Calculating Deflection in Ring Fixed at Center with Point Force

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Discussion Overview

The discussion revolves around calculating the deflection of a ring fixed at its center when subjected to a radial point force. Participants explore the application of boundary conditions in the context of a fourth-order differential equation, as well as the implications of the ring's geometry and support conditions. The scope includes theoretical modeling and mathematical reasoning related to static deflection.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks assistance in applying boundary conditions to a derived set of equations for deflection, noting the need for four conditions but facing challenges due to unspecified displacements.
  • Another participant questions the initial premise, suggesting that a ring does not have a material center and asks for clarification, possibly indicating a misunderstanding of the geometry involved.
  • A participant introduces the concept of a thin ring on an elastic foundation, likening it to a tire's geometry, and asks how to calculate static deflection when a point load is applied at the circumference.
  • Concerns are raised about the equilibrium of the system, particularly regarding the support of the elastic center and the implications of the applied load.
  • Further clarification is provided about the setup, emphasizing the role of the elastic membrane in maintaining equilibrium and the conditions under which the radial point force is applied.

Areas of Agreement / Disagreement

Participants express differing views on the initial problem setup, with some questioning the feasibility of the described conditions and others attempting to clarify the scenario. The discussion remains unresolved regarding the correct interpretation of the ring's geometry and the application of boundary conditions.

Contextual Notes

There are limitations regarding the assumptions made about the ring's material properties and support conditions, as well as the need for clearer definitions of the geometrical setup. The discussion does not resolve the mathematical steps needed to apply the boundary conditions effectively.

vishal007win
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How to calculate the deflection in ring, fixed at center, with a radial point force acting at some point(let say at theta=0) ?
I have derived complete set of equations. Now how to apply boundary conditions ?
to get a complete solution of 4th order equation, I need 4 B.C's, but here displacements are not specified at any point. Imposing periodicity I can get 2 more conditions
u(0)=u(2*pi)
w(0)=w(2*pi)
where u and w are radial and tangential displacements respectively
and how to apply force boundary condition in this case?

Can someone please help?
 
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How to calculate the deflection in ring, fixed at center, with a radial point force acting at some point(let say at theta=0) ?

This does not make sense.

A ring has no material centre, do you mean a disk?

Perhaps a diagram?
 
consider the thin ring on elastic foundation(shaded portion shown in the figure is an elastic foundation(elastic membrane) which provide a support from center to ring)
The inner circle shown can be taken as mass-less and inertia-less disc.
actually resembling the tyre geometry with side walls and inner wheel drum.
now if a point load is applied at the circumference of ring. How to calculate the static deflection in this case?
 

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If h is your applied load, and the wheel does not touch the ground I still don't see equilibrium in your diagram.

How is the elastic centre supported?
 
Studiot said:
If h is your applied load, and the wheel does not touch the ground I still don't see equilibrium in your diagram.

How is the elastic centre supported?
h is the width of the ring.
the attachment in the above post actually resembles kind of set up shown in the figure below.. where elastic membrane resembles the sidewalls of tires and the ring represents the belt of tire and the inner disc represents the wheel drum. Now equilibrium is maintained by the reaction from the centre which is fixed(force getting transmitted by elastic membrane). i hope that makes the problem clear.
now instead of modelling contact region, if radial point force is applied at bottom, assuming the centre of wheel fixed, how to find the static deflection of ring?
 

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    tailwhl1.jpg
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