Vibrations of Spinning Rings With Radial and Circumferential Displacement Constraints

1991 ◽  
Author(s):  
Shyh-Chin Huang ◽  
Chen-Kai Su
Keyword(s):  
Natural Frequencies ◽  
Point Contact ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Complex Conjugate ◽  
Constrained System ◽  
Form Complex ◽  
Circumferential Displacement ◽  
On The Road ◽  
Displacement Constraints

Abstract The frequencies and mode shapes of rolling rings with radial and circumferential displacement constraints are investigated. The displacement constraints practically come from the point contact, e.g., rolling tire on the road, or other applications. The proposed approach to analysis is calculating the natural frequencies and modes of a non-contacted spinning ring, then employing the receptance method for displacement constraints. The frequency equation for the constrained system is hence obtained, and it can be solved numerically or graphically. The receptance matrix developed for the spinning ring is surprisingly found not symmetric as usual. Moreover, the cross receptances are discovered to form complex conjugate pairs. That is a feature that has never been described in literature. The results show that the natural frequencies for the spinning ring in contact, as expected, higher than those for the non-contacted ring. The variance of frequencies to rotational speeds are then illustrated. The analytic forms of mode shapes are also derived and sketched. The traveling modes are then shown for cases.

A Unified Frequency Equation and the Motion Analysis of a Moving Flexible Beam Carrying a Concentrated Mass

1999 ◽  
Author(s):  
S. Park ◽  
J. W. Lee ◽  
Y. Youm ◽  
W. K. Chung
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Open Loop ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Lumped Mass ◽  
Concentrated Mass ◽  
Forcing Function ◽  
The Mathematical Model

Abstract In this paper, the mathematical model of a Bernoulli-Euler cantilever beam fixed on a moving cart and carrying an intermediate lumped mass is derived. The equations of motion of the beam-mass-cart system is analyzed utilizing unconstrained modal analysis, and a unified frequency equation which can be generally applied to this kind of system is obtained. The change of natural frequencies and mode shapes with respect to the change of the mass ratios of the beam, the lumped mass and the cart and to the position of the lumped mass is investigated. The open-loop responses of the system by arbitrary forcing function are also obtained through numerical simulations.

Free Vibration of In-Planar Rotating Ring: An Analytical Solution

2000 ◽  
Author(s):  
Hamid R. Hamidzadeh
Keyword(s):  
Analytical Solution ◽  
High Speed ◽  
Systematic Approach ◽  
Natural Frequencies ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Stress Theory ◽  
Wave Numbers ◽  
Annular Disks ◽  
Circumferential Wave

Abstract An analytical method is presented to consider the vibration characteristics of high speed rotating rings. More specifically, a systematic approach based on an established solution for linear in-plane vibration of spinning annular disks is used to compute natural frequencies and mode shapes of rotating rings. The medium is considered to be homogenous, and elastic isotropic. The developed analytical solution is achieved by implementing the two-dimensional plane stress theory. The modal displacements and stresses at both inner and outer boundaries are determined, and the required boundary conditions are satisfied to obtain the frequency equation. The dimensionless natural frequencies for different modes, rotating speeds, and thickness ratios are computed. In addition, variations of dimensionless critical speeds for several circumferential wave numbers versus radius ratio of the ring are presented. The provided results are for the two different cases of clamped-free and free-free rings.

Analysis for Vibration of Laminated Shallow Shells with Non-Uniform Curvature

2007 ◽  
Vol 334-335 ◽  
pp. 85-88
Author(s):  
Daisuke Narita ◽  
Yoshihiro Narita
Keyword(s):  
Natural Frequencies ◽  
Ritz Method ◽  
Frequency Equation ◽  
Total Potential Energy ◽  
Surface Shape ◽  
Mode Shapes ◽  
Commercial Vehicles ◽  
Shallow Shells ◽  
Total Potential ◽  
Door Panel

Despite a large number of technical papers on vibration of composite shallow shells, all the previous papers have dealt with shallow shells with uniform curvature to avoid difficulty in the analysis. Recent composite products, however, require various surface designs of thin panels from the viewpoint of industrial design, for example, in the fender and door panel designs of commercial vehicles. The present study proposes an analytical method to deal with vibration of shallow shells with non-uniform curvature. An interpolating function is introduced to represent the required surface shape and the corresponding curvature is derived as a function of the position (x,y). The obtained curvature is substituted into the total potential energy of the shell, and the procedure is shown to derive a frequency equation in the Ritz method. Numerical examples clarifies the effects of non- uniform curvature on the natural frequencies and mode shapes.

scholarly journals Vibration Modes and Parameter Analysis of V-Shaped Electrothermal Microactuators

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Zhuo Zhang ◽  
Yueqing Yu ◽  
Xuping Zhang
Keyword(s):  
Modal Analysis ◽  
Natural Frequencies ◽  
Frequency Equation ◽  
Parameter Analysis ◽  
Mode Shapes ◽  
Element Analysis ◽  
Shaped Beam ◽  
The Finite Element Analysis ◽  
Parametric Studies ◽  
First Time

Comprehensive analysis on the modal characteristics of V-shaped electrothermal microactuators is presented in this paper for the first time. Considering the unique geometric characteristics of the V-shaped beam, that is, two inclined beams supporting a movable shuttle, both the lateral and longitudinal deflections are taken into account in the modal analysis. Boundary and continuity conditions are employed to obtain the frequency equation. Natural frequencies are then obtained by solving the frequency equation. Mode shapes corresponding to their natural frequencies are also calculated analytically. The theoretical modal analysis is verified with the finite element analysis using ANSYS software. Based on the model analysis, this paper further investigates the relationship between natural frequencies and the volume scaling of the V-shaped beam. Finally, comprehensive parametric studies in terms of material properties and structural dimensions are conducted to provide insights and guidance in designing the V-shaped beam electrothermal microactuators.

Exact Solutions for Longitudinal Vibration of Multi-Step Bars with Varying Cross-Section

1999 ◽  
Vol 122 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Q. S. Li
Keyword(s):  
Natural Frequencies ◽  
Longitudinal Vibration ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Transmission Tower ◽  
Power Functions ◽  
Closed Form Solutions ◽  
High Rise ◽  
One Step ◽  
Area Variation

Using appropriate transformations, the equation of motion for free longitudinal vibration of a nonuniform one-step bar is reduced to an analytically solvable equation by selecting suitable expressions, such as power functions and exponential functions, for the area variation. Exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step nonuniform bar are derived and used to obtain the frequency equation of multi-step bars. The new exact approach is presented which combines the transfer matrix method and closed form solutions of one step bars. A numerical example demonstrates that the calculated natural frequencies and mode shapes of a television transmission tower are in good agreement with the corresponding experimental data, and the selected expressions are suitable for describing the area variation of typical high-rise structures. [S0739-3717(00)00302-0]

Three-Dimensional Vibration Analysis of a Truncated Quadrangular Pyramid

1987 ◽  
Vol 54 (1) ◽  
pp. 115-120 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Tagawa
Keyword(s):  
Vibration Analysis ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Three Dimensional ◽  
Frequency Equation ◽  
The Body ◽  
Mode Shapes ◽  
Algebraic Polynomials ◽  
Transformation Of Variables ◽  
Unit Edge

An analysis is presented for the three-dimensional vibration problem of determining the natural frequencies and the mode shapes of a truncated quadrangular pyramid. For this purpose, the body is transformed into a right quadrangular prism with unit edge lengths by a transformation of variables. With the displacements of the transformed prism assumed in the forms of algebraic polynomials, the dynamical energies of the prism are evaluated, and the frequency equation is derived by the Ritz method. This method is applied to quadrangular pyramids in which the base is clamped and the other sides are free, and the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically, from which the vibration characteristics arising in the pyramids are studied.

Free Vibration of a Point-Supported Spherical Shell

1985 ◽  
Vol 52 (4) ◽  
pp. 890-896 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Muramoto
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Lagrangian Multiplier ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Lagrangian Multiplier Method

An analysis is presented for the free vibration of an elastically or a rigidly point-supported spherical shell. For this purpose, the deflection displacements of the shell are written in a series of the products of the associated Legendre functions and the trigonometric functions. The dynamical energies of the shell are evaluated, and the frequency equation is derived by the Ritz method. For a rigidly point-supported shell, the Lagrangian multiplier method is conveniently employed. The method is applied to a closed spherical shell supported at equispaced four points located along a parallel of latitude; the natural frequencies and the mode shapes are calculated numerically, and the effects of the point supports on the vibration are studied.

A beam with arbitrarily placed lateral dampers: Evolution of complex modes with damping

2016 ◽  
Vol 24 (2) ◽  
pp. 379-392 ◽  
Author(s):  
Y Chen ◽  
DM McFarland ◽  
BF Spencer ◽  
LA Bergman
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Numerical Study ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Dynamic Features ◽  
Complex Modes ◽  
Orthogonality Conditions ◽  
Boundary Condition Method ◽  
Damped Systems

Free vibration of a beam with multiple arbitrarily placed lateral viscous dampers is investigated to gain insight into the intrinsic dynamic features of non-proportional damped systems. In terms of virtual boundary condition method, complex modes of a damper-beam system are achieved, and the solution is also suitable for the beams that have different boundary conditions. The features of the wave numbers satisfying the frequency equation were discussed in theory. The orthogonality analysis conducted in this paper provides two orthogonality conditions for complex modes. Pseudoundamped natural frequencies, damping ratios and complex modes are surveyed via numerical study. The analysis on the evolution of complex modes shows that the increasing damping would lead to over damped modes, and the mode shape that corresponds to the small one of a pair of real-valued natural frequencies is close to the static deformation shape of a beam subjected to static forces located at the positions of the dampers. For the rest modes that would never be over damped with increasing damping, the mode shapes and corresponding psuedoundamped natural frequencies will converge to that of a beam with rolling supports located at where dampers are placed. The exact solution of free vibration of a multiple-span beam is presented in addition.

Tire Vibrations

1977 ◽  
Vol 5 (4) ◽  
pp. 202-225 ◽  
Author(s):  
G. R. Potts ◽  
C. A. Bell ◽  
L. T. Charek ◽  
T. K. Roy
Keyword(s):  
Natural Frequencies ◽  
Standing Waves ◽  
Resonant Mode ◽  
Mode Shapes ◽  
Resonant Vibrations ◽  
Resonant Frequencies ◽  
Radial Tire ◽  
Analysis Techniques ◽  
Thin Ring ◽  
Good Agreement

Abstract Natural frequencies and vibrating motions are determined in terms of the material and geometric properties of a radial tire modeled as a thin ring on an elastic foundation. Experimental checks of resonant frequencies show good agreement. Forced vibration solutions obtained are shown to consist of a superposition of resonant vibrations, each rotating around the tire at a rate depending on the mode number and the tire rotational speed. Theoretical rolling speeds that are upper bounds at which standing waves occur are determined and checked experimentally. Digital Fourier transform, transfer function, and modal analysis techniques used to determine the resonant mode shapes of a radial tire reveal that antiresonances are the primary transmitters of vibration to the tire axle.

Dynamic model for high-speed rotors based on their experimental characterization

2017 ◽  
Vol 2 (4) ◽  
pp. 25
Author(s):  
L. A. Montoya ◽  
E. E. Rodríguez ◽  
H. J. Zúñiga ◽  
I. Mejía
Keyword(s):  
Dynamic Model ◽  
High Speed ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Experimental Characterization ◽  
Rotating Systems ◽  
Computing Performance ◽  
The Impact ◽  
Stiffness Distribution ◽  
Good Agreement

Rotating systems components such as rotors, have dynamic characteristics that are of great importance to understand because they may cause failure of turbomachinery. Therefore, it is required to study a dynamic model to predict some vibration characteristics, in this case, the natural frequencies and mode shapes (both of free vibration) of a centrifugal compressor shaft. The peculiarity of the dynamic model proposed is that using frequency and displacements values obtained experimentally, it is possible to calculate the mass and stiffness distribution of the shaft, and then use these values to estimate the theoretical modal parameters. The natural frequencies and mode shapes of the shaft were obtained with experimental modal analysis by using the impact test. The results predicted by the model are in good agreement with the experimental test. The model is also flexible with other geometries and has a great time and computing performance, which can be evaluated with respect to other commercial software in the future.

Sign in / Sign up

Export Citation Format

Share Document