Vibration Frequencies for a Uniform Beam With One End Elastically Supported and Carrying a Mass at the Other End

1975 ◽  
Vol 42 (4) ◽  
pp. 878-880 ◽  
Author(s):  
D. A. Grant
Keyword(s):  
Normal Modes ◽  
Frequency Equation ◽  
The Other ◽  
Concentrated Mass ◽  
Mass Moment ◽  
Wide Range ◽  
Modes Of Vibration ◽  
Mass Moment Of Inertia ◽  
Elastically Supported ◽  
Range Of Values

In this paper the author obtains the frequency equation for the normal modes of vibration of uniform beams with linear translational and rotational springs at one end and having a concentrated mass at the other free end. The eigenfrequencies for the fundamental mode are given for a wide range of values of mass ratio, mass moment of inertia ratios, and stiffness ratios.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Free Vibrations of Constrained Beams

1952 ◽  
Vol 19 (4) ◽  
pp. 471-477
Author(s):  
Winston F. Z. Lee ◽  
Edward Saibel
Keyword(s):  
Natural Frequencies ◽  
Degree Of Approximation ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Continuous Beam ◽  
Concentrated Mass ◽  
Simple Manner ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Rigid Supports

Abstract A general expression is developed from which the frequency equation for the vibration of a constrained beam with any combination of intermediate elastic or rigid supports, concentrated masses, and sprung masses can be found readily. The method also is extended to the case where the constraint is a continuous elastic foundation or uniformly distributed load of any length. This method requires only the knowledge of the natural frequencies and natural modes of the beam supported at the ends in the same manner as the constrained beam but not subjected to any of the constraints between the ends. The frequency equation is obtained easily and can be solved to any desired degree of approximation for any number of modes of vibration in a quick and simple manner. Numerical examples are given for a beam with one concentrated mass, for a beam with one sprung mass, and a continuous beam with one sprung mass.

A study of the free vibration of flexible-link flexible-joint manipulators

2011 ◽  
Vol 225 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
M Vakil ◽  
R Fotouhi ◽  
P N Nikiforuk ◽  
F Heidari
Keyword(s):  
Joint Stiffness ◽  
Sensitivity Study ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Flexible Link ◽  
Flexible Joint ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Payload Mass ◽  
Analytical Expressions

In this article, explicit expressions for the frequency equation, mode shapes, and orthogonality of the mode shapes of a Single Flexible-link Flexible-joint manipulator (SFF) are presented. These explicit expressions are derived in terms of non-dimensional parameters which make them suitable for a sensitivity study; sensitivity study addresses the degree of dependence of the system’s characteristics to each of the parameters. The SFF carries a payload which has both mass and mass moment of inertia. Hence, the closed-form expressions incorporate the effect of payload mass and its mass moment of inertia, that is, the payload mass and its size. To check the accuracy of the derived analytical expressions, the results from these analytical expressions were compared with those obtained from the finite element method. These comparisons showed excellent agreement. By using the closed-form frequency equation presented in this article, a study on the changes in the natural frequencies due to the changes in the joint stiffness is performed. An upper limit for the joint stiffness of a SFF is established such that for the joint stiffness above this limit, the natural frequencies of a SFF are very close to those of its flexible-link rigid-joint counterpart. Therefore, the value of this limit can be used to distinguish a SFF from its flexible-link rigid-joint manipulator counterpart. The findings presented in this article enhance the accuracy and time-efficiency of the dynamic modeling of flexible-link flexible-joint manipulators. These findings also improve the performance of model-based controllers, as the more accurate the dynamic model, the better the performance of the model-based controllers.

Experimental Determination of the Mass Moment of Inertia of a Flywheel Using Dynamics and Statistical Methods

2015 ◽  
Author(s):  
Pezhman Hassanpour ◽  
Monica Weaser ◽  
Ray Colquhoun ◽  
Khaled Alghemlas ◽  
Abdullah Alrashdan
Keyword(s):  
Experimental Determination ◽  
Accurate Result ◽  
Moment Of Inertia ◽  
The Other ◽  
Experimental Measurements ◽  
Ball Bearings ◽  
Experiment Data ◽  
Mass Moment ◽  
Mass Moment Of Inertia

This paper presents the analysis of the mass moment of inertia (MMI) of a flywheel using experiment data. This analysis includes developing two models for determining the MMI of the flywheel. The first model considers the effect of mass moment of inertia only, while the second model takes the effect of friction in the ball bearings into consideration. The experiment results have been used along with both models to estimate the MMI of the flywheel. It has been demonstrated that while the model with no friction can be used for estimating the MMI to some extent, the model with friction produces the most accurate result. On the other hand, an effective application of the model with friction requires several experimental measurements using different standard masses. This translates into more expensive method in terms of experiment time and equipment cost.

scholarly journals Application of the Electronic Differential Analyzer to the Oscillation of Beams, Including Shear and Rotary Inertia

1955 ◽  
Vol 22 (1) ◽  
pp. 13-19
Author(s):  
C. E. Howe ◽  
R. M. Howe
Keyword(s):  
Transverse Shear ◽  
Normal Modes ◽  
Bending Moment ◽  
Accurate Method ◽  
Rotary Inertia ◽  
Lateral Vibration ◽  
Wide Range ◽  
Modes Of Vibration ◽  
Set Up ◽  
Differential Analyzer

Abstract The equations for normal modes of lateral vibration of beams are set up on the electronic differential analyzer. Beam deflections due to transverse shear and rotary-inertia forces are included. The differential analyzer is shown to be a fast and accurate method for solving the problem. Analyzer outputs include mode shape, slope, bending moment, and shear force along the beam. Curves showing the normal-mode frequencies for the first three modes of vibration of a uniform free-free beam are presented for a wide range of transverse shear and rotary-inertia parameters. The electronic differential analyzer also is utilized to solve the problem of lateral vibration of nonuniform beams.

Connecting Rod Dynamics

2014 ◽  
Author(s):  
Samuel Doughty
Keyword(s):  
The Other ◽  
Connecting Rod ◽  
Complex Motion ◽  
Additional Mass ◽  
Mass Model ◽  
Usual Model ◽  
Automotive Engines ◽  
Correct Analysis ◽  
Mass Moment ◽  
Mass Moment Of Inertia

The complex motion of a slider-crank connecting rod has motivated analysts to work in terms of an “equivalent link,” comprised of two point masses at the ends of a massless link, where one end is located at the crank pin and the other end is at the wrist pin. It has long been known that this limited model is not fully equivalent in the dynamic sense, but the practice persists and errors are routinely introduced into torsional vibration and shaking force calculations. The purpose of this paper is to expose this error and show the nature of its effects. This is accomplished by means of a fully correct analysis, based on the two point mass model extended to include a massless additional mass moment of inertia, and then examining the terms that the usual model omits. Numerical results are given for several actual automotive engines.

scholarly journals Stability Analysis for Unsymmetrical Shaft With Flexible Bladed-Disk

1997 ◽  
Author(s):  
Samer Masoud ◽  
Naim Khader
Keyword(s):  
Equations Of Motion ◽  
Rotor Speed ◽  
Governing Equations ◽  
Bladed Disk ◽  
Shaft System ◽  
Special Cases ◽  
Mass Moment ◽  
Wide Range ◽  
Mass Moment Of Inertia ◽  
Bearing Stiffness

The governing equations of motion for a rotating flexible blade-rigid disk-flexible shaft system are derived. The bladed disk is attached at one end of an asymmetric shaft with uniformly distributed mass, mass moment of inertia, and stiffness. The shaft is held by two isotropic supports; one at the far end from the bladed disk, modeled by two translational and two rotational springs, and an intermediate support, modeled by two translational springs only. The effect of shaft asymmetry on the dynamics behavior of the rotating bladed disk shaft system is examined over a wide range of rotational speed, and for different combinations of springs’ stiffness, which determines the type of shaft supports. The cantilever, and the simply supported shaft with an over hang can be looked upon as special cases of the described above shaft configuration, since the former is obtained by assigning large stiffness for both translational and rotational springs at the end support, and zero spring stiffness at the intermediate one, whereas the latter is obtained by assigning large stiffness for the translational springs at both supports and zero stiffness for the rotational springs. Stability boundaries are calculated, and presented in terms of shaft asymmetry and rotor speed for given bearing stiffness.

Uniqueness in the inversion of inaccurate gross Earth data

1970 ◽  
Vol 266 (1173) ◽  
pp. 123-192 ◽  
Keyword(s):  
Normal Mode ◽  
Normal Modes ◽  
Variance Matrix ◽  
The Earth ◽  
Mass Moment ◽  
Frequency Independent ◽  
Finite Set ◽  
Mass Moment Of Inertia ◽  
Limits Of Error ◽  
Local Average

A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation ( Q ) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.

Vibration of a Beam With Concentrated Mass, Spring, and Dashpot

1948 ◽  
Vol 15 (1) ◽  
pp. 65-72
Author(s):  
Dana Young
Keyword(s):  
Analytical Method ◽  
Composite System ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Concentrated Mass ◽  
Orthogonal Functions ◽  
Numerical Examples ◽  
Modes Of Vibration ◽  
Concentrated Masses ◽  
Mass Spring

Abstract An analytical method is developed for determining the natural frequencies of a composite system which consists of a uniform beam with a concentrated mass, spring, and dashpot attached at any point along the length of the beam. The method makes use of a series expansion in terms of the set of orthogonal functions which represent the normal modes of vibration of the beam alone. Numerical examples are given for a beam with a combined concentrated mass and spring, for a beam with two concentrated masses, and for a beam with an attached dashpot.

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