CN112257213B - Method for describing large-deflection vibration of rubber cylindrical shell - Google Patents
Method for describing large-deflection vibration of rubber cylindrical shell Download PDFInfo
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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 of establishing a mathematical model to be capable of researching nonlinear vibration.
Description
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 class of superelastic materials, some important properties of which (such as high elasticity, large deformations, etc.) are an integral part of engineering. Accordingly, the range of products made of these materials is also wide, such as tires, gaskets, oil pipes, and the like. The cylindrical shell is widely applied to the fields of spacecrafts, conveying pipes, engine drums and the like due to the excellent mechanical properties. For thin shell structures and flexible structures, they produce a typical nonlinear response to disturbance stimuli. [1] . In addition, the super-elastic cylindrical shell has the characteristics of geometric nonlinearity and physical nonlinearity, and has important significance in researching the nonlinearity, particularly chaotic vibration characteristics.
In the field of mechanics, a plate-shell structure is the most economical structural form in practical application, and research on chaotic vibration of the plate-shell has attracted wide attention. Wang et alHuman body [2] The natural frequency, complex mode and critical velocity of the axially moving rectangular plate were studied. The motion equation of plate vibration is established by using classical sheet theory, and the influence of the distance ratio, the motion speed, the immersion depth ratio, the boundary conditions, the rigidity ratio, the length-width ratio and the fluid plate density ratio of the plate on the free vibration of the moving plate-fluid system is discussed. Hao et al [3] And the nonlinear dynamics behavior of a rectangular plate of the simply supported functional gradient material in a thermal environment under transverse and planar excitation is analyzed, and periodic solution, quasi-periodic solution and chaotic motion are given. Yang et al [4] The simplified Flu gge shell theory and modal orthogonalization are utilized to give a unified solution for cylindrical shell vibration analysis under the general stress distribution. Li et al [5] The global bifurcation and multi-pulse chaotic dynamics behavior of the simply supported rectangular thin plate is researched by using an extended Melnikov method. Bich et al [6] Based on the improved Donnell shell theory, nonlinear vibration of the functionally graded cylindrical shell under axial and transverse loading was analyzed, and the effects of preloaded axial compression, functionally graded material properties and dimensional ratios on shell behavior were discussed. By first order shear deformation theory and perturbation method, sofiyev et al [7] Nonlinear free vibration of interaction of the functional gradient orthotropic cylindrical shell and the dual-parameter elastic foundation was studied. Zhu et al [8] The nonlinear free vibration behavior of the orthotropic piezoelectric cylinder shell was studied and the effect of surface parameters and geometric properties was analyzed. Amabili et al [9] The nonlinear vibration of the water-filled cylindrical shell under radial simple harmonic excitation is researched by using an experiment and a numerical method, and chaotic motion is found in a frequency domain with traveling wave response. Yamaguchi et al [10] The 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 studied by using a kinetic equation containing quadratic and cubic nonlinear terms, and the effectiveness of the analysis of the chaotic motion by using a single-mode model is briefly described. Kryskoa et al [12] Chaotic vibration of the closed cylindrical shell in a temperature field was studied. Zhang et al [13] The resonant response of the composite laminated cylindrical shell was studiedAnd chaotic dynamics, a twinning phenomenon exists between Pomeasu-Manneville type intermittent chaos and double period bifurcation. Li et al [14] Based on Reddy third-order shearing theory for the first time, a new displacement field is utilized to analyze nonlinear transient dynamic response of the functional gradient material sandwich isotropic homogeneous material and the functional gradient panel hyperbolic shell. Amabili et al [15] The geometric nonlinear response of the water-filled cylindrical shell is studied, and the result shows that the response of the water-filled cylindrical shell is subjected to cycle doubling bifurcation, subharmonic response, quasi-cycle response and chaotic behavior along with the increase of the excitation amplitude.
Generally, chaotic motion is a part of nonlinear vibration, and research on nonlinear behavior is the basis for analyzing chaotic phenomenon. The origin of the superelastic cylindrical shell nonlinear response is largely twofold: firstly, large deformation of the shell; and secondly, the nonlinear constitutive relation of the super-elastic material. Large deformations of the structure often introduce geometrical nonlinearities, which are major difficulties in studying the nonlinear vibrations of the cylindrical shell. Amabili [16] The original consistent first-order shear deformation theory of all nonlinear terms in-plane displacement and rotation is reserved, and the numerical application of the nonlinear forced vibration of the laminated cylindrical shell of the simply supported composite material is proposed. In the framework of classical nonlinear theory, krysko et al [17] The problem of complex vibration of the closed infinite cylindrical shell under the action of transverse local load is studied. Breslavsky et al [18] Nonlinear response of the water-filled simply supported cylindrical thin shell under multi-harmonic excitation is studied, and nonlinear dynamic behaviors are discussed. Du et al [19] Nonlinear forced vibration of an infinitely long FGM cylindrical shell is researched by utilizing Lagrangian theory and a multi-scale method, and a chaotic path of the system is disclosed. In addition to geometric nonlinearity caused by large deformations, material nonlinearity is also an important factor in causing nonlinear response. Because the super-elastic material has physical nonlinearity, the corresponding structure is easy to deform greatly, and geometrical nonlinearity can be generated after deformation. Thus, studies on superelastic structures generally involve geometric and physical nonlinearities. Thus, there is a great deal of interest in the nonlinear response of superelastic structures. Iglesias et al [20] Large-amplitude axisymmetric free vibration of an incompressible super-elastic orthotropic cylindrical structure is studied, and the fact that the motion of the structure can evolve from periodic motion to quasi-periodic and chaotic motion is proved.Et al [21] The nonlinear vibration behavior of the pre-stretched superelastic round film under the action of limited deformation and variable transverse pressure is analyzed in detail. Breslavsky et al [22] Free and forced nonlinear vibration of the superelastic rectangular sheet was studied, and the structure found that the frequency shift phenomenon between low and large amplitude vibration of the sheet was reduced with increasing initial deflection.
In the field of dynamics, there are many studies on the cylindrical shell or the superelastic structure, but there are few studies on the dynamic behavior of the superelastic cylindrical shell. Shahinboost et al [23] Based on the finite deformation theory, large-amplitude radial vibration of the super-elastic thin-wall tube is analyzed, and an accurate solution of the simplification problem is obtained. Wang et al [24] The symmetrical and nonlinear movements of the super-elastic cylindrical tube formed by the axial transverse isotropy compressible neo-Hookean material in the radial direction and the axial direction are studied. 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 driving mode and accompanying mode resonance state was found.
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 gives a nonlinear differential equation system describing the motion of the cylindrical shell by utilizing an energy principle.
The beneficial effects are that:
the invention researches the problem of nonlinear vibration of a thin-walled rubber cylindrical shell composed of classical incompressible materials under radial harmonic excitation. Based on the Donnell nonlinear thin shell theory, the Lagrange equation and the small strain hypothesis, a nonlinear differential equation describing the large deflection vibration of the rubber cylindrical shell is obtained.
Drawings
Fig. 1: a cylindrical shell schematic; (a) sign definition of relative size and displacement; (b) cross-section schematic in the x-direction.
Fig. 2: natural frequencies of shells corresponding to different circumferential wave numbers n are alpha=0.02 and beta=0.5.
Fig. 3: when the parameters α and β take different values, (a) m=1, and (b) m=3.
Fig. 4: relationship between natural frequency and structural parameters, (a) α to ω, β=0.5, m=1, (b) α to ω, β=2, m=1, (c) β to ω, α=0.005, m=1, (d) β to ω, α=0.02, m=1.
Fig. 5: cylindrical shell radial motion and excitation amplitude F z Is a bifurcation diagram of (1).
Fig. 6: when the cylindrical shell generates chaotic motion, poincare sections corresponding to different excitation amplitudes, (a) F z ＝5.05N， (b)F z ＝7.3N，(c)F z ＝8.55N，(d)F z ＝9.8N。
Fig. 7: the bifurcation diagram describing the radial movement of the housing is not depicted at the excitation frequency.
Fig. 8: poincare cross section describing radial chaotic motion of shell at different excitation frequencies, (a) omega=0.909 omega 14 ，(b) Ω＝0.998ω 14 ，(c)Ω＝1.001ω 14 ，(d)Ω＝1.0375ω 14 。
Fig. 9: and when the thickness-diameter ratio takes different values, a bifurcation diagram of the radial movement of the cylindrical shell is described.
Fig. 10: when the thickness-to-diameter ratio takes different values, poincare cross-section describing 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, (f) α= 0.017095.
Fig. 11: a bifurcation diagram describing the radial movement of the housing when the material parameters take different values, (a) mu 1 Influence of (b) mu 2 Is a function of (a) and (b).
Fig. 12: uncoupled fromThe situation is as follows: given the excitation amplitude, F z Time response and poincare cross section, (a) modality (m=1, n=4), (b) modality (m=1, n=0), (c) modality (m=3, n=0).
Fig. 13: coupling condition: given the excitation amplitude, F z Time response and poincare cross section, (a) modality (m=1, n=4), (b) modality (m=1, n=0), (c) modality (m=3, n=0).
Detailed Description
Summary of the invention
At present, the research on the nonlinear vibration of the cylindrical shell is more, but most of the nonlinear vibration is based on the linear constitutive relation. In particular, few literature reports are available based on the non-linear motion of the cylindrical shell in the superelastic constitutive relationship. The focus of the research of the invention is mainly focused on the super elasticity of rubber materials and other properties (such as viscoelasticity and the like), and the rubber cylindrical shell characterized by the incompressible Mooney-Rivlin material structure is researched for some interesting movements such as period, quasi-period and chaos under the radial simple harmonic excitation effect. Section 2 gives the relevant tensor knowledge, and furthermore, based on the Donnell nonlinear thin shell theory, gives a control differential equation describing the rubber cylinder shell motion; section 3 solves the nonlinear differential equation set by using a Runge-Kutta method, and analyzes the influence of periodic excitation, structural parameters and material parameters on radial vibration of the shell through a bifurcation diagram and a Poincare section respectively. Finally, several conclusions of the invention are presented in section 4.
Formula 2
2.1 tensor basis
Let X be original configuration χ 0 One particle in the matrix is marked with the coordinate mapping relation of X=χ 0 (X). Accordingly, in the current configuration χ t When the current position coordinate of the particle is x, the relationship is as follows
To analyze χ from the initial configuration 0 To the current configuration χ t For the variant of formula (1) with respect to X:
wherein f=dx/dX is the deformation gradient tensor, which is convenient for the next analysis, and defines a standard mark [26]
J＝detF (3)
Green-Lagrange strain tensor of
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 (either the cauchy tensor U or cauchy tensor V), and then the general deformation is decomposed into pure stretching and rotation. The polar decomposition theorem states that the deformation gradient tensor F has a unique form of left and right decomposition.
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＝U 2 (6)
Substituting formula (6) into formula (4)
The invariant of the right Cauchy-Green deformation tensor can be noted as
2.2 Strain energy function of superelastic Material
As a typical class of superelastic material models, the Mooney-Rivlin model is often used to characterize the nonlinear elastic behavior of rubber materials, with the following form of strain energy function
Wherein mu 1 ,μ 2 The specific expression of the main invariant is as follows, which is the material constant and combines the formula (7) and the formula (8)
Taking into account the incompressible properties of the rubber material, i.e. under incompressible conditions j=1 [26] . Can be obtained by combining (10)
Considering the small strain hypothesis, ε can be obtained zz The polynomial expression of (2) is as follows:
substituting formula (12) and formula (10) into formula (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 ε is considered herein xx ，ε θθ ，ε zz ，ε xθ ，ε xz And epsilon θz The four-order expansion of (a) is as follows:
2.3 Shell theory and Displacement discretization
A column coordinate system (x, θ, z) is built in the middle face of the thin-walled rubber cylinder shell as shown in fig. 1. x, θ, z are axial, circumferential, and radial, respectively. Fig. 1 (b) shows a cross section of the cylindrical shell perpendicular to the axial direction x, u, v and w representing the displacement of the mid-plane in the x, θ and z directions, respectively. u (u) 1 ,u 2 And u 3 Respectively representing 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) isSymbol definition of relevant size and displacement; fig. 1 (b) is a schematic cross-sectional view in the x-direction.
According to Kirchhoff-Love hypothesis [27] Any particle displacement (u 1 ,u 2 ,u 3 ) The displacement (u, v, w) from a point on the mid-plane satisfies the following relationship
Based on the Donnell nonlinear shallow shell theory, the corresponding strain-displacement relationship can be obtained as follows [28]
Generally speaking, there is ε for a thin shell zz ≈0,ε xz ≈0,ε θz And 0. Then an expression for kinetic and elastic strain energy can be derived as follows:
where ρ and Φ are the material density and strain energy function, respectively.
For the simple support boundary condition of the shell, when x= 0,l, then there is
v＝w＝0 (18)
To simplify the problem, an approximation function is used to discretize an infinite degree of freedom continuous system into a finite degree of freedom system. The following basis functions of surface displacement satisfying the same geometric boundary conditions are used to discretize the continuous system.
Wherein m and n are respectively axial halfWave number and circumferential wave number, lambda m =mpi/l, t is time, u mn (t),v mn (t),w mn And (t) is a generalized coordinate related to time.
2.4 Lagrangian equation, external excitation and damping
Let W e For the work done by periodic external force, the Rayleigh dissipation function is introduced to describe the work W done by non-conservative damping force d . Its corresponding expression is as follows
Wherein F is x ,F θ ,F z The unit distribution forces acting in the x, θ and z directions of the housing, respectively, and c is the damping coefficient. Further calculate and get
Wherein the method comprises the steps ofc m,n The damping coefficient of the corresponding mode can be usually determined by experiments.
Order theω m,n For the natural frequency of the corresponding mode ρ m,n Is the modal mass of the modality. Let q= (u) m,n ,v m,n ,w m,n ) T The elements being time-dependent displacements q i (i=1, 2, …, 9). Generalized force G i (i=1, 2, …, 9) can be derived from differentiation of the dissipation function and the virtual work of the external force, namely:
now, the Lagrange equation describing the cylindrical shell motion is given, namely:
where l=t-P is the Lagrangian function of the system and i is the number of modes.
Substituting the relevant expression into Lagrange's equation (23) to obtain a nonlinear differential equation set describing cylindrical shell motion
Wherein M is a mass matrix, K is a linear stiffness matrix, K 2 And K 3 The two-and three-dimensional nonlinear stiffness matrices are respectively, C is a rayleigh damping matrix, and c=βk+γm, β, γ is an experimentally determined constant, q= { q 1 ,q 2 ,…,q 9 } T For time dependent displacement, f= { F x1 ,F x2 ,F x3 ,F θ1 ,F θ2 ,F θ3 ,F z1 ,F z2 ,F z3 } T For excitation amplitude, elements of the mass and linear stiffness matrix are given in the appendix.
The invention only considers the radial vibration of the cylindrical shell under the radial periodic load, namely F xj ＝F θj =0 (j=1, 2, 3). The following notations are now introduced
Multiplying M simultaneously on both sides of (25) -1 From the formula (25) and the formula (26):
wherein the method comprises the steps of
Q＝{Q 1 ,Q 2 ,…,Q 9 } T
Wherein ζ i ＝ω m,n,i ζ m,n,i ，ω m,n,i And zeta 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 corresponding in-plane inertia and damping terms are negligible. Currently, most documents simplify the differential equation described above into a radial motion differential equation by introducing a stress function, ignoring the surface inertia term and the damping term. However, the calculation process becomes more complex after introducing the stress function. To simplify this process, the present invention deals with equation (27) based on the degree-of-freedom condensation method, ignoring the planar inertia term and the damping term. Equation (27)) may give an approximate motion and deformation relationship,
in addition, in the case of the optical fiber,
wherein the method comprises the steps of
The following differential equation of motion can be obtained in combination with equation (31):
wherein the method comprises the steps of
3 numerical examples and results
To be communicated withNumerical simulation of the nonlinear vibration problem considered by the invention requires selection of material parameters and structural parameters, i.e., μ 1 ＝416185.5Pa,μ 2 ＝-498.8Pa,ρ＝1100kgm -3 (document [22 ]]Linearization material parameters for the incompressible Mooney-Rivlin material are given). R=150×10 -3 m (radius of thin-walled cylindrical shell). Zeta type m,n,3 =0.0005 (damping ratio ζ) m,n,i ＝ζ m,n,1 ω m,n,i /ω m,n,1 See document [30 ]]). To discuss the effect of structural parameters, the following two parameters are now introduced:
wherein alpha and beta are thickness-diameter ratio and diameter-length ratio respectively.
3.1 natural frequency
The structural damping coefficient is now determined by analyzing the natural frequency of the housing. The relation of natural frequencies is obtained by the equation (24), and furthermore, the influence of different parameters on the trend of natural frequency change is discussed, as shown in fig. 2.
Natural frequencies of shells corresponding to different circumferential wave numbers n in fig. 2, α=0.02, β=0.5
Fig. 2 illustrates that the natural frequency of radial motion is generally small compared to the axial and circumferential natural frequencies. Therefore, to examine the fundamental frequency characteristics of the cylindrical shell, further analysis is now made of the radial natural frequency of the cylindrical shell.
By analysing different combinations of structural parameters, the relation between structural parameters and radial natural frequencies 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, for the middle-long shell β=0.5, comparing α=0.005, α=0.01, and α=0.02, the larger the thickness-diameter ratio α is, the larger the corresponding modal frequency is, the larger the circumferential wave number n is, and the more significant the effect is. By comparing β=0.5, β=1.0, and β=1.5, the larger the diametral length ratio is, the larger the corresponding modal frequency is, the smaller the circumferential wave number n is, and the more remarkable the influence is.
Fig. 3 is a graph showing the radial natural frequency variation trend when the structural parameters α and β take different values, wherein (a) m=1 and (b) m=3, and further, as can be seen from fig. 3, the fundamental frequency of the shell is generally not equal to the natural frequency for the modes m=1 and n=0. The influence of the aspect ratio on the fundamental frequency is more remarkable as compared with the influence of the thick-diameter ratio alpha and the aspect ratio beta on the natural frequency. As the aspect ratio β increases, the circumferential wavenumber n of the mode in which the low frequency is located increases significantly. But the higher order mode shape is difficult to excite, the invention only focuses on the effect of the thick-to-diameter ratio alpha. And then further analyzed using a circumferential wave n=0 to 4. Next consider the effect of structural parameters on five modal frequencies. Fig. 4 shows the relationship between natural frequencies and structural parameters, (a) α to ω, β=0.5, m=1, (b) α to ω, β=2, m=1, (c) β to ω, α=0.005, m=1, and (d) β to ω, α=0.02, m=1. As shown in fig. 4 (a) and 4 (b), the thickness-to-diameter ratio has a slightly different effect on the first 5 modal frequencies of the rubber cylinder shell. In general, the modal influence of the larger thickness-to-diameter ratio α on the circumferential wave number n is larger, which is consistent with the analysis result of fig. 3. Furthermore, the intersection of the curves in fig. 4 (a) shows that for a cylindrical shell of medium length, the frequencies with different circumferential wavenumber modes can also be equal if appropriate parameters are chosen, which generally means that there is an internal resonance of 1:1. Furthermore, fig. 4 (b) shows that when the diameter is relatively large, the frequencies of the low-loop wavenumber modes will be very close. Fig. 4 (c) and 4 (d) show that the diametral length ratio β has an extremely complex influence on the natural frequency of the cylindrical shell. This also verifies the conclusion drawn in fig. 3 that the frequencies of the low-loop wavenumber modes are almost equal when the ratio of radial length β is greater than 1. When the radius-to-length ratio β e (0.5, 2), the frequency of the mode (m=1, n=4) is substantially lowest, i.e., when the thickness-to-diameter ratio α is large, 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 thereto. Furthermore, the intersection of the curves corresponding to different circumferential wave numbers with the curves corresponding to a certain ratio of diametral length indicates that the internal resonance is related to the ratio of diametral length of the shell.
3.2 non-Linear vibration without coupling Effect
Obviously, based on the nonlinear differential equation set (32), the vibration behavior of the rubber cylindrical shell studied by the invention is strongly nonlinear. And secondly, adopting a fourth-order Runge-Kutta method to numerically solve an equation set, and analyzing the influence of various parameters on the vibration characteristics of the rubber cylindrical shell. Furthermore, from the above analysis, it can be seen that the fundamental frequency of a cylindrical shell is typically determined by its radial natural frequency. Therefore, the present invention only considers the case where the external excitation is equal to the radial natural frequency. From the analysis of fig. 3, it was found that when the circumferential wave number n=4, the frequency corresponding to the combination of different structural parameters was the lowest, and the influence of the structural parameters was also apparent. Thus, the following study only considers modalities (m=1, n=4). In addition, as shown in fig. 3, as the ratio of the diameter to the length increases, the fundamental frequency mode is a higher-order natural mode with a larger circumferential wave number.
3.2.1 influence of excitation amplitude and excitation frequency
In general, the amplitude of the external stimulus has the most direct relationship to the response of the structure. Therefore, the chaotic dynamics of the cylindrical shell at different excitation amplitudes needs to be analyzed first. And (3) 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, and selecting Poincare sections under different excitation amplitudes to obtain a bifurcation diagram of the radial motion of the cylindrical shell and the external excitation amplitude. FIG. 5 is a cylinder radial motion and excitation amplitude F z Is a bifurcation diagram of (1).
FIG. 5 shows that when the excitation amplitude F z The radial movement of the cylindrical shell is periodic at < 5N, when the excitation amplitude F z About 5.05N, the radial motion will first exhibit chaotic behavior. Subsequently, as the excitation amplitude increases, the radial vibration exhibits an alternating periodic and chaotic motion. Notably, the movement of cycle 3 can be observed in the bifurcation diagram. And is located between two chaotic areas, and the path branched by cycle doubling is changed into cycle vibration from chaotic evolution. This is in accordance with document [30 ]]Is consistent with the view of (a).
FIG. 6 is a cross section of Poincare corresponding to different excitation amplitudes when the cylindrical shell produces chaotic motion, where F of (a) z =5.05n, F of (b) z =7.3N, F of (c) z =8.55N, F of (d) z ＝9.8N。
Fig. 6 shows poincare cross sections at different excitation amplitudes. Some structures with very interesting fractal features were found, called singular attractors. In general, attractors can be classified into four types, i.e., point attractors, limit cycle attractors, torus attractors, and singular attractors. Here we mainly introduce singular attractors, attractors in the phase space, where the points do not repeat nor the tracks intersect, but they remain always in the same region of the phase space. Unlike limit cycle or point attractors, singular attractors are non-periodic. Singular attractors may take a wide variety of different forms. As the bifurcation parameters change, they will rotate and stretch to different degrees, appearing to be complex in structure and different in shape, but in fact, there is self-similarity between locally and globally. Furthermore, the presence of the singular attractor also indicates that even though the movement of the cylindrical shell is irregular and unpredictable, its movement area is still determined for high deflection vibrations. In other words, the radial movement of any point of the shell is limited to the area defined by the singular attractors, 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 a singular attractor, although they will separate from each other over time, they will not escape from the area defined by the singular attractor.
Further, to investigate the influence of the excitation frequency, the excitation amplitude was taken as F z =5.05n, gives a bifurcation diagram of the effect of the excitation frequency on the cylindrical shell movement. FIG. 7 is a bifurcation diagram depicting radial movement of the housing without using an excitation frequency.
FIG. 7 depicts a bifurcation diagram of radial motion of the cylindrical shell with a fixed excitation amplitude and different excitation frequencies. Obviously, with the change of frequency, the periodic motion can be evolved from the chaotic distinction and vice versa. Furthermore, when the ratio of natural frequency to 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 depicting radial chaotic motion of a housing at different excitation frequencies, wherein (a) Ω=0.909 ω 14 Omega = 0.998ω of (b) 14 Omega = 1.001 omega of (c) 14 Omega = of (d)1.0375ω 14 . Fig. 8 illustrates a poincare cross section with a specific excitation frequency. The shape characteristics of these singular attractors are similar to those of fig. 6 (a), in particular fig. 8 (c). The elongate structural features distinguish it from fig. 8 and 6, which means that singular attractors with smaller excitation amplitudes may have an elongate shape.
3.2.2 influence of structural parameters
This section discusses the effect of the aspect ratio α on the radial movement of the cylindrical shell. For thin shells, the diametral length ratio beta=0.5 and the excitation amplitude F are set z =5.05n. Fig. 9 is a bifurcation diagram describing radial movement of the cylindrical shell at different thickness to radius ratios, as shown in fig. 9, where radial movement of the cylindrical shell exhibits frequent alternating between periodic and chaotic movement when the thickness to radius ratio is less than 0.02. When the thickness-to-diameter ratio is greater than 0.02, the radial movement of the shell is substantially periodic, i.e., within a certain range, increasing the thickness-to-diameter ratio is beneficial to improving the movement stability of the thin-walled cylindrical shell. Fig. 10 is a poincare cross section depicting radial chaotic motion of a shell at different thickness to diameter ratios, where (c) α= 0.0134951, (d) α= 0.0135199, (e) α= 0.0135220, and (f) α= 0.017095. Fig. 10 shows poincare sections at different thick radii. Again, the presence of the singular attractor was demonstrated. The results show that the singular attractors under different thickness-to-diameter ratio conditions are similar to the attractors described above, and that the attractors exhibit different structures within different parameter ranges for systems of discontinuous chaotic regions. In particular, fig. 10 (b) shows a period 9 motion, and as the thickness radius ratio α increases by a small amount, it can be seen from fig. 10 (c) that the radial motion is immediately converted into a chaotic motion having 3 isolated attractive domains. In addition, fig. 10 (d) shows a periodic 3 motion, and the periodic motion also evolves into chaos with an increase in the aspect ratio. The results show that when the thickness to diameter ratio parameter of the cylindrical shell is between the period and the chaotic region, a slight correction thereof may cause a drastic change in the vibration characteristics.
3.2.3 influence of Material parameters
Since the material parameters of rubber materials are typically obtained by fitting experimental data, the values obtained for different fitting methods may be slightly different.It is assumed that the material parameters will float 10% above and below the initial values given in the literature, namely: mu' 1 ＝k 1 μ 1 ,μ′ 2 ＝k 2 μ 2 Wherein μ' 1 ,μ′ 2 Is a variable material parameter, and k 1 ,k 2 ∈[0.9,1.1]In the literature, the following analysis of the movement characteristics of a variable parameter housing is given. FIG. 11 is a bifurcation diagram depicting radial movement of the housing with different material parameters (a) is μ 1 (b) is mu 2 Is a function of (a) and (b). FIG. 11 shows that the material parameter μ 1 The change of (2) will affect the motion characteristics of the cylindrical shell, the chaotic response will evolve from chaotic motion to periodic motion through periodic double bifurcation path, and the material parameter mu 2 The response to the cylindrical shell has less impact. Thereby embodying the material parameter mu 1 Fitting accuracy of (d) to material parameter mu 2 The fitting accuracy of (2) is more important.
3.3 nonlinear vibration of coupling Effect
To analyze interactions between modes, three modes (m=1, n=4; m=1, n= 0;m =3, n=0) were chosen for discretized displacement, and then some interesting nonlinear dynamics of the radial motion of the shell were studied. A simple comparative analysis was performed using the time response and poincare cross section. Fig. 12 is a no coupling case: time response and Poincare cross section at a given excitation amplitude, F z =9.8n, the mode (m=1, n=4) of (a), (the mode (m=1, n=0) of (b), (the mode (m=3, n=0) of (c).
Fig. 12 (a, b, c) shows the solution for the uncoupled case obtained using the fourth-order ringe-Kutta method. Interestingly, under multi-modal discretization conditions, there may also be chaos in the uncoupled case, but its fine structural features are no longer apparent. This may be due to an increase in the degree of freedom resulting in an increase in computational complexity, and thus a decrease in accuracy. Furthermore, since the coupling effect is not considered, the response between modalities is independent. This means that there is a chaotic response in one modality, and the response in the other modality can still be periodic. Furthermore, the amplitude of the axisymmetric mode is much smaller than the non-axisymmetric mode. This means that when the coupling effect is not considered, the pairUniaxially symmetric modal analysis is possible. Fig. 13 is a coupling case: time response and Poincare cross section at a given excitation amplitude, F z =9.8n, the mode (m=1, n=4) of (a), (the mode (m=1, n=0) of (b), (the mode (m=3, n=0) of (c).
Fig. 13 (a, b, c) shows the solution for the coupling case obtained by the fourth-order ringe-Kutta method. In axisymmetric modes, quasi-periodic motion replaces chaotic motion, making the response more regular. This means to some extent that the coupling effect between the modes can improve the stability of the motion. Meanwhile, due to the coupling effect, each order mode also has a synchronous effect, namely, the response characteristics of each order mode are similar. As shown in fig. 13, the poincare cross section of the three modes has five isolated areas.
Conclusion 4
The invention researches the problem of nonlinear vibration of a rubber cylindrical shell formed by incompressible Mooney-Rivlin materials under the radial simple harmonic excitation. Notably, in general, the mechanical behavior of rubber materials can be approximated by their relative superelastic constitutive relationships under certain conditions [32] (the present invention only considers the superelasticity of rubber materials, but does not consider other properties such as its viscoelasticity). Based on the Donnell thin shell theory and the small strain hypothesis, a differential equation set describing the nonlinear vibration of the shell is established. The nonlinear dynamics of the housing were investigated with corresponding bifurcation diagrams and poincare cross sections. The following five main conclusions were drawn:
(1) The increase of the thickness-diameter ratio and the diameter-length ratio can both improve the natural frequency of the mode, the thickness-diameter ratio has obvious influence on the mode with larger circumferential wave number, and the diameter-length ratio has obvious influence on the mode with smaller circumferential wave number.
(2) When the excitation amplitude is greater than a certain critical value, the radial motion alternates between the chaotic motion and the periodic motion in the form of double-period bifurcation. In addition, when the excitation amplitude is sufficiently large, the periodic motion can be branched from the chaotic section as the excitation frequency is changed.
(3) The aspect 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 that ratio when that value is less than a critical value. This means that small changes in the ratio can translate chaotic motion into periodic motion and vice versa.
(4) For the incompressible Mooney-Rivlin model, material parameter μ 1 Impact ratio mu on nonlinear vibration behavior of rubber cylindrical shell 2 More remarkable;
(5) For the multi-modal case, the response of the cylindrical shell is similar to that of the single mode case when the coupling effect between the different modes is not considered. However, when the coupling effect is considered, a different conclusion will be drawn; in addition, the coupling effect may improve stability of the structural response.
Credit is given: the invention is funded by national natural science foundation (Nos. 11672069, 1170539, 11872145).
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Appendix
Linear stiffness matrix:
K 17 ＝K 71 ＝-π 2 h(μ 1 -μ 2 ),K 12 ＝K 13 ＝K 15 ＝K 16 ＝K 18 ＝K 19 ＝0,
K 39 ＝K 93 ＝-6π 2 h(μ 1 -μ 2 ),K 31 ＝K 32 ＝K 34 ＝K 35 ＝K 36 ＝K 37 ＝K 38 ＝0,
K 61 ＝K 62 ＝K 63 ＝K 64 ＝K 65 ＝K 67 ＝K 68 ＝K 69 ＝0,
K 91 ＝K 92 ＝K 94 ＝K 95 ＝K 96 ＝K 97 ＝K 98 ＝0。
Claims (2)
1. a method for describing large-deflection vibration of a rubber cylindrical shell is characterized by comprising the following steps of: a system of nonlinear differential equations describing cylindrical shell motion is given using the energy principle: let W e For the work done by periodic external force, the Rayleigh dissipation function is introduced to describe the work W done by non-conservative damping force d The corresponding expression is as follows:
wherein F is x ,F θ ,F z The unit distribution forces acting in the x, theta and z directions of the shell are respectively shown, and c is a damping coefficient; further calculate and get
c m,n Damping coefficients of corresponding modes;
order theω m,n For the natural frequency of the corresponding mode ρ m,n A modal mass for the modality; let q= (u) m,n ,v m,n ,w m,n ) T The elements being time-dependent q i (i=1, 2, …, 9); generalized force G i (i=1, 2, …, 9) derived from differentiation of the dissipation function and the virtual work of the external force, namely:
lagrange's equation describing cylindrical shell motion:
l=t-P is a Lagrangian function of the system, i is the number of modes;
a system of nonlinear differential equations describing cylindrical shell motion, namely:
m is a mass matrix, K is a linear stiffness matrix, K 2 And K 3 The two-and three-dimensional nonlinear stiffness matrices are respectively, C is a rayleigh damping matrix, and c=βk+γm, β, γ is an experimentally determined constant, q= { q 1 ,q 2 ,…,q 9 } 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 high deflection vibration of a rubber cylinder shell as recited in claim 1, wherein:
the following signs were introduced
Multiplying M simultaneously on both sides of (25) -1 From the formula (25) and the formula (26):
wherein the method comprises the steps of
ζ i ＝ω m,n,i ζ m,n,i ，ω m,n,i And zeta m,n,i (i=1, 2, …, 9) are natural frequency and damping ratio, respectively;
under the condition of neglecting plane inertia and damping terms, the equation (27) is processed based on a degree-of-freedom condensation method, and the following approximate motion and deformation relation exists:
in addition, in the case of the optical fiber,
wherein the method comprises the steps of
The following differential equation of motion is obtained in conjunction with equation (30):
wherein the method comprises the steps of
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