Rotary Inertia and Shear in Beam Vibration Treated by the Ritz Method

Summary The natural frequencies of vibration of a cantilever beam allowing for rotary inertia and shear deformation are obtained by the approximate Ritz method. The workability of the method is dependent upon the approximating functions chosen for the dynamic displacement curves. A series of characteristic functions representing the normal modes of vibration of cantilever beams in simple flexure is used as the approximating functions for both deflections due to flexure and shear deformation. Good agreement is shown between frequencies obtained by the Ritz method and those resulting from an analytical solution. The effect upon the natural frequencies of allowing for rotary inertia alone is shown and it is seen to increase rapidly with mode number.

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The Normal Modes of Vibrations of Beams Having Noncollinear Elastic and Mass Axes

Journal of Applied Mechanics ◽

10.1115/1.4009049 ◽

1940 ◽

Vol 7 (3) ◽

pp. A97-A105

Author(s):

Clyne F. Garland

Keyword(s):

Physical Properties ◽

Natural Frequencies ◽

Ritz Method ◽

Normal Modes ◽

Longitudinal Axis ◽

Cantilever Beams ◽

Amplitude Ratios ◽

Numerical Example ◽

Modes Of Vibration ◽

Axis Passing

Abstract This analysis deals with vibration characteristics of cantilever beams in which the longitudinal axis, passing through the mass centers of the elementary sections, is not collinear with the longitudinal axis about which the beam tends to twist under the influence of an applied torsional couple. Expressions are derived from which the natural frequencies and normal modes of vibration of such a beam can be determined. The Rayleigh-Ritz method is employed to determine the frequencies and amplitude ratios. Following the development of the general expressions, more specific equations are derived which express the natural frequencies and relative amplitudes of motion in each of two normal modes of vibration. The theoretical relationships of the several physical properties of the beam to the natural frequencies of vibration are shown graphically. Finally a numerical example is presented for a particular beam, and the computed natural frequencies and normal modes are compared with those determined experimentally.

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Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1971_013_008_02 ◽

1971 ◽

Vol 13 (1) ◽

pp. 51-59 ◽

Cited By ~ 22

Author(s):

B. Dawson ◽

N. G. Ghosh ◽

W. Carnegie

Keyword(s):

Differential Equations ◽

Shear Deformation ◽

Cross Section ◽

Natural Frequencies ◽

Rectangular Cross Section ◽

Slenderness Ratio ◽

Twist Angle ◽

Equations Of Motion ◽

Rotary Inertia ◽

Cantilever Beams

This paper is concerned with the vibrational characteristics of pre-twisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotary inertia. A method of solution of the differential equations of motion allowing for shear deformation and rotary inertia is presented which is an extension of the method introduced by Dawson (1)§ for the solution of the differential equations of motion of pre-twisted beams neglecting shear and rotary inertia effects. The natural frequencies for the first five modes of vibration are obtained for beams of various breadth to depth ratios and lengths ranging from 3 to 20 in and pre-twist angle in the range 0–90°. The results are compared with those obtained by an alternative method (2), where available, and also to experimental results.

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The Effects of Shear Deformation and Rotary Inertia on the Lateral Frequencies of Cantilever Beams in Bending

Journal of Engineering for Industry ◽

10.1115/1.3428121 ◽

1972 ◽

Vol 94 (1) ◽

pp. 267-278 ◽

Cited By ~ 18

Author(s):

W. Carnegie ◽

J. Thomas

Keyword(s):

Finite Difference ◽

Shear Deformation ◽

Matrix Equation ◽

Equations Of Motion ◽

Flexural Vibration ◽

Experimental Results ◽

Rotary Inertia ◽

Cantilever Beams ◽

Algebraic Equations ◽

Good Agreement

This paper deals with the effect of shear deformation and rotary inertia on the frequencies of flexural vibration of pre-twisted and non-pre-twisted uniform and tapered cantilever beams. The equations of motion are derived and transformed into a set of linear simultaneous algebraic equations by using finite-difference solutions for the derivatives. The resulting eigenvalue matrix equation is solved for the frequency parameters by a QR transformation. The effects of various tapers, depth-to-length ratios and pre-twist angles on the frequencies of vibration are investigated for the first five modes. Results obtained are compared with those presented by other investigators where available and show good agreement. The experimental results presented also show good agreement with the corresponding theoretical values.

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Vibration of Rectangular Plates by the Ritz Method

Journal of Applied Mechanics ◽

10.1115/1.4010175 ◽

1950 ◽

Vol 17 (4) ◽

pp. 448-453 ◽

Cited By ~ 6

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.

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Vibration of Triangular Cantilever Plates by the Ritz Method

Journal of Applied Mechanics ◽

10.1115/1.4010935 ◽

1954 ◽

Vol 21 (4) ◽

pp. 365-370

Author(s):

B. W. Andersen

Keyword(s):

Cantilever Beam ◽

Natural Frequencies ◽

Ritz Method ◽

Plan View ◽

Modes Of Vibration ◽

Accuracy Of Results ◽

Isosceles Triangles

Abstract Using the method published by Ritz in 1909, natural frequencies and corresponding node lines have been determined for two symmetric and two antisymmetric modes of vibration of isosceles triangular plates clamped at the base and having length-to-base ratios of 1, 2, 4, and 7 and for the two lowest modes of right triangular plates clamped along one leg and having ratios of the length of the free leg to that of the clamped one of 2, 4, and 7. A nonorthogonal co-ordinate system was used which gave constant limits of integration over the area of the triangle. The co-ordinate transformation made it necessary to modify the functions used by Ritz in approximating deflections and to consider cross products in the integration. The integration was done numerically, using tables compiled by Young and Felgar in 1949. To check the accuracy of results, a solution was obtained to the problem of a vibrating cantilever beam of uniform depth and triangular plan view. The results obtained were found to be consistent with those obtained for the plates by using an eight-term series to approximate the deflections of the symmetric plates (isosceles triangles) and a six-term series to approximate the deflections of the unsymmetric plates (right triangles).

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The Natural Frequencies of Free and Constrained Non-Uniform Beams

Journal of the Royal Aeronautical Society ◽

10.1017/s0368393100073703 ◽

1960 ◽

Vol 64 (599) ◽

pp. 697-699 ◽

Cited By ~ 2

Author(s):

R. P. N. Jones ◽

S. Mahalingam

Keyword(s):

Degrees Of Freedom ◽

Natural Frequencies ◽

Ritz Method ◽

Normal Modes ◽

Iteration Process ◽

Conservative System ◽

Basic System ◽

Orthogonal Functions ◽

Added Masses ◽

The Given

The Rayleigh-Ritz method is well known as an approximate method of determining the natural frequencies of a conservative system, using a constrained deflection form. On the other hand, if a general deflection form (i.e. an unconstrained form) is used, the method provides a theoretically exact solution. An unconstrained form may be obtained by expressing the deflection as an expansion in terms of a suitable set of orthogonal functions, and in selecting such a set, it is convenient to use the known normal modes of a suitably chosen “ basic system.” The given system, whose vibration properties are to be determined, can then be regarded as a “ modified system,” which is derived from the basic system by a variation of mass and elasticity. A similar procedure has been applied to systems with a finite number of degrees of freedom. In the present note the method is applied to simple non-uniform beams, and to beams with added masses and constraints. A concise general solution is obtained, and an iteration process of obtaining a numerical solution is described.

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INFLUENCE OF A LIQUID ON THE NATURAL FREQUENCIES OF ALMOST CIRCULAR PLATES

International Journal of Applied Mechanics ◽

10.1142/s1758825114500525 ◽

2014 ◽

Vol 06 (05) ◽

pp. 1450052 ◽

Cited By ~ 2

Author(s):

MANUEL GASCÓN-PÉREZ ◽

PABLO GARCÍA-FOGEDA

Keyword(s):

Dynamic Characteristics ◽

Natural Frequencies ◽

Vibration Mode ◽

Normal Modes ◽

Virtual Mass ◽

Circular Plates ◽

Hankel Transformation ◽

Fluid System ◽

Surrounding Fluid ◽

Good Agreement

In this work, the influence of the surrounding fluid on the dynamic characteristics of almost circular plates is investigated. First the natural frequencies and normal modes for the plates in vacuum are calculated by a perturbation procedure. The method is applied for the case of elliptical plates with a low value of eccentricity. The results are compared with other available methods for this type of plates with good agreement. Next, the effect of the fluid is considered. The normal modes of the plate in vacuum are used as a base to express the vibration mode of the coupled plate-fluid system. By applying the Hankel transformation the nondimensional added virtual mass 2 increment (NAVMI) are calculated for elliptical plates. Results of the NAVMI factors and the effect of the fluid on the natural frequencies are given and it is shown that when the eccentricity of the plate is reduced to zero (circular plate) the known results of the natural frequencies for circular plates surrounded by liquid are recovered.

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The Out-Of-Plane Natural Frequencies of Curved Composite Beams Including the Effect of the Rotary Inertia and Shear Deformation

Proceedings of the Ninth International Conference on Computational Structures Technology ◽

10.4203/ccp.88.308 ◽

2009 ◽

Author(s):

B. Ayhan ◽

F. Kadioglu

Keyword(s):

Shear Deformation ◽

Natural Frequencies ◽

Composite Beams ◽

Rotary Inertia ◽

Out Of Plane

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Natural Frequency Characteristics of the Beam with Different Cross Sections Considering the Shear Deformation Induced Rotary Inertia

Applied Sciences ◽

10.3390/app10155245 ◽

2020 ◽

Vol 10 (15) ◽

pp. 5245

Author(s):

Chunfeng Wan ◽

Huachen Jiang ◽

Liyu Xie ◽

Caiqian Yang ◽

Youliang Ding ◽

...

Keyword(s):

Shear Deformation ◽

Cross Sections ◽

Timoshenko Beam ◽

High Frequency ◽

Natural Frequencies ◽

Beam Theory ◽

Equation Of Motion ◽

Timoshenko Beam Theory ◽

Rotary Inertia ◽

Cross Sectional

Based on the classical Timoshenko beam theory, the rotary inertia caused by shear deformation is further considered and then the equation of motion of the Timoshenko beam theory is modified. The dynamic characteristics of this new model, named the modified Timoshenko beam, have been discussed, and the distortion of natural frequencies of Timoshenko beam is improved, especially at high-frequency bands. The effects of different cross-sectional types on natural frequencies of the modified Timoshenko beam are studied, and corresponding simulations have been conducted. The results demonstrate that the modified Timoshenko beam can successfully be applied to all beams of three given cross sections, i.e., rectangular, rectangular hollow, and circular cross sections, subjected to different boundary conditions. The consequence verifies the validity and necessity of the modification.

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