Vibration of Rectangular Plates by the Ritz Method

Young, Dana

doi:10.1115/1.4010175

Cited by 352 publications

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“…An important component in the application of the Rayleigh±Ritz method is the selection of appropriate shape functions to describe the modal shapes of the vibrating blade. Initially characteristic beam functions were used, generally following the method of Young [9], and comparable results with those of Young [9] and Barton [10] were obtained for test cases where gravitational acceleration was not included. Simply supported plate functions, degenerated beam functions and orthogonal polynomials were also considered.…”

Section: Vibration Of Blade As a Thin Plate In The Rayleigh±ritz Methodsmentioning

confidence: 99%

The dynamics and limits on the scaling of a flexible kinetic sculpture

Gooch

Raine

2000

Proceedings of the Institution of Mechanical Engineers, Part C:

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Highly¯exible structures may be scaled up by dierent criteria that lead to dierent degrees of structural integrity as the scale increases. This paper ®rst explores the eects of dierent scaling rules, focusing in particular on a double-size version of the Len Lye kinetic sculpture Blade, whose aesthetic performance characteristics must be preserved as the size is increased. Stresses in the sculpture increase, reaching ®rst an economic limit beyond which frequent fatigue failure makes the sculpture too costly to operate. As the scale further increases, a size is reached where the sculpture will collapse. A dimensional parameter for comparing the values of the life to failure for dierent construction materials is also developed. The second part of the paper summarizes the results of dierent analytical approaches to the dynamics of Blade, both with and without the inclusion of an axial acceleration load due to gravity. The results of analyses are found to be in good agreement with experimental data. The paper closes by discussing some of the design implications of implementing the drive system for a double-size version of Blade that has been built and is now on public display. Keywords: kinetic sculpture,¯exible structure dynamics, dynamic scaling of structures NOTATION a blade length (m) A mn coecient used in the series representation of transverse blade de¯ection b blade width (m) B k Y C k constant terms used in the Gram±Schmidt process C m Blade material cost ($akg) D¯exural rigidity of a plate Eh 3 a121 À # 2 (N m) E Young's modulus (N/m 2 ) g gravitational constant (m/s 2 ) h blade thickness (m) I second moment of area (m 4 ) k 1 Y k 2 Y k 3 constants in the blade life and life±cost equations m mass per unit length (kg/m) N in-plane load per unit width (N/m) N f number of blade stress cycles to failure N pf number of blade performances to failure N pfa$ number of blade performances to failure per dollar cost P dimensionless blade life parameter R radius of curvature (m) t time (s) T 3 length of Blade performance in the third-order mode shape (s) w transverse blade displacement (m) x axial blade coordinate (m) y coordinate across the width dimension of the blade (m) stability factor crit stability factor at the critical buckling load ik Kronecker delta loss coecient ! roots of the characteristic frequency equation for a clamped±free beam " frequency parameter # Poisson's ratio & density (kg/m 3 ' normal axial or bending stress (N/m 2 ) 0 characteristic functions of a clamped±free beam 2 characteristic functions of a free±free beam 3 frequency (r/s) characteristic value 3 2 &habaD Subscripts iY kY mY nY pY qY rY s positive integers o original scale sc scaled The MS was

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Section: Vibration Of Blade As a Thin Plate In The Rayleigh±ritz Methodsmentioning

confidence: 99%

The dynamics and limits on the scaling of a flexible kinetic sculpture

Gooch

Raine

2000

Proceedings of the Institution of Mechanical Engineers, Part C:

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“…It has been shown 7,13,14 that the method of Rayleigh-Ritz is advantageous to other numerical methods, such as the finite element method for finding the modal characteristics, including damping, of rectangular plates and, for that reason, this method is utilized in this study. In the Rayleigh-Ritz method, 14,15 adequate flexibility in the expression for the plate lateral deflection w is assured by employing a double-summation series of the product of two onedimensional, independent functions of the plate coordinates x and y, thus:…”

Section: Theorymentioning

confidence: 99%

The effect of layup and boundary conditions on the modal damping of FRP composite panels

Maheri

2010

Journal of Composite Materials

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The degree of the modal damping in a fiber-reinforced plastic (FRP) composite panel depends on a number of factors including the nature of the layup, the modal deformation, the geometry and the boundary conditions of the panel. In this article, using the theoretical predictions of modal response of a square layered FRP panel, the variation of damping with the plate layup under a combination of clamped and free boundary conditions is studied. Particular attention is paid to the variation of modal damping with the fiber orientation of the two outmost layers of the panel with a view to possibly manipulation of this orientation in order to maximisze the modal damping.

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“…The classical laminate theory neglects the transverse shear effects of the plate and is sufficient to describe the flexural behaviour of plates of large dimensions (length and width) with respect to its thickness. Expression (2) of the strain energy introduces the bending stiffnesses D ij which are related to the in-plane reduced stiffnesses Q ij of the plate by the expressions: 11,12,16,22,26,66…”

Section: Formulation Of the Flexural Vibrations Of A Rectangular Platementioning

confidence: 99%

“…So as to solve the system of equations (11), we have to choose the admissible functions X m and Y n which satisfy the appropriate boundary conditions and then to calculate the integrals (8) and (9). As introduced by Young [16], the beam functions (the characteristic functions of the flexural vibrations of beams) can be used as admissible functions. Also we have considered the use of polynomial functions [13,17] satisfying the essential boundary conditions.…”

Section: Mode Shapesmentioning

confidence: 99%

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Measuring the Bending Stiffnesses of Orthotropic and Symmetric Laminates from Flexural Vibrations

Berthelot¹,

Angoulvant

2002

Journal of Composite Materials

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The paper develops an extensive analysis of the problem of determining the bending stiffnesses of orthotropic and symmetric laminates from flexural vibrations of rectangular plates with different boundary conditions consisting of clamped or free edges. Rayleigh’s approximation and Ritz method are considered for evaluating the flexural vibrations of plates. The experimental results and an analysis of the sensibility show clearly the tendency for the evaluation of the bending stiffnesses from the natural frequencies of the flexural vibrations of an orthotropic or symmetric plate to be an ill conditioned problem. It results that iterative procedures associated with Ritz method or with numerical method as finite element analysis cannot be used for evaluating the bending stiffnesses from the natural frequencies of a single plate in industrial applications. For orthotropic laminates, the paper shows how the Rayleigh’s approximation allows to overcome the problem of the divergence of the iterative processes. This approximation suggests another methods for evaluating the bending stiffnesses and their variations with the frequency. Next for symmetric laminates, the evaluation of the stiffnesses from the flexural vibrations of plates is implemented using the measurement of the mode shapes deduced from experimental modal analysis.

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Vibration of Rectangular Plates by the Ritz Method

Cited by 352 publications

References 0 publications

The dynamics and limits on the scaling of a flexible kinetic sculpture

The dynamics and limits on the scaling of a flexible kinetic sculpture

The effect of layup and boundary conditions on the modal damping of FRP composite panels

Measuring the Bending Stiffnesses of Orthotropic and Symmetric Laminates from Flexural Vibrations

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