scholarly journals Transverse Vibration of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Zhongmin Wang ◽  
Rongrong Li
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Cantilever Beam ◽  
Cross Section ◽  
Transverse Vibration ◽  
Slenderness Ratio ◽  
Circular Cross Section ◽  
Dimensionless Parameters ◽  
Inner Radius ◽  
Tapered Cantilever Beam

Problems related to the transverse vibration of a rotating tapered cantilever beam with hollow circular cross-section are addressed, in which the inner radius of cross-section is constant and the outer radius changes linearly along the beam axis. First, considering the geometry parameters of the varying cross-sectional beam, rotary inertia, and the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. Next, dimensionless parameters and variables are introduced for the differential equation and boundary conditions, and the differential quadrature method (DQM) is employed to solve this differential equation with variable coefficients. Combining with discretization equations for the differential equation and boundary conditions, an eigen-equation of the system including some dimensionless parameters is formulated in implicit algebraic form, so it is easy to simulate the dynamical behaviors of rotating tapered beams. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.

scholarly journals Small Symmetrical Deformation of Thin Torus with Circular Cross-Section

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cross Section ◽  
Circular Cross Section ◽  
Complex Form ◽  
Real Form ◽  
Element Analysis ◽  
Numerical Studies ◽  
Ode System ◽  
Symmetrical Deformation

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner's formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell.

The Transverse Vibration of a Doubly Tapered Beam

1977 ◽  
Vol 44 (1) ◽  
pp. 123-126 ◽  
Author(s):  
D. O. Banks ◽  
G. J. Kurowski
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Cross Section ◽  
Transverse Vibration ◽  
Eigenvalues And Eigenfunctions ◽  
Free Boundary Conditions ◽  
Tapered Beam ◽  
Transverse Vibrations ◽  
The Cross ◽  
Homogeneous Beam

We analyze the transverse vibrations of a thin homogeneous beam which is symmetric with respect to the x-y and x-z planes. The cross section of the beam at x is assumed to have the form D(x)={(x,y,z)|x∈[0,1],y=xαy1,z=xβz1,(y1,z1)∈D1} where D1 is the cross section at x = 1. Expressions are obtained from which the eigenvalues and eigenfunctions can be easily found for 0 ≤ α < 2 and all combinations of clamped, hinged, guided, and free boundary conditions at both ends of the beam.

Natural Frequencies of In-Plane Vibration of Arcs

1983 ◽  
Vol 50 (2) ◽  
pp. 449-452 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
K. Tanaka
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Plane Vibration

The natural frequencies of in-plane vibration are presented for uniform arcs with circular cross section under all combinations of boundary conditions.

Calculation of the Natural Frequencies of Transverse Vibration of Complex Beams Using the Differential-Matrix Equations

2011 ◽  
Vol 243-249 ◽  
pp. 284-289
Author(s):  
Yu Zhang
Keyword(s):  
Iterative Method ◽  
Boundary Conditions ◽  
Cross Section ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Composite Beams ◽  
Matrix Equations ◽  
Generalized Differential ◽  
Set Up ◽  
Differential Matrix

The generalized differential-matrix equations of transverse vibration of the beams were set up and they were solved by means of Cauchy sequence iterative method. Then according to the boundary conditions at two ends of the beams the natural frequencies of the transverse vibration of the different beams including the complex beams of non-uniform section and composite beams under different boundary conditions were figured out. The form of the differential-matrix is simple. The calculation of the sequence iterations can be accomplished by simple computer program. Using the method in this paper, the amount of work of calculation is reduced greatly and the results are accurate compared with the approximate method in which a beam of non-uniform section is replaced by many small segments of equal cross-section.

Finite amplitude convection cells

1967 ◽  
Vol 30 (3) ◽  
pp. 577-600 ◽  
Author(s):  
J. L. Robinson
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Boundary Layers ◽  
Cross Section ◽  
Circular Cross Section ◽  
Maximum Heat ◽  
Mean Field Equations

In this paper we consider two-dimensional steady cellular motion in a fluid heated from below at large Rayleigh number and Prandtl number of order unity. This is a boundary-layer problem and has been considered by Weinbaum (1964) for the case of rigid boundaries and circular cross-section. Here we consider cells of rectangular cross-section with three sets of velocity boundary conditions: all boundaries free, rigid horizontal boundaries and free vertical boundaries (referred to here as periodic rigid boundary conditions), and all boundaries rigid; the vertical boundaries of the cells are insulated. It is shown that the geometry of the cell cross-section is important, such steady motion being not possible in the case of free boundaries and circular cross-section; also that the dependence of the variables of the problem on the Rayleigh number is determined by the balances in the vertical boundary layers.We assume only those boundary layers necessary to satisfy the boundary conditions and obtain a Nusselt number dependence $N \sim R^{\frac{1}{3}}$ for free vertical boundaries. For the periodic rigid case, Pillow (1952) has assumed that the buoyancy torque is balanced by the shear stress on the horizontal boundaries; this is equivalent to assuming velocity boundary layers beside the vertical boundaries (rather than the vorticity boundary layers demanded by the boundary conditions) and leads to a Nusselt number dependence N ∼ R¼. If it is assumed that the flow will adjust itself to give the maximum heat flux possible the two models are found to be appropriate for different ranges of the Rayleigh number and there is good agreement with experiment.An error in the application of Rayleigh's method in this paper is noted and the correct method for carrying the boundary-layer solutions round the corners is given. Estimates of the Nusselt numbers for the various boundary conditions are obtained, and these are compared with the computed results of Fromm (1965). The relevance of the present work to the theory of turbulent convection is discussed and it is suggested that neglect of the momentum convection term, as in the mean field equations, leads to a decrease in the heat flux at very high Rayleigh numbers. A physical argument is given to derive Gill's model for convection in a vertical slot from the Batchelor model, which is appropriate in the present work.

scholarly journals Thermoelastic Coupling Vibration and Stability Analysis of Rotating Annular Sector Plates

2019 ◽  
Vol 2019 ◽  
pp. 1-18
Author(s):  
Yongqiang Yang ◽  
Zhongmin Wang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Transverse Vibration ◽  
Differential Quadrature Method ◽  
Angular Speed ◽  
Thermoelastic Coupling ◽  
Coupling Vibration ◽  
Annular Sector ◽  
Sector Plate ◽  
Annular Sector Plate

This study investigates the thermoelastic coupling vibration and stability of rotating annular sector plates. Based on Hamilton’s principle and thermal conduction equation with deformation effect, the differential equation of transverse vibration for a rotating annular sector plate is established. The differential equation of vibration and corresponding boundary conditions are discretized by the differential quadrature method. Then, the thermoelastic coupling transverse vibrations under three different boundary conditions are calculated. The change curve of the first three order dimensionless complex frequencies of the rotating annular sector plate with the dimensionless angular speed are analyzed in the case of the thermoelastic coupling and uncoupling. The effects of the dimensionless angular speed, the ratio of inner to outer radius, the sector angle, and the dimensionless thermoelastic coupling coefficient on transverse vibration and stability of the annular sector plate are discussed. Finally, we obtained the type of instability and corresponding critical speed of the rotating annular sector plate in the case of the thermoelastic coupling and uncoupling.

Size-Dependent Free Vibration of Microbeams Submerged in Fluid

2020 ◽  
Vol 20 (12) ◽  
pp. 2050131
Author(s):  
H. C. Li ◽  
L. L. Ke ◽  
J. Yang ◽  
S. Kitipornchai
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Couple Stress Theory ◽  
Slenderness Ratio ◽  
Circular Cross Section ◽  
Coupled System ◽  
Modified Couple Stress Theory ◽  
Mode Shapes ◽  
Size Dependent

The size-dependent free vibration of microbeams submerged in fluid is presented in this paper based on the modified couple stress theory. Two different cross-section shapes of microbeams are considered, i.e. the circular cross-section and rectangular cross-section. This nonclassical microbeam model is introduced for capturing the size effect of microstructures. In this fluid and structure coupled system, the effect of hydrodynamic loading on microbeams can be expressed by the added mass method. By using Hamilton’s principle and differential quadrature (DQ) method, we can derive governing equations of microbeams in fluid, and then rewrite them in the discretized form. The frequencies and mode shapes for microbeams are determined by proposing an iterative method. Numerical examples are given to show the effect of fluid depth, fluid density, length scale parameter, slenderness ratio, boundary condition and cross-section shape on the vibration characteristics.

Dynamic Deflection of a Cantilever Beam Carrying Moving Mass

2014 ◽  
Vol 592-594 ◽  
pp. 1040-1044
Author(s):  
Shakti P. Jena ◽  
D.R. Parhi
Keyword(s):  
Differential Equation ◽  
Numerical Analysis ◽  
Boundary Conditions ◽  
Cantilever Beam ◽  
Continuum Mechanics ◽  
Fourth Order ◽  
Moving Mass ◽  
Kutta Method ◽  
Beam Structure ◽  
Runge Kutta Method

In the present work, the dynamic deflection of a cantilever beam subjected to moving mass has been investigated theoretically and numerically. The mass is moved by an external force. The effects of mass magnitude and the speed of the moving mass on the response of the beam structure have been investigated. Using continuum mechanics the differential equation for the systems have been developed and solved by fourth order Runge-Kutta method with different boundary conditions. Numerical analysis has been carried out with different examples to describe the response of the beam structure.

scholarly journals Ludwick Cantilever Beam in Large Deflection Under Vertical Constant Load

2016 ◽  
Vol 10 (1) ◽  
pp. 23-37 ◽  
Author(s):  
Alberto Borboni ◽  
Diego De Santis ◽  
Luigi Solazzi ◽  
Jorge Hugo Villafañe ◽  
Rodolfo Faglia
Keyword(s):  
Differential Equation ◽  
Cantilever Beam ◽  
Cross Section ◽  
Large Deflection ◽  
External Action ◽  
Constant Load ◽  
Large Deflections ◽  
Eulerian Method ◽  
Vertical Displacements ◽  
Newton Raphson

The aim of this paper is to calculate the horizontal and vertical displacements of a cantilever beam in large deflections. The proposed structure is composed with Ludwick material exhibiting a different behavior to tensile and compressive actions. The geometry of the cross-section is constant and rectangular, while the external action is a vertical constant load applied at the free end. The problem is nonlinear due to the constitutive model and to the large deflections. The associated computational problem is related to the solution of a set of equation in conjunction with an ODE. An approximated approach is proposed here based on the application Newton-Raphson approach on a custom mesh and in cascade with an Eulerian method for the differential equation.

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