# Engineering Update

The relevant design theories that will be used to perform the engineering analysis include stress calculations and applying failure theories under static condition and under fatigue condition for the components. In addition, calculations of torque related to rotational speed will be used to represent operating conditions. The analytical calculations will be compared to finite element analysis performed using ANSYS.

Stress Calculations
The stress acting on the system will be predominantly shear stress, which is calculated using:
$$τ = {Tr \over J}$$where is the shear stress, T is the torque, and J is the polar moment of inertia [1]. The normal stress will be calculated using:
$$σ = {F \over A}$$where F is the force in the axial direction and A is the cross-sectional area in the axial direction. The stress calculations are necessary, as it will be used to analyze the analysis involving the failure theories.

Torque Calculations
The torque will be compared to the rotational speed using the following equation:
$$T = {I α}$$where net torque, T is directly proportional to the angular acceleration, by the mass moment of inertia, I. The torque calculations are need to calculate the amount of torque need to obtain a certain angular velocity.

Static Failure Analysis (Distortion Energy Theory)
The Distortion Energy theory will be used to analyze stresses and determine means for yielding and potential failure under static condition using the corresponding equations:

$$σ' = { \sqrt{{(σ_1-σ_2)^2} +{(σ_2-σ_3)^2}+{(σ_3-σ_1)^2}\over 2} }$$
where ' is the von Mises stress; σ1, σ2, and σ3 are principal stress; Sy is the yield strength; and n is the factor of safety [1]. The principal stresses are the eigenvalues of the Cauchy stress tensor, which is a three by three matrix [1]. The distortion energy theory was chosen for the static failure analysis because that theory is meant for ductile material and it is more accurate than the maximum shear stress theory from experimental testing [1].

Fatigue Failure Analysis (Modified Goodman Theory)
The Modified Goodman theory will be used to analyze stresses and determine means for yielding and potential failure under fatigue condition using the corresponding equation:
$${1 \over n} = {{ σ_a \over S_y} + { σ_m \over S_{ut}}}$$ where a is the alternating stress, m is the mean stress, Seis the endurance limit, Sut is the ultimate tensile strength, and n is the factor of safety [1]. The ultimate tensile strength is a materials properties and the endurance limit depends on the operating conditions and stress concentrations. The amplitude stress, and the midrange stress, are defined as:
$$σ_a = {\vert {{σ_{max}-σ_{min}} \over 2 } \vert}$$
$$σ_a = {{{σ_{max}+ σ_{min}} \over 2 }}$$ where is the maximum stress and is the minimum stress [1]. The modified Goodman theory is chosen for the fatigue failure analysis because it is the most conservative fatigue failure theory. The group believes that the most conservative fatigue failure theory should be used since the coupling hub will be used in fatigue condition in the industry.

Finite Element Analysis
Finite Element Analysis performed from programming computer codes or performed by hand are very time consuming. Due to the time constraints of this project, using a commercial software would be ideal. The finite element analysis (FEA) will be performed using the commercial software, ANSYS. ANSYS is a popular FEA software used in the industries. The 3D solid model will be created using PTC Creo Parametric and then imported into ANSYS for the analysis. The group decided to use ANSYS software because that is the software Dresser-Rand preferred for analysis of their coupling hub. The independent coupling hub should also be analyzed using ANSYS since the two coupling hub will be compared.

Validation
For Dresser-Rand prototype: Dresser-Rand already have results from experimental testing. The FEA performed will be compared to those.
Original Coupling: The FEA will be compared to the analytical calculations, which includes the stress calculation, the torque calculation, static failure analysis, and fatigue failure analysis. References

[1] Budynas R. and Nisbett J., 2011, Shigleys Mechanical Engineering Design, McGraw-Hill, New York NY.