Frequency equations for the flexural vibration of samples with a uniform cross-section

Abstract This paper calls attention to a new method of dealing with deflections of beams, the cross sections of which vary by steps. It is shown that the effect of this variation on the shape of the deflection curve can be represented by a properly chosen force system acting on a beam of uniform cross section. There is no approximation involved in this substitution, whereby the original problem is reduced to one of computing deflections of beams of constant cross section.

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The multipurpose optimal design of elastic structures with a piecewise uniform cross section

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/bf01591812 ◽

1971 ◽

Vol 22 (5) ◽

pp. 834-855 ◽

Cited By ~ 2

Author(s):

J. -M. Chern ◽

J. B. Martin

Keyword(s):

Optimal Design ◽

Cross Section ◽

Elastic Structures ◽

Uniform Cross Section

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Effects of the wall curvature on the flow in a duct of uniform cross-section

Acta Mechanica ◽

10.1007/s00707-003-0071-0 ◽

2004 ◽

Vol 167 (3-4) ◽

pp. 123-130 ◽

Cited By ~ 1

Author(s):

M. E. Erdogan ◽

C. E. Imrak

Keyword(s):

Cross Section ◽

Uniform Cross Section

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Rayleigh–Bloch surface waves along periodic gratings and their connection with trapped modes in waveguides

Journal of Fluid Mechanics ◽

10.1017/s0022112099004425 ◽

1999 ◽

Vol 386 ◽

pp. 233-258 ◽

Cited By ~ 65

Author(s):

R. PORTER ◽

D. V. EVANS

Keyword(s):

Finite Number ◽

Surface Waves ◽

Cross Section ◽

Electromagnetic Waves ◽

Mode Shapes ◽

Special Choice ◽

Trapped Mode ◽

Uniform Cross Section ◽

New Type ◽

Incident Waves

Rayleigh–Bloch surface waves are acoustic or electromagnetic waves which propagate parallel to a two-dimensional diffraction grating and which are exponentially damped with distance from the grating. In the water-wave context they describe a localized wave having dominant wavenumber β travelling along an infinite periodic array of identical bottom-mounted cylinders having uniform cross-section throughout the water depth. A numerical method is described which enables the frequencies of the Rayleigh–Bloch waves to be determined as a function of β for an arbitrary cylinder cross-section. For particular symmetric cylinders, it is shown how a special choice of β produces results for the trapped mode frequencies and mode shapes in the vicinity of any (finite) number of cylinders spanning a rectangular waveguide or channel. It is also shown how one particular choice of β gives rise to a new type of trapped mode near an unsymmetric cylinder contained within a parallel-sided waveguide with locally-distorted walls. The implications for large forces due to incident waves on a large but finite number of such cylinders in the ocean is discussed.

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Particular Solutions of the Equation of Conduction of Heat in One Dimension

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500036312 ◽

1924 ◽

Vol 43 ◽

pp. 50-63

Author(s):

Marion C. Gray

Keyword(s):

Cross Section ◽

Integral Method ◽

Series Solution ◽

Arbitrary Element ◽

Initial Distribution ◽

One Dimension ◽

Operational Method ◽

Boundary Values ◽

Thermal Disturbance ◽

Uniform Cross Section

The problem of the conduction of heat in one dimension is usually concerned with the propagation of a thermal disturbance along a bar or rod of uniform cross section. The solution of the problem is required for a given initial distribution of temperature, and given boundary values, usually at each end of the rod. In most cases this solution is found by assuming a series solution and then proving that the series satisfies the equation of the disturbance well as all the assigned conditions. Other methods, for example the contour integral method developed by Carslaw, also introduce this arbitrary element of choice in choosing the integrand and the contour of integration. The object of the present paper is to develop the application of Heaviside's Operational method to the solution of the problem, and to show that it leads in all cases to solutions equivalent to the known forms, although initially no assumptions are made regarding the nature of the solution.

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Shape Optimization of a Compliant Mechanism for an Actively Conformable Rotor Airfoil

Volume 2: 28th Biennial Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2004-57617 ◽

2004 ◽

Author(s):

Andrew Nissly ◽

Phuriwat Anusonti-Inthra ◽

Mary Frecker ◽

Farhan Gandhi

Keyword(s):

Shape Optimization ◽

Cross Section ◽

Strain Energy ◽

Rotation Angle ◽

Compliant Mechanism ◽

Aerodynamic Loads ◽

Optimal Geometry ◽

Uniform Cross Section ◽

Narrow Space ◽

Actuation Strain

In the present study, the optimal shape of a limited amount of passive material in a compliant mechanism of predetermined topology was determined. The simple compliant mechanism with a small number of actuators can be packaged in the long, narrow space of the rotor airfoil cross-section. The compliant mechanism is designed for maximum rotation angle under actuation loads, and minimum deflection under aerodynamic loads. Rotation angle (RA) and Strain Energy (SE), are used as measures of the deflections created by the actuation and aerodynamic loads, respectively. The design objectives are achieved by maximizing a multi-criteria objective function that represents a ratio of the RA to SE. Shape optimization of the compliant mechanism is conducted and the results indicate that the optimal compliant mechanism consists of a passive substructure with uniform cross section. The optimal geometry of the compliant mechanism is also determined in a parametric study (optimal ratio of the length to height of 0.3), and this structure can produce rotation angle of 13 Deg/m. when the actuators provide 1% actuation strain. The deflection due to aerodynamic loads is extremely small. The performance of the mechanism is examined further with variations in material and actuator properties. Additional results include an analysis of a compliant mechanism structure based on a modified topology, which is introduced to reduce numbers of actuators.

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