Non-linear free vibrations of Kelvin–Voigt visco-elastic beams

A full visco-elastic non-linear beam with cubic non-linearities is considered, and the governing equations of motion of the system for large amplitude vibrations are derived. By using the method of multiple scales, the non-linear mode shapes and natural frequencies of the beam are then analytically formulated. The resulting formulations for amplitude, non-linear natural frequencies and mode shapes can be used for any type of boundary conditions. Next, method of Galerkin is used to separate the time and space variables. The equations of motion show the presence of a non-linear damping term in addition to the ones with non-linear inertia and geometry. As it is known, the presence of non-linear inertia and the geometric terms make the non-linear natural frequencies to be dependent on constant amplitude of vibration. But, when damping non-linearities are present, it is seen that the amplitude is exponentially time-dependent, and so, the non-linear natural frequencies will be logarithmically time-dependent. Additionally, it is shown that the mode shapes will be dependent on the third power of time-dependent amplitude. The analytical results are applied to hinged–hinged and hinged–clamped boundary conditions and the results are compared with numerical simulations. The results match very closely for both cases specially for the case of hinged–hinged beam.     

A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations

In this paper, the generalized differential quadrature (GDQ) method is applied to study the dynamic behavior of functionally graded materials (FGMs) and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner-Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Firstly, the differential quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner-Mindlin and the Toorani-Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal those are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of shell and panel structures considered.     

频率修正在超声变幅杆设计中的研究与应用

超声变幅杆在沿轴线方向作纵向振动时,由于存在泊松效应,纵向振动会引起变幅杆沿径向方向的横向振动。传统变幅杆在设计过程中,忽略横向振动的影响则会导致计算出的谐振频率高于实际频率。为了让变幅杆理论设计谐振频率与实际频率相符合,分别对简单圆柱杆和阶梯形变幅杆进行研究,推导出常用半径比范围内简单阶梯形变幅杆的频率修正公式,使其满足设计和使用要求,并可以减少能量传输过程中的能量损耗。     

周转斜盘发动机模态分析及减振方案设计

针对周转斜盘发动机结构复杂,有限元模型规模大且分析效率较低等问题,引入子结构模态综合法用于发动机振动特性分析和减振设计。基于Ansys软件建立了发动机模态综合法分析模型,计算了发动机模态。分析了发动机振动激励源和振动传递路径,讨论了发动机结构减振措施,提出了发动机关键轴承处采用柔性支撑的减振方案,拟定了柔性支撑减振器可行结构形式。基于模态综合法计算模型,将减振器作为子结构建立采取减振措施的动力学分析模型。谐响分析表明:减振方案在低频段很好的效果。     

A comparative study of the eigenvalue solutions for mass-loaded beams under classical boundary conditions

A comparative study of the eigenfrequency analysis for an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented in this short note. The dimensionless frequency equation for different combinations of classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions. Two formulation methods have been commonly used for the boundary-value problem. One is to adopt a single frame originated from the beam's left-end, while another is by dual frames associated with the concentrated mass. It is found that the forms derived by dual frames are more compact than the corresponding expressions by using the single frame. Nevertheless, the comparison for all the cases shows that the dual-frame expressions need more time to obtain the same set of eigenvalues if compared with the time by using the single-frame expressions.     

Natural frequencies of composite plates with random material properties using higher-order shear deformation theory

Composites are known to display a considerable amount of scatter in their material properties due to large number of parameters associated with the manufacturing and fabrication processes. In the present work, the material properties have been taken as random variables for accurate prediction of the system behavior. Higher order shear theory including rotatory inertia effects has been accounted for in the system dynamic equations. A first order perturbation technique has been employed to obtain the solution of the governing equations. An approach has been outlined for obtaining closed form expressions for the variances of eigen solutions. The effects of side to thickness ratio and variation in standard deviation of the material properties have been investigated for cross-ply symmetric and anti-symmetric laminates. The mean and standard deviations of the first five natural frequencies have been worked out for laminated rectangular plates with all edges simply supported. The higher order shear deformation theory results have been validated with Monte Carlo simulation results and compared with the results based on classical laminate and first order shear deformation theories.