Optimal Forms of Shallow Shells With Circular Boundary, Part 1: Maximum Fundamental Frequency

1984 ◽  
Vol 51 (3) ◽  
pp. 526-530 ◽  
Author(s):  
R. H. Plaut ◽  
L. W. Johnson ◽  
R. Parbery
Keyword(s):  
Boundary Conditions ◽  
Fundamental Frequency ◽  
Iterative Procedure ◽  
Material Surface ◽  
Uniform Thickness ◽  
Shallow Shells ◽  
Simply Supported ◽  
Circular Boundary ◽  
Axisymmetric Shell ◽  
Boundary Part

Thin, shallow, elastic shells with given circular boundary are considered. We seek the axisymmetric shell form which maximizes the fundamental frequency of vibration. The boundary conditions, material, surface area, and uniform thickness of the shell are specified. We employ a bimodal formulation and use an iterative procedure based on the optimality condition to obtain optimal forms. Results are presented for clamped and simply supported boundary conditions. For the clamped case, the optimal forms have zero slope at the boundary. The maximum fundamental frequency is significantly higher than that for the corresponding spherical shell if the boundary is clamped, but only slightly higher if it is simply supported.

Optimal Forms of Shallow Shells With Circular Boundary, Part 3: Maximum Enclosed Volume

1984 ◽  
Vol 51 (3) ◽  
pp. 536-539 ◽  
Author(s):  
R. H. Plaut ◽  
L. W. Johnson
Keyword(s):  
Spherical Shell ◽  
Surface Area ◽  
Material Surface ◽  
Uniform Thickness ◽  
Base Plane ◽  
Shallow Shells ◽  
Simply Supported ◽  
Circular Boundary ◽  
Boundary Part ◽  
Uniformly Distributed Loads

In Parts 1 and 2, we determined optimal forms of shallow shells with respect to vibration and stability, respectively. In this final part, we consider a given load and find the shell form for which the volume between the base plane and the deflected shell is a maximum. As before, the shell is assumed to be thin, elastic, and axisymmetric, with a given circular boundary that is either clamped or simply supported. The material, surface area, and uniform thickness of the shell are specified. Both uniformly distributed loads and concentrated central loads are treated. In the numerical results, the maximum enclosed volume is on the order of 10 percent higher than that for the corresponding spherical shell.

Optimal Forms of Shallow Shells With Circular Boundary, Part 2: Maximum Buckling Load

1984 ◽  
Vol 51 (3) ◽  
pp. 531-535 ◽  
Author(s):  
R. H. Plaut ◽  
L. W. Johnson
Keyword(s):  
Critical Load ◽  
Elastic Shell ◽  
Material Surface ◽  
Fundamental Vibration ◽  
Uniform Loading ◽  
Concentrated Load ◽  
Shallow Shells ◽  
Simply Supported ◽  
Circular Boundary ◽  
Boundary Part

In Part 1, optimal forms were determined for maximizing the fundamental vibration frequency of a thin, shallow, axisymmetric, elastic shell with given circular boundary. Our objective in this part is to maximize the critical load for buckling under a uniformly distributed load or a concentrated load at the center. Again, the shell form is varied and the material, surface area, and uniform thickness of the shell are specified. Both clamped and simply supported boundary conditions are considered for the case of uniform loading, while one example is presented involving a concentrated load acting on a clamped shell. The optimality condition leads to forms that have zero slope at the boundary if it is clamped. The maximum critical load is sometimes associated with a limit point and sometimes with a bifurcation point. It is often substantially higher than the critical load for the corresponding spherical shell.

Optimal Forms of Shallow Cylindrical Panels With Respect to Vibration and Stability

1986 ◽  
Vol 53 (1) ◽  
pp. 135-140 ◽  
Author(s):  
R. H. Plaut ◽  
L. W. Johnson
Keyword(s):  
Surface Area ◽  
Fundamental Frequency ◽  
Axial Load ◽  
Cylindrical Panel ◽  
Buckling Load ◽  
Material Surface ◽  
Uniform Thickness ◽  
Simply Supported ◽  
Cylindrical Panels

Thin, shallow, elastic, cylindrical panels with rectangular planform are considered. We seek the midsurface form which maximizes the fundamental frequency of vibration, and the form which maximizes the buckling value of a uniform axial load. The material, surface area, and uniform thickness of the panel are specified. The curved edges are simply supported, while the straight edges are either simply supported or clamped. For the clamped case, the optimal panels have zero slope at the edges. In the examples, the maximum fundamental frequency is up to 12 percent higher than that of the corresponding circular cylindrical panel, while the buckling load is increased by as much as 95 percent. Most of the solutions are bimodal, while the rest are either unimodal or trimodal.

scholarly journals On the Free Vibration Modeling of Spindle Systems: A Calibrated Dynamic Stiffness Matrix

2021 ◽  
Author(s):  
Omar Gaber ◽  
Seyed M. Hashemi
Keyword(s):  
Boundary Conditions ◽  
Fundamental Frequency ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Dynamic Stiffness Matrix ◽  
Simply Supported ◽  
Multiple Components ◽  
Stiffness Matrices ◽  
Linear Spring ◽  
Vibration Modeling

The effect of bearings on the vibrational behavior of machine tool spindles is investigated. This is done through the development of a calibrated dynamic stiffness matrix (CDSM) method, where the bearings flexibility is represented by mass less linear spring elements with tuneable stiffness. A dedicated MAT LAB code is written to develop and to assemble the element stiffness matrices for the system’s multiple components and to apply the boundary conditions.The developed method is applied to an illustrative example of spindle system.When the spindle bearings are modeled as simply supported boundary conditions, the DSM model results in a fundamental frequency much higher than the system’s nominal value.The simply supported boundary conditions are then replaced by linear spring elements, and the spring constants are adjusted such that the resulting calibrated CDSM model leads to the nominal fundamental frequency of the spindle system.The spindle frequency results are also validated against the experimental data.The proposed method can be effectively applied to predict the vibration characteristics of spindle systems supported by bearings.

Symmetrical Bending of Circular Plates of Constant Radial Bending Stress

1962 ◽  
Vol 29 (4) ◽  
pp. 696-700 ◽  
Author(s):  
J. P. Lee
Keyword(s):  
Boundary Conditions ◽  
Variable Thickness ◽  
Integrodifferential Equation ◽  
Thin Plates ◽  
Bending Stress ◽  
Circular Plates ◽  
Uniform Thickness ◽  
Simply Supported ◽  
Equation Boundary ◽  
Plates Of Variable Thickness

Bending of simply supported circular plates of constant radial bending stress subjected to uniformly distributed loading is investigated by solving a nonlinear integrodifferential equation. Boundary conditions are satisfied by joining the central portion of the plates of variable thickness to an annular rim along the boundary with uniform thickness. Usual assumptions for bending of thin plates of small deflections are assumed valid.

scholarly journals On the Free Vibration Modeling of Spindle Systems: A Calibrated Dynamic Stiffness Matrix

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Omar Gaber ◽  
Seyed M. Hashemi
Keyword(s):  
Boundary Conditions ◽  
Fundamental Frequency ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Dynamic Stiffness Matrix ◽  
Spindle System ◽  
Simply Supported ◽  
Multiple Components ◽  
Stiffness Matrices ◽  
Linear Spring

The effect of bearings on the vibrational behavior of machine tool spindles is investigated. This is done through the development of a calibrated dynamic stiffness matrix (CDSM) method, where the bearings flexibility is represented by massless linear spring elements with tuneable stiffness. A dedicated MATLAB code is written to develop and to assemble the element stiffness matrices for the system’s multiple components and to apply the boundary conditions. The developed method is applied to an illustrative example of spindle system. When the spindle bearings are modeled as simply supported boundary conditions, the DSM model results in a fundamental frequency much higher than the system’s nominal value. The simply supported boundary conditions are then replaced by linear spring elements, and the spring constants are adjusted such that the resulting calibrated CDSM model leads to the nominal fundamental frequency of the spindle system. The spindle frequency results are also validated against the experimental data. The proposed method can be effectively applied to predict the vibration characteristics of spindle systems supported by bearings.

scholarly journals On the Free Vibration Modeling of Spindle Systems: A Calibrated Dynamic Stiffness Matrix

2021 ◽  
Author(s):  
Omar Gaber ◽  
Seyed M. Hashemi
Keyword(s):  
Boundary Conditions ◽  
Fundamental Frequency ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Dynamic Stiffness Matrix ◽  
Simply Supported ◽  
Multiple Components ◽  
Stiffness Matrices ◽  
Linear Spring ◽  
Vibration Modeling

The effect of bearings on the vibrational behavior of machine tool spindles is investigated. This is done through the development of a calibrated dynamic stiffness matrix (CDSM) method, where the bearings flexibility is represented by mass less linear spring elements with tuneable stiffness. A dedicated MAT LAB code is written to develop and to assemble the element stiffness matrices for the system’s multiple components and to apply the boundary conditions.The developed method is applied to an illustrative example of spindle system.When the spindle bearings are modeled as simply supported boundary conditions, the DSM model results in a fundamental frequency much higher than the system’s nominal value.The simply supported boundary conditions are then replaced by linear spring elements, and the spring constants are adjusted such that the resulting calibrated CDSM model leads to the nominal fundamental frequency of the spindle system.The spindle frequency results are also validated against the experimental data.The proposed method can be effectively applied to predict the vibration characteristics of spindle systems supported by bearings.

The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching

1961 ◽  
Vol 28 (2) ◽  
pp. 288-291 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Flexural Vibration ◽  
Frequency Equation ◽  
Numerical Results ◽  
Special Technique ◽  
Point Matching ◽  
Buckling Loads ◽  
Simply Supported ◽  
Flexural Vibrations

The bending by uniform lateral loading, buckling by two-dimensional hydrostatic pressure, and the flexural vibrations of simply supported polygonal plates are investigated. The method of meeting the boundary conditions at discrete points, together with the Marcus membrane analog [1], is found to be very advantageous. Numerical examples include the calculation of the deflections and moments, and buckling loads of triangular square, and hexagonal plates. A special technique is then given, whereby the boundary conditions are exactly satisfied along one edge, and an example of the buckling of an isosceles, right-angled triangle plate is analyzed. Finally, the frequency equation for the flexural vibrations of simply supported polygonal plates is shown to be the same as that for buckling under hydrostatic pressure, and numerical results can be written by analogy. All numerical results agree well with the exact solutions, where the latter are known.

A semi-analytical three-dimensional free vibration analysis of functionally graded curved panels integrated with piezoelectric layers

2014 ◽  
Vol 21 (4) ◽  
pp. 571-587 ◽  
Author(s):  
Hamid Reza Saeidi Marzangoo ◽  
Mostafa Jalal
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Simply Supported ◽  
Piezoelectric Layers ◽  
Curved Panels

AbstractFree vibration analysis of functionally graded (FG) curved panels integrated with piezoelectric layers under various boundary conditions is studied. A panel with two opposite edges is simply supported, and arbitrary boundary conditions at the other edges are considered. Two different models of material property variations based on the power law distribution in terms of the volume fractions of the constituents and the exponential law distribution of the material properties through the thickness are considered. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. For the simply supported boundary conditions, closed-form solution is given by making use of the Fourier series expansion, and applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free-end conditions. Natural frequencies of the hybrid curved panels are presented by solving the eigenfrequency equation, which can be obtained by using edges boundary conditions in this state equation. The results obtained for only FGM shell is verified by comparing the natural frequencies with the results obtained in the literature.

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