedom continuous system is discretized into a finite-degree-of-freedom system using an approximation function. The continuous system is discretized using the following basis functions for mid-surface displacements that satisfy the same geometric boundary conditions.

Wherein m and n are axial half wave number and circumferential wave number, lambda_mN.pi/l, t is time, u_mn(t),v_mn(t),w_mn(t) is a generalized coordinate related to time.

2.4 Lagrange's equation, external excitation and damping

Let W_eIntroducing Rayleigh dissipation function to describe work W of non-conservative damping force for work of periodic external force_d. The corresponding expression is as follows

Wherein F_x,F_θ,F_zThe unit distribution forces acting in the x, theta and z directions of the housing, respectively, and c is the damping coefficient. Further calculation can obtain

Wherein

c_m,nThe damping coefficient for the corresponding mode can generally be determined experimentally.

Order to

ω_m,nNatural frequency, p, of the corresponding mode_m,nIs the modal quality of the mode. Let q be (u)_m,n,v_m,n,w_m,n)^TThe element being a time-dependent displacement q_i(i-1, 2, …, 9). Generalized force G_i(i ═ 1,2, …,9) can be derived from the differential of the dissipation function and the imaginary work of the external force, i.e.:

the Lagrange equation describing the motion of the cylindrical shell is given, namely:

where L ═ T-P is the Lagrangian function of the system and i is the mode number.

Substituting the relevant expressions into Lagrange's equation (23) can obtain a nonlinear differential equation system for describing the motion of the cylindrical shell

Where M is the mass matrix, K is the linear stiffness matrix, K₂And K₃Are the quadratic and cubic nonlinear stiffness matrices, respectively, C is the rayleigh damping matrix, and C ═ β K + γ M, β, γ are the experimental measurement constants, q ═ { q ═ γ M₁,q₂,…,q₉}^TFor time-dependent displacement, F ═ F_x1,F_x2,F_x3,F_θ1,F_θ2,F_θ3,F_z1,F_z2,F_z3}^TThe elements of the mass and linear stiffness matrices for the excitation amplitude are given in the appendix.

The invention only considers the radial vibration of the cylindrical shell under the action of radial periodic load,i.e. F_xj＝F_θj0( j 1,2, 3). Now introduce the following notations

Multiplying by M simultaneously on both sides of equation (25)^-1From equations (25) and (26), it is possible:

wherein

Q＝{Q₁,Q₂,…,Q₉}^T

Wherein ζ_i＝ω_m,n,iζ_m,n,i，ω_m,n,iAnd ζ_m,n,i(i ═ 1,2, …,9) are the natural frequency and damping ratio, respectively.

Furthermore, since the in-plane displacement is small relative to the radial displacement, the inertial and damping terms in the respective planes are negligible. At present, most documents reduce the above differential equation into a radial motion differential equation by introducing a stress function, ignoring surface inertia terms and damping terms. However, with the introduction of the stress function, the calculation process becomes more complicated. In order to simplify the process, the method processes the formula (27) based on a degree-of-freedom condensation method under the condition of neglecting a plane inertia term and a damping term. Equation (27)) may give an approximate motion and deformation relationship,

in addition to this, the present invention is,

wherein

The following differential equation of motion can be obtained in conjunction with equation (31):

wherein

3 numerical examples and results

In order to numerically simulate the nonlinear vibration problem considered in the present invention, the material and structure parameters, i.e., μ₁＝416185.5Pa,μ₂＝-498.8Pa,ρ＝1100kgm^-3(document [22 ]]The linearization material parameters for the incompressible Mooney-Rivlin material are given). R is 150X 10^-3m (radius of thin-walled cylindrical shell). Zeta_m,n,30.0005 (damping ratio ζ)_m,n,i＝ζ_m,n,1ω_m,n,i/ω_m,n,1From the literature [30 ]]). To discuss the effect of structural parameters, the following two parameters are introduced:

wherein alpha and beta are respectively the thickness-diameter ratio and the diameter-length ratio.

3.1 natural frequency

The structural damping coefficient is now determined by analyzing the natural frequency of the housing. The relation of the natural frequency can be obtained by using the equation (24), and further, the influence of different parameters on the variation trend of the natural frequency is discussed, as shown in fig. 2.

Fig. 2 shows the natural frequency of the casing corresponding to different hoop wave numbers n, where α is 0.02 and β is 0.5

Fig. 2 illustrates that the natural frequency of radial motion is typically small compared to the axial and hoop natural frequencies. Therefore, to examine the fundamental frequency characteristics of the cylindrical shell, further analysis was made on the radial natural frequency of the cylindrical shell.

By analyzing different combinations of structural parameters, the relationship between the structural parameters and the radial natural frequency is given, as shown in fig. 3. In general, the aspect ratio of the short shell satisfies β > 1, whereas the long shell is the opposite. As shown in the graph of fig. 3, comparing α to 0.005, α to 0.01, and α to 0.02 for the long and medium shells β to 0.5, it is found that the larger the aspect ratio α is, the larger the corresponding modal frequency is, and the larger the hoop wave number n is, the more significant the influence thereof is. Comparing β to 0.5, β to 1.0, and β to 1.5, it is understood that the larger the aspect ratio is, the larger the corresponding modal frequency is, and the smaller the ring wave number n is, the more significant the influence thereof is.

Fig. 3 is a radial natural frequency trend graph when the structural parameters α and β take different values, where m of (a) is 1 and m of (b) is 3, and it can be seen from fig. 3 that the fundamental frequency of the shell is generally not equal to the natural frequency subtended by the modes m 1 and n 0. The influence of the ratio of the thickness to the diameter to the natural frequency is more obvious than the influence of the ratio of the length to the diameter to the fundamental frequency. The ring wave number n of the mode in which the low frequency is located is remarkably increased along with the increase of the length-diameter ratio beta. However, the higher-order mode is difficult to excite, and the invention only focuses on the influence of the thickness-diameter ratio alpha. And then further analyzing by using the ring wave n as 0-4. The influence of the structural parameters on the five modal frequencies is considered next. Fig. 4 shows the relationship between the natural frequency and the structural parameter, where α to ω and β in (a) are 0.5 and m is 1, α to ω and β in (b) are 2 and m is 1, β to ω and α in (c) are 0.005 and m is 1, and β to ω and α in (d) are 0.02 and m is 1. As shown in fig. 4(a) and 4(b), the thickness-diameter ratio slightly differs from the influence of the first 5 modal frequencies of the rubber cylindrical shell. In general, the aspect ratio α has a large influence on the mode with a large hoop wave number n, which is consistent with the analysis result of fig. 3. Furthermore, the intersection of the curves in fig. 4(a) shows that for a medium length cylindrical shell, the frequencies of different ring wavenumber modes can also be equal if the appropriate parameters are chosen, which typically means that there is a 1:1 internal resonance. Furthermore, fig. 4(b) shows that the frequencies of the low ring wavenumber modes will be very close when the radial length is large. Fig. 4(c) and 4(d) show that the aspect ratio β has an extremely complex effect on the natural frequency of the cylindrical shell. This also verifies the conclusion drawn in fig. 3 that the frequencies of the low ring wavenumber modes are almost equal when the ratio of the radial length β is greater than 1. When the aspect ratio β ∈ (0.5,2), the frequency of the mode (m ∈ 1, n ∈ 4) is substantially lowest, that is, when the aspect ratio α is larger, the frequency of the mode (m ∈ 1, n ═ 3) may be lowest, but the frequency of the mode (m ∈ 1, n ∈ 4) is very close to it. Furthermore, the intersection of the curves corresponding to different hoop wavenumbers with the curve corresponding to a certain aspect ratio indicates that the internal resonance is related to the aspect ratio of the shell.

3.2 non-Linear vibration without coupling Effect

Obviously, the vibration behavior of the rubber cylindrical shell studied by the present invention is strongly non-linear based on the system of nonlinear differential equations (32). Secondly, the four-order Runge-Kutta method is adopted to solve the equation set, and the influence of various parameters on the vibration characteristic of the rubber cylindrical shell is analyzed. Furthermore, from the above analysis, it can be seen that the fundamental frequency of the cylindrical shell is generally determined by its radial natural frequency. The invention therefore only considers the case where the external excitation is equal to the radial natural frequency. Through the analysis of fig. 3, it is found that when the hoop wave number n is 4, the frequency corresponding to different structural parameter combinations is the lowest, and the influence of the structural parameters is also obvious. Therefore, the following studies only consider the modes (m 1, n 4). As shown in fig. 3, as the radial-to-length ratio increases, the fundamental mode is a higher-order eigenmode in which the ring wave number is large.

3.2.1 Effect of excitation amplitude and excitation frequency

Generally, the amplitude of the external stimulus has the most direct relationship to the response of the structure. Therefore, it is first necessary to analyze the chaotic dynamic behavior of the cylindrical shell under different excitation amplitudes. Setting the structural parameters as alpha 0.02 and beta 0.5, solving a nonlinear differential equation set by adopting a fourth-order Runge-Kutta method, selecting Poincare sections under different excitation amplitudes, and obtaining the radial motion and beta of the cylindrical shellBifurcation diagram of the external excitation amplitude. FIG. 5 shows radial motion and excitation amplitude F of a cylindrical shell_zThe bifurcation diagram of (2).

FIG. 5 shows the amplitude F when excited_z< 5N, the radial motion of the cylindrical shell is periodic, when the excitation amplitude F is_zApproximately equal to 5.05N, the radial motion will first exhibit chaotic behavior. Subsequently, as the excitation amplitude increases, the radial vibrations exhibit an alternation of periodic and chaotic movements. Notably, the motion of cycle 3 can be observed in the bifurcation diagram. And is positioned between two chaotic regions, and evolves to periodic vibration through a path with doubled period and bifurcation. This is in contrast to document [30 ]]The view of (a) is consistent.

FIG. 6 is a Poincare cross section corresponding to different excitation amplitudes when the cylindrical shell generates chaotic motion, where F in (a)_z(ii) 5.05N, F of (b)_z7.3N, F of (c)_zF of (d) 8.55N_z＝9.8N。

Fig. 6 shows Poincar é cross-sections for different excitation amplitudes. Some structures with very interesting fractal characteristics were found, called strange attractors. Generally, attractors can be classified into four types, namely, point attractors, limit ring attractors, torus attractors, and fanciful attractors. Here we mainly describe singular attractors, attractors in phase space, where the points inside do not repeat and the tracks do not intersect, but they remain in the same area of phase space. Unlike limit-ring or point attractors, the strange attractors are aperiodic. The strange attractor can have an infinite number of different forms. As the bifurcation parameters change, they rotate and stretch to different degrees, appearing as complex structures and varying shapes, but in practice there is a self-similarity between local and global. Furthermore, the presence of the strange attractors also indicates that even if the motion of the cylindrical shell is irregular and unpredictable, its motion area is still definite for large deflection vibrations. In other words, the radial motion of any point of the shell is limited to the area determined by the singular attractor, but the specific location of that point cannot be determined. This is also an important feature of systems with singular attractors: locally unstable but globally stable. For two adjacent points in the singular attractor, although they may separate from each other over time, they do not escape from the region defined by the singular attractor.

Further, to investigate the effect of the excitation frequency, the excitation amplitude was taken to be F_zThe bifurcation diagram of the effect of the excitation frequency on the cylindrical shell motion is given at 5.05N. FIG. 7 is a bifurcation diagram depicting radial motion of the housing without the use of an excitation frequency.

FIG. 7 depicts a bifurcation diagram of the radial motion of the cylindrical shell with a fixed excitation amplitude and different excitation frequencies. Obviously, as the frequency changes, the periodic motion can evolve from chaotic differentiation and vice versa. In addition, when the ratio of the natural frequency to the excitation frequency is within the range (1.04,1.05), the motion of cycle 3 can be observed.

FIG. 8 is a Poincare cross section illustrating the radial chaotic motion of the housing at different excitation frequencies, where Ω of (a) is 0.909 ω₁₄Omega of (b) is 0.998 omega₁₄(c) is 1.001. omega₁₄Omega of (d) 1.0375 omega₁₄. Fig. 8 illustrates a Poincar é cross-section with a specific excitation frequency. The shape characteristics of these singular attractors are similar to fig. 6(a), particularly fig. 8 (c). The elongated structural features distinguish them from fig. 8 and 6, which means that a singular attractor with a smaller excitation amplitude may have an elongated shape.

3.2.2 influence of structural parameters

This subsection discusses the effect of thickness to diameter ratio α on the radial motion of the cylindrical shell. For a thin shell, the radial-to-length ratio β is set to 0.5, and the excitation amplitude F is set_z5.05N. Fig. 9 is a bifurcation diagram describing radial motion of the cylindrical shell under different thickness-to-radius ratios, and as shown in fig. 9, when the thickness-to-radius ratio is less than 0.02, the radial motion of the cylindrical shell exhibits frequent alternation between periodic motion and chaotic motion. When the thickness-diameter ratio is larger than 0.02, the radial movement of the shell is basically periodic, namely within a certain range, the increase of the thickness-diameter ratio is beneficial to improving the movement stability of the thin-wall cylindrical shell. FIG. 10 is a Poincare cross-section depicting the radial chaotic motion of the hull at different aspect ratios, where α of (c) is 0.0134951, α of (d) is 0.0135199, and (e)0.0135220, and 0.017095 for α in (f). Fig. 10 shows poincare cross sections at different thickness radii. Again demonstrating the presence of a strange attractor. The result shows that the singular attractors under the condition of different thickness-diameter ratios are similar to the attractors, and the attractors can be found to present different structures in different parameter ranges for the system of the discontinuous chaotic region. In particular, fig. 10(b) shows a cycle 9 motion, with a small amplitude increase in thickness radius ratio α, it can be seen from fig. 10(c) that the radial motion immediately translates into a chaotic motion with 3 isolated attraction domains. In addition, fig. 10(d) shows the periodic 3 motion, and the periodic motion also evolves to chaos as the aspect ratio increases. The results show that when the thickness-diameter ratio parameter of the cylindrical shell is between the period and the chaotic region, slight correction of the thickness-diameter ratio parameter may cause drastic change of the vibration characteristics.

3.2.3 Effect of Material parameters

Since the material parameters of rubber materials are usually obtained by fitting experimental data, the values obtained by different fitting methods may be slightly different. It is assumed that the material parameters will fluctuate by 10% from the initial values given in the literature, i.e.: mu's'₁＝k₁μ₁,μ′₂＝k₂μ₂Of which is mu'₁,μ′₂Is a varying material parameter, and k₁,k₂∈[0.9,1.1]The following analysis of the movement behavior of a variable-parameter housing is given in the literature. FIG. 11 is a bifurcation diagram depicting radial motion of the shell under different material parameters, (a) is μ₁Is (a), (b) is₂The influence of (c). FIG. 11 shows the material parameter μ₁The change of the parameter (mu) influences the motion characteristic of the cylindrical shell, the chaotic response of the cylindrical shell is evolved into periodic motion from chaotic motion through a periodic multiple-bifurcation path, and the material parameter (mu) is₂The response to the cylindrical shell has less influence. Thereby embodying the material parameter mu₁Is compared with the material parameter mu₂The fitting accuracy of (2) is more important.

3.3 nonlinear vibration of coupling effect

To analyze the interaction between the modes, three modes were chosen (m-1, n-4, m-1, n-0, m-0)3, n ═ 0) were discretized and then some interesting nonlinear dynamic behavior of the radial movement of the shell was studied. A simple comparative analysis was performed using the time response and Poincar é sections. Fig. 12 is a no coupling case: time response and Poincare cross section at a given excitation amplitude, F_zThe mode of (a) is (m) 1, and N is 4, (b) is (m) 1, and N is 0, and (c) is (m) 3, and N is 0).

FIG. 12(a, b, c) shows a solution obtained by the fourth-order Runge-Kutta method in the case of no coupling. Interestingly, under the condition of multi-modal discretization, chaos phenomenon can exist under the non-coupling condition, but the fine structural features of the chaos phenomenon are not obvious any more. This may be due to an increase in the degree of freedom leading to an increase in computational complexity and thus a decrease in accuracy. Furthermore, since coupling effects are not taken into account, the response between the modalities is independent. This means that when there is a chaotic response in one modality, the response of the other modality can still be periodic. Furthermore, the amplitude of the axisymmetric mode is much smaller than the non-axisymmetric mode. This shows that uniaxial symmetry mode analysis is feasible without considering the coupling effect. Fig. 13 is a coupling case: time response and Poincare cross section at a given excitation amplitude, F_zThe mode of (a) is (m) 1, and N is 4, (b) is (m) 1, and N is 0, and (c) is (m) 3, and N is 0).

Fig. 13(a, b, c) shows a solution obtained by the fourth-order Runge-Kutta method in the case of coupling. In the axisymmetric mode, the quasi-periodic motion replaces the chaotic motion, so that the response is more regular. This means to some extent that coupling effects between the modes can improve the stability of the motion. Meanwhile, due to the coupling effect, each order of mode also has a synchronization effect, that is, each order of mode has similar response characteristics. As shown in fig. 13, the poincare cross section of three modes has five isolated regions.

4 conclusion

The invention researches the problem of nonlinear vibration of a rubber cylindrical shell consisting of incompressible Mooney-Rivlin materials under the action of radial simple harmonic excitation. It is noted that, in general, the related superelasticity constitutive relation of rubber material under certain conditions can be used to approximately describeMechanical behavior of the rubber material^[32](the invention only considers the super elasticity of the rubber material, but does not consider other properties such as viscoelasticity and the like). Based on the Donnell thin shell theory and the small strain assumption, a system of differential equations describing the nonlinear vibration of the shell is established. The nonlinear dynamic behavior of the housing was investigated using the corresponding bifurcation diagram and Poincar section. The following five main conclusions are reached:

(1) the increase of the thickness-diameter ratio and the diameter-length ratio can improve the natural frequency of the mode, the thickness-diameter ratio has obvious influence on the mode with larger ring wave number, and the diameter-length ratio has obvious influence on the mode with smaller ring wave number.

(2) When the excitation amplitude is larger than a certain critical value, the radial motion is alternately performed between the chaotic motion and the periodic motion in a form of multiple period bifurcation. In addition, when the excitation amplitude is sufficiently large, the periodic motion may branch off from the chaotic region as the excitation frequency varies.

(3) The thickness to diameter ratio has a significant effect on the chaotic behavior of the cylindrical shell, and for a given ratio, the vibration of the rubber cylindrical shell is highly sensitive to the ratio when the value is less than a critical value. This means that small changes in the ratio can convert chaotic motion into periodic motion and vice versa.

(4) For the incompressible Mooney-Rivlin model, the material parameter μ₁Influence ratio mu on nonlinear vibration behavior of rubber cylindrical shell₂Is more remarkable;

(5) for the multi-modal case, the response of the cylindrical shell is similar to that of the single-modal case, when the coupling effect between the different modes is not considered. However, when the coupling effect is considered, a different conclusion will be reached; in addition, coupling effects may improve the stability of the structural response.

Thank you: the invention obtains the subsidy of national science fund (Nos.11672069,11702059, 11872145).

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Appendix

Linear stiffness matrix:

K₁₇＝K₇₁＝-π²h(μ₁-μ₂),K₁₂＝K₁₃＝K₁₅＝K₁₆＝K₁₈＝K₁₉＝0,

K₃₉＝K₉₃＝-6π²h(μ₁-μ₂),K₃₁＝K₃₂＝K₃₄＝K₃₅＝K₃₆＝K₃₇＝K₃₈＝0,

K₆₁＝K₆₂＝K₆₃＝K₆₄＝K₆₅＝K₆₇＝K₆₈＝K₆₉＝0,

K₉₁＝K₉₂＝K₉₄＝K₉₅＝K₉₆＝K₉₇＝K₉₈＝0。

Claims (2)

1. a method for describing large-deflection vibration of a rubber cylindrical shell is characterized by comprising the following steps: the non-linear micro-motion describing the motion of the cylindrical shell is given by using the energy principleDividing into an equation set: let W_eIntroducing Rayleigh dissipation function to describe work W of non-conservative damping force for work of periodic external force_dThe corresponding expression is as follows:

wherein F_x,F_θ,F_zThe unit distribution forces acting on the shell in the x direction, the theta direction and the z direction respectively, and c is a damping coefficient; further calculation can obtain

c_m,nDamping coefficient for the corresponding mode;

order to

ω_m,nNatural frequency, p, of the corresponding mode_m,nThe modal mass for that mode; let q be (u)_m,n,v_m,n,w_m,n)^TThe element being time-dependent q_i(i ═ 1,2, …, 9); generalized force G_i(i ═ 1,2, …,9), which results from the differentiation of the dissipation function and the imaginary work of the external force, i.e.:

lagrange's equation describing cylindrical shell motion:

L-T-P is a Lagrangian function of the system, and i is a modal number;

a system of nonlinear differential equations describing the motion of the cylindrical shell, namely:

m is a mass matrix, K is a linear stiffness matrix, K₂And K₃Are the quadratic and cubic nonlinear stiffness matrices, respectively, C is the rayleigh damping matrix, and C ═ β K + γ M, β, γ are the experimental measurement constants, q ═ { q ═ γ M₁,q₂,…,q₉}^T，F＝{F_x1,F_x2,F_x3,F_θ1,F_θ2,F_θ3,F_z1,F_z2,F_z3}^T，F_xj＝F_θj＝0(j＝1,2,3)。

2. A method of describing large deflection vibrations of a rubber cylindrical shell as claimed in claim 1 wherein:

introduce the following notations

Multiplying by M simultaneously on both sides of equation (25)^-1From equations (25) and (26), it is possible:

wherein

ζ_i＝ω_m,n,iζ_m,n,i，ω_m,n,iAnd ζ_m,n,i(i ═ 1,2, …,9) are natural frequency and damping ratio, respectively;

neglecting the plane inertia and damping terms, and processing equation (27) based on the degree of freedom condensation method, there is the following approximate motion and deformation relationship:

in addition to this, the present invention is,

wherein

Combining equation (30) to obtain the following differential equation of motion:

wherein

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