Nonlinear vibrations of a polar-orthotropic thin circular plate subjected to circularly moving point load

2021 ◽  
Vol 256 ◽  
pp. 112953
Author(s):  
Amit K. Rai ◽  
Shakti S. Gupta
Keyword(s):  
Circular Plate ◽  
Nonlinear Vibrations ◽  
Point Load ◽  
Thin Circular Plate ◽  
Polar Orthotropic

scholarly journals Green's function of the clamped punctured disk

1977 ◽  
Vol 20 (2) ◽  
pp. 175-181 ◽  
Author(s):  
Mitsuru Nakai ◽  
Leo Sario
Keyword(s):  
Green’S Function ◽  
Circular Plate ◽  
Green's Function ◽  
American Mathematical Society ◽  
Point Load ◽  
Thin Elastic Plate ◽  
Constant Sign ◽  
Boundary Circle ◽  
Simply Supported ◽  
Image Position

If a thin elastic circular plate B: ∣z∣ < 1 is clamped (simply supported, respectively) along its edge ∣z∣ = 1, its deflection at z ∈ B under a point load at ζ ∈ B, measured positively in the direction of the gravitational pull, is the biharmonic Green's function β(z, ζ) of the clamped plate (γ(z, ζ) of the simply supported plate, respectively). We ask: how do β(z, ζ) and γ(z, ζ) compare with the corresponding deflections β0(z, ζ) and γ0(z, ζ) of the punctured circular plate B0: 0 < ∣ z ∣ < 1 that is “clamped” or “simply supported”, respectively, also at the origin? We shall show that γ(z, ζ) is not affected by the puncturing, that is, γ(·, ζ) = γ0(·, ζ), whereas β(·, ζ) is:on B0 × B0. Moreover, while β((·, ζ) is of constant sign, β0(·, ζ) is not. This gives a simple counterexmple to the conjecture of Hadamard [6] that the deflection of a clampled thin elastic plate be always of constant sign:The biharmonic Gree's function of a clampled concentric circular annulus is not of constant sign if the radius of the inner boundary circle is sufficiently small.Earlier counterexamples to Hadamard's conjecture were given by Duffin [2], Garabedian [4], Loewner [7], and Szegö [9]. Interest in the problem was recently revived by the invited address of Duffin [3] before the Annual Meeting of the American Mathematical Society in 1974. The drawback of the counterexample we will present is that, whereas the classical examples are all simply connected, ours is not. In the simplicity of the proof, however, there is no comparison.

Nonlinear vibrations of a thin isotropic circular plate subjected to moving point loads

2020 ◽  
pp. 095440622097615
Author(s):  
Amit K Rai ◽  
Shakti S Gupta
Keyword(s):  
Boundary Conditions ◽  
Frequency Response ◽  
Circular Plate ◽  
Nonlinear Vibrations ◽  
Moving Load ◽  
Harmonic Balance Method ◽  
Angular Speed ◽  
Transverse Displacement ◽  
Governing Equations ◽  
Rotating Load

Here, we have studied the linear and nonlinear vibrations of a thin circular plate subjected to circularly, radially, and spirally moving transverse point loads. We follow Kirchoff’s theory and then incorporate von Kármán nonlinearity and employ Hamilton’s principle to obtain the governing equations and the associated boundary conditions. We solve the governing equations for the simply-supported and clamped boundary conditions using the mode summation method. Using the harmonic balance method for frequency response and Runge-Kutta method for time response, we solve the resulting coupled and cubic nonlinear ordinary differential equations. We show that the resonance instability due to a circularly moving load can be avoided by splitting it into multiple loads rotating at the same radius and angular speed. With the increasing magnitude of the rotating load, the frequency response of the transverse displacement shows jumps and modal interaction. The transverse response collected at the centre of the plate shows subharmonics of the axisymmetric frequencies only. The spectrum of the linear response due to spirally moving load contains axisymmetric frequencies, the angular speed of the load, their combination, and superharmonics of axisymmetric frequencies.

On Dynamic Cycle Collapse of Circular Plates

1999 ◽  
Vol 66 (1) ◽  
pp. 250-253 ◽  
Author(s):  
P. D. Chinh
Keyword(s):  
Circular Plate ◽  
Upper Bound ◽  
Point Load ◽  
Circular Plates ◽  
Load Effect ◽  
Collapse Mechanisms ◽  
Kinematic Method ◽  
Dynamic Cycle ◽  
Incremental Collapse

The upper bound kinematic method, which is based on a reduced kinematic formulation and involves construction of fictitious elastic moment fields and potential incremental collapse mechanisms, is used to evaluate the dynamic cycle collapse loads for a symmetrically loaded circular plate. The respective nonshakedown curves are constructed, A point load effect is discussed.

scholarly journals Mobility Matrix of a Thin Circular Plate Carrying Concentrated Masses Based on Transverse Vibration Solution

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Desheng Li ◽  
Junhong Zhang
Keyword(s):  
Circular Plate ◽  
Transverse Vibration ◽  
Vibration Mode ◽  
Sound Power ◽  
Vibration Source ◽  
Thin Circular Plate ◽  
Mobility Matrix ◽  
Solution Point ◽  
Concentrated Masses ◽  
Mobility Solution

When calculating the vibration or sound power of a vibration source, it is necessary to know the point mobility of the supporting structure. A new method is presented for the calculation of point mobility matrix of a thin circular plate with concentrated masses in this paper. Transverse vibration mode functions are worked out by utilizing the structural circumferential periodicity of the inertia excitation produced by the concentrated masses. The numerical vibratory results, taking the clamped case as an instance, are compared to the published ones to validate the method for ensuring the correctness of mobility solution. Point mobility matrix, including the driving and transfer point mobility, of the titled structure is computed based on the transverse vibration solution. After that, effect of the concentrated masses on the mechanical point mobility characteristics is analyzed.

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