环形陀螺敏感结构的平面振动分析

在薄壁圆环振动特性基础上,研究了振动环式微机械陀螺的支撑梁对环的振动模态及自然频率的影响。对一种外支撑式环形微机械陀螺敏感结构进行了ANSYS模态仿真,得到工作振动模态的变形量。以薄壁环2节点变形模态函数为参考函数对仿真变形量用最小二乘法拟合,拟合误差在4.5%以内,各函数系数一致性误差小于1.5%。基于支撑梁对环结构的模态函数影响较小的条件,用能量法和速度积分法得到结构的应变能和动能函数,进一步得到具有支撑梁环结构自然频率的近似解。选取4组支撑梁尺寸,其近似解与仿真结果的相对误差在±3%以内。     

FGM环扇形板的面内自由振动分析

假定功能梯度材料(FGM)的物性参数沿环扇形板径向按照幂律梯度变化,基于平面线弹性理论,建立了FGM环扇形板面内自由振动的运动控制微分方程。采用二维微分求积法(DQM)对FGM环扇形板面内自由振动的无量纲运动控制微分方程进行离散,数值求解了不同边界条件下FGM环扇形板面内自由振动的无量纲固有频率,同时也给出了FGM环扇形板扇形角为!/4时有限元商用软件ANSYS的部分计算结果,验证了本文方法的正确性。结果表明,在相应边界条件下,FGM环扇形板的梯度指标、内外半径比以及扇形角对无量纲固有频率均有影响,其计算结果和分析方法可供设计和研究参考。     

In-plane vibration of a circular ring with arbitrary concentrated elements by an analytical method

Archive of Applied Mechanics - This paper investigates the vibration characteristics of a circular ring with an arbitrary number of concentrated elements based on the Hamilton principle. The shear...     

In-plane and out-of-plane free vibration and stability of a curved rod in flow

This paper investigates the free vibration and stability of a curved rod in flow. The equations of the three-dimensional motions of the rod are derived by the Newtonian approach. The differential quadrature method (DQM) is introduced to formulate the discrete forms of the governing equations of the inextensible rod with clamped–clamped supports. Based on numerical calculations, the effects of several system parameters, especially the flow velocity, on the natural frequencies and stability of the system are discussed. Buckling and flutter instability are detected as the flow velocity is varied in a certain range. Moreover, a derivation of the generalized slender-body theory for such a deformable curved rod is given in Appendix A.     

射线寻迹法分析旋转不等截面圆环结构的动态响应

用射线寻迹法分析了旋转不等截面圆环的自由响应和受迫响应问题。首先,建立了单元局域坐标系,用行波、快衰和近场波动表示波在单元结构内的振动位移;其次,将激励力视为一种特殊的不连续节点,激励点两端为两个不同单元,考虑不同单元之间通过转递系数和转换矩阵耦合,并引入反射矩阵和透射矩阵处理行波在单元间的传播;再次,将结构的稳态响应表达为波动所有迹的叠加,而将自由振动表达为波幅矢量初始迹为零的情况;最后,根据波传递规则将弹性波动的反射矩阵和透射矩阵进行集成,对独立波动单元进行总装,从所得的动力学模型得到旋转不等截面圆环的响应。在给出了行波在该动态模型中传播特性的基础上,由数值算例可以发现旋转状态下圆环的各阶响应频率都高于静止状态下的响应频率。研究内容有助于高速轴承保持架、MEMS转子等的动态特性的精确计算。     

In-plane free vibration and stability of loaded and shear-deformable circular arches

The first known equations governing vibrations of preloaded, shear-deformable circular arches are derived according to a variational principle for dynamic problems concerning an elastic body under equilibrium initial stresses. The equations are three partial differential equations with variable coefficients. The governing equations are solved for arches statically preloaded with a uniformly distributed vertical loading, by obtaining a static, closed-form solution and an analytical dynamic solution from series solutions and dynamic stiffness matrices. Convergence to accurate results is obtained by increasing the number of elements or by increasing both the number of terms in the series solution and the number of terms in the Taylor expansion of the variable coefficients. Graphs of non-dimensional frequencies and buckling loads are presented for preloaded clamped arches. They clarify the effects of opening angle and thickness-to-radius ratio on vibration frequencies and buckling loads. The effects of static deformations on vibration frequencies are also investigated. This work also compares the results obtained from the proposed governing equations with those obtained from the classical theory neglecting shear deformation.     

In-plane modal testing of a free isotropic rectangular plate

In-plane vibration modes of an aluminum panel were experimentally identified from frequency response tests. Responses were measured on the panel edges and at selected locations on the panel surface. The measurements on the surface were made by attaching accelerometers oriented parallel to the panel plane. Resonance frequencies, relative damping ratios and mode shapes were established for the lowest 12 in-plane modes found in the frequency range between 1600 and 7000 Hz. A damping ratio of less than 0.05 percent of critical damping is proved to be valid for the aluminum panel. A finite element software was used to calculate 12 corresponding theoretical in-plane eigenfrequencies and mode shapes. An outline for a nondestructive procedure is suggested to estimate the input data for the elastic constants of an isotropic plate model. Two of the modes were used in analogy with the flexural vibration of beams and plates. The modes illustrate the deformation pattern including shear deformations, through the thickness, for the bending modes of thick beams or plates. The Rayleigh-Timoshenko theory also was used for the calculation of these two eigenfrequencies.     

高性能对称解耦结构的线振动陀螺

为了最大限度克服微机电陀螺的两个模态的相互耦合作用,提高微机电陀螺的综合性能指标,采用国内现有MEMS标准工艺方法,设计和制作了一种高性能单晶硅对称解耦结构的线振动陀螺。采用对称结构形式和保证陀螺驱动和检测模态振型都是弯曲振动模式,易于模态匹配;由于采用驱动模态和检测模态结构解耦方式,从微结构设计上大大降低了正交耦合误差影响,使陀螺具有输出零位小、零偏稳定性好的优点。测试结果表明：初次加工的样机,在大气中驱动和检测模态固有频率分别在2430Hz和2580Hz左右,在150Hz带宽内具有0.1～0.5（°）/s的分辨率;随着加工精度的提高和检测电路的改进,该陀螺在大气中15Hz带宽内实现0.008（°）/s的分辨率,在真空状态下,这种高性能单晶硅对称解耦结构的线振动陀螺性能会有进一步的提高。     

In-plane forced vibration of curved pipe conveying fluid by Green function method

The Green function method(GFM) is utilized to analyze the in-plane forced vibration of curved pipe conveying fluid, where the randomicity and distribution of the external excitation and the added mass and damping ratio are considered. The Laplace transform is used, and the Green functions with various boundary conditions are obtained subsequently. Numerical calculations are performed to validate the present solutions, and the effects of some key parameters on both tangential and radial displacements are further investigated. The forced vibration problems with linear and nonlinear motion constraints are also discussed briefly. The method can be radiated to study other forms of forced vibration problems related with pipes or more extensive issues.     

Identification of zones in a free evolving vortex ring

A direct numerical simulation of a three-dimensional free evolving vortex ring has been performed to understand some features of the dynamics of vortex rings. In previous experiments, a persistent wake has been observed and the direct simulation has clarified the causes of this wake formation. A kinematic alignment analysis of the vorticity vector with the eigenvectors of the strain-rate tensor, together with vorticity visualisations, have been employed to explain the growth of the Widnall instabilities. The analysis has shown the presence of two different zones, the first characterised by intense vorticity and low strain rate and the second with opposite features. In the former, there is a ring core which maintains its coherence even in the long term, in the latter there are elongated structures contributing to the deformation of the ring.     

In-plane vibration analysis of plates in curvilinear domains by a differential quadrature hierarchical finite element method

Free in-plane vibration analysis of plates is carried out by a differential quadrature hierarchical finite element method (DQHFEM). The NURBS (Non-Uniform Rational B-Splines) patches of geometries were first transformed into differential quadrature hierarchical (DQH) patches, and then the elastic field was discretized by the same DQH basis. The DQHFEM solved the compatibility problem caused by different parametrization of neighbouring patches of isogeometric analysis using NURBS. And mesh refinement in DQHFEM does not propagate from patch to patch. The DQHFEM matrices also have the embedding property as the hierarchical finite element method (HFEM). In-plane vibration analyses of plates of several planforms showed that the DQHFEM is similar as the fixed interface mode synthesis method that can analyse a structure using a few nodes on the boundary of substructure elements and only several clamped modes inside each substructure element, but the DQHFEM does not need modal analysis and is of high accuracy. The accuracy and convergence of the DQHFEM were validated through comparison with exact and approximate results in literatures and computed by the authors.     

On the non-linear vibration of a thin ring with steady pressurization

Large amplitude, flexural oscillations of an inextensible, linearly elastic, pressurized ring are analyzed. Non-linear governing equations describing the planar motion of a thin rod curved in its undeformed state and subject to a distributed load applied normal to the neutral axis are developed using Hamilton's extended principle. The equations are specialized to study the behavior of a circular ring, and approximate solutions are obtained for a single mode response by a perturbation technique. Free, undamped oscillations and forced response of the ring near resonance are discussed. The influence of the magnitude of pressurization on the non-linear character of the motion is investigated.     

Laminar free convection heat transfer from a horizontal ring

The analytical solution of laminar free convective heat transfer in an unlimited space from an isothermal horizontal ring with an adiabatic plug is presented. The results of theoretical considerations are presented as relation of the Nusselt and Rayleigh numbers: $$Nu_D = 1.151 \cdot (Ra_D )^{1/5} \cdot \Phi (\phi _0 )$$ \] where Φ(φ₀) is a function of shape coefficient of the ring (φ₀=d/D). The solution presented has been verified experimentally with rings of constant external diameter (D=0.06 [m]) and various internal diameters (d=0, 0.01, 0.02, 0.04 and 0.05 [m]). The fluid tested was glycerin. The theoretical predictions agree well with the experimental results.     

Frequency analysis of nonlinear free vibration of a cross string

This work concerns nonlinear free vibration of a cross string under large amplitude. The equations governing the nonlinear vibration of the cross string are derived at first from the Hamilton principle, and they take the form of Duffing equation. Then the perturbation method is used to solve the nonlinear coupled natural frequency of the cross string. The nonlinear natural frequency not only has the characteristic of nonlinearity, but also reflects the coupled characteristic, i.e., the natural frequency of the cross string varying with that of its constituent strings. The results show that the overall effect on the cross string is somehow averaged due to the nonlinearity of each constituent string, i.e., the natural frequencies of the cross string contain both the linear natural frequencies of the constituent strings and the nonlinear parts that depend upon the vibration amplitude, the diameter of one constituent string, the length ratio of the two strings, etc., but the contribution of each constituent string to the natural frequency is in different proportions.     

Flow stability of a flat plastic ring with free boundaries

The problem of two-dimensional unstable flow of an ideally plastic ring acted upon by internal pressure is formulated. The determination of the law of motion for the boundaries and of the time change of pressure is reduced to an ordinary nonlinear differential equation of the second order. For this equation a particular solution of the Cauchy problem is determined; this corresponds to a widening of the ring boundaries with a negative acceleration. For the field of initial velocities an estimate from above is available, expressed in terms of the original parameters. The very particular unstable flow obtained for an ideally plastic ring is also investigated with respect to stability to small harmonic perturbations of the velocity vector, the pressure, or the boundaries of the ring. It is shown that the fundamental flow is stable irrespective of the wave number. This result has been obtained by assuming that the inertial forces in the perturbed flow are small compared to the lasting ones.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 94–101, March–April, 1975.     

Thermal buckling and free vibration of FG truncated conical shells with stringer and ring stiffeners using differential quadrature method

In this paper, thermal buckling and free vibration of orthogonally stiffened functionally graded truncated conical shells in thermal environment is investigated. Conical shell has been stiffened by rings and stringers, and the influences of the stiffeners are evaluated by the aid of smearing method. The material properties of the structure are assumed to be changed continuously in the thickness direction. First, the initial thermal stresses are obtained accurately by solving the thermoelastic equilibrium equations. Then, by taking into account the initial thermal stresses, equations of motion as well as boundary conditions are obtained, applying the Hamilton’s principle and the first-order shear deformation theory. The natural frequencies of the system have been achieved, solving these governing equations with considering Differential Quadrature Method (DQM). In addition to Eigen frequency analysis, the critical buckling-temperature of the conical shell has been computed. Moreover, the effects of geometrical parameters, number of stiffeners, thermal environment and various boundary conditions on natural frequency of the system have been investigated. Finally, in order to validate the present work, the results are compared with those of other researches available in literature.     

In-plane bending vibration analysis of a rotating beam with multiple edge cracks by using the transfer matrix method

Meccanica - In this study, the effects of cracks on the natural frequencies of a rotating Bernoulli–Euler beam are investigated using a new numerical method in which these effects can be...     

Gradient single-crystal plasticity with free energy dependent on dislocation densities

This study develops a small-deformation theory of strain-gradient plasticity for single crystals. The theory is based on: (i) a kinematical notion of a continuous distribution of edge and screw dislocations; (ii) a system of microscopic stresses consistent with a system of microscopic force balances, one balance for each slip system; (iii) a mechanical version of the second law that includes, via the microscopic stresses, work performed during viscoplastic flow; and (iv) a constitutive theory that allows:

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the free energy to depend on densities of edge and screw dislocations and hence on gradients of (plastic) slip;

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the microscopic stresses to depend on slip-rate gradients.

The microscopic force balances when augmented by constitutive relations for the microscopic stresses results in a system of nonlocal flow rules in the form of second-order partial differential equations for the slips. When the free energy depends on the dislocation densities the microscopic stresses are partially energetic, and this, in turn, leads to backstresses in the flow rules; on the other hand, a dependence of these stresses on slip-rate gradients leads to a strengthening. The flow rules, being nonlocal, require microscopic boundary conditions; as an aid to numerical solutions a weak (virtual power) formulation of the flow rule is derived.