A shear spring is made by bonding the rubber annulus to a...

A shear spring is made by bonding the rubber annulus to a rigid fixed ring and a plug. When an axial load $\mathbf{P}$ is placed on the plug, show that the slope at point $y$ in the rubber is $d y / d r=-\tan \gamma=-\tan (P /(2 \pi h G r))$. For small angles we can write $d y / d r=-P /(2 \pi h G r)$. Integrate this expression and evaluate the constant of integration using the condition that $y=0$ at $r=r_\sigma$. From the result compute the deflection $y=\delta$ of the plug.

Key Concepts

Shear Deformation

In this problem, the rubber annulus is deforming in shear as the axial load produces a twist-like effect between the fixed ring and the plug. Shear deformation is characterized by a change in shape without a change in volume, and it relates the applied load to the angular displacement (or slope) in the deformed material. The use of tan(?) in the formulation connects this angular measure to the local shear strain in the rubber.

Small-Angle Approximation

The small-angle approximation is applied here to simplify the mathematical formulation of the problem. When the deformation angles are sufficiently small, the tangent of the angle (tan ?) can be approximated by the angle itself (? in radians). This simplification makes the differential equation linear (dy/dr ? -P/(2?hGr)) and much easier to integrate, which is a common technique in engineering problems involving small displacements or rotations.

Integration of Differential Equations

The process involves integrating the differential equation that relates the slope (dy/dr) to the radial position (r). By integrating the expression, one obtains the deflection y as a function of r. This step is fundamental in mechanics and mathematical physics, where differential relationships need to be integrated to find global displacements or fields from local derivatives.

Boundary Conditions

The integration of a differential equation inherently introduces an undetermined constant. In this problem, the boundary condition provided (y = 0 at r = r?) is used to evaluate this constant. Applying boundary conditions is essential in solving physical problems because they ensure that the mathematical solution corresponds to the actual physical constraints or configurations of the system.

Material Shear Modulus (G)

The shear modulus G is a material property that relates shear stress to shear strain and appears in the formulation of the differential equation. It quantifies the material's resistance to shear deformation. In the context of the shear spring, the shear modulus is critical in determining how much the rubber will deform for a given load, thereby linking the applied load to the observed deflection.

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