Stresses and deformations of an eccentric cylindrical cam during hydro-joining

The stresses and deformations of an eccentric cylindrical cam during the process of hydro-joining have been calculated in orthogonal curvilinear coordinates according to the mechanical model of the cam and governing equations in terms of appropriate complex potentials with suitable boundary conditions. The radial and shearing stress coefficients determined for an example cam are far less than that of the tangential stress during hydro-joining. The tangential stress coefficient of the outer surface of the cam is greater than that of the internal surface when the polar angle exceeds a particular value. The position of the maximum value of the radial stress coefficient is located on the internal surface of the cam, and the maximum shear stress coefficient is located between the inside and outside surfaces of the cam. The cam deformations on the internal and external surfaces under internal pressure respectively attain maximum values at particular angles. The maximum values of the radial and y-directional deformations are located at the position of the minimum wall thickness. The radial deformations determined for an example cam are far larger that the tangential deformations during hydro-joining. The errors between the theoretical and numerical solutions for the tangential stress and the y-directional deformation are both very small.


Introduction
When the stresses exceed the strength of cam material under the action of the contact pressure, the failure of the cam will occur [1][2]. An eccentric cylindrical cam (ECC) has a cylindrical structure of variable cross-section, and the basic equations governing stresses and deformations are therefore variable coefficient differential equations, which are very difficult to solve [3][4]. To obtain the stresses and deformations of an ECC, many researchers have adopted numerical methods based on the finite element model of the cam [5][6][7][8].
In this work, we determine the positions and values of the maximum stress and deformation of an example ECC. We also develop the finite element model of the cam and determine the numerical solutions for the stress and deformation of the cam.

Calculation formulas for deformation
Taking the same complex potentials as employed in the calculation of the stress components, we calculate the radial and tangential deformations of an ECC and obtain the expressions for the radial and tangential deformations of an ECC owing to the expansion of the axial tube in the process of hydro-joining.
4 Examples 4.1 Stresses of the ECC As an example, we set the radius of the outer circle of an ECC at 48 mm, the radius of the inner circle at 30 mm, the distance between their centers at 5.4 mm, and the height of the ECC at 18 mm, as shown in Figure 1. The tangential stress coefficients of the internal and external surfaces of the ECC are shown in Figure 2. The shearing stress coefficient at the polar angle π/2 is shown in Figure 3. The radial stress coefficient at the position of the minimum wall thickness is shown in Figure 4. The tangential stress of the ECC at the position of the minimum wall thickness is shown in Figure 5.

Conclusions
The tangential stress and deformation of the ECC are a linear function of the uniform loads acting on the internal surface of the cam during hydro-joining. The radial and shearing stress coefficients determined for an example cam were far less than that of the tangential stress during hydro-joining. The tangential stress coefficient on the external surface was found to be greater than that on the inner surface for a polar angle greater than 152°. The maximum value of the tangential deformations on the internal and external surfaces of the cam are 2.49 and 8.27 μm, respectively, under the action of an internal pressure of 50 MPa, which are located at the polar angles 42° and 62°. Deformations in the radial and y direction at the position of the minimum wall thickness are 23.15 and 21.46 μm, respectively. Deformations in the radial and x direction are 16.59 and 12.15 μm, respectively, which are located in the polar angles 114° and 96°, respectively. Under the action of a hydrostatic pressure of 50 MPa, the differences between the theoretical and numerical solutions of the tangential stress is 1.6% on the external surface of cam, and that of the deformation is 2.0% in the y direction. The errors between the theoretical and numerical solutions of the stresses and deformations of the cam were found to be very small. Therefore, we conclude that the derived theoretical formulas can be applied to the failure analysis of an ECC during hydrojoining.