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\title{Failure Analysis of a Universal Coupling using Finite Element Method}
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\begin{document}

             \author[1]{Muhammad Ziaur  Rahman}

             \author[2]{Tazmin Rashid  Mumu}

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\date{\small \em Received: 15 December 2012 Accepted: 1 January 2013 Published: 15 January 2013}

\maketitle


\begin{abstract}
        


Generation of stress, displacement and strain in a universal coupling has been analyzed.Circumferential stress is applied at the yoke slot and also on the hub and simulated separately. The simulation is carried out with the help of SolidWorks 2010. To show the effect of temperature rise due to friction at the yoke slot, thermal load is gradually increased at the slot. The results are demonstrated both in the form of surface contour and graph. It has been showed that friction between yoke slot and hub can increase the temperature, which can eventually increase the thermal stress paving the way to failure of yoke or hub material. It is also found that the hub experiences a larger stress compared to the yoke when loaded under same pressure. Thus, the hub has the higher probability to fail than the yoke. At the end of the paper, some recommendations regarding universal coupling building material and reduction of friction have been made. Finally, the results obtained here are highly accurate and conform to the physical and loading conditions.

\end{abstract}


\keywords{universal coupling, finite element analysis, solidworks, stress, strain, displacement.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
niversal coupling is commonly used in rotating shaft that transmits rotary motion. It is a specialized rotary joint used to allow a rotating split shaft to deflect along its axis in any direction. This flexibility is achieved by constructing the joint with two U-shaped yokes (coupler) joined by a cross shaped hub (pin). One yoke is attached to the end of each portion of the split shaft and joined with the cross hub, with the U-sections oriented at 90 o to each other. This arrangement allows the horizontal primary shaft to drive the inclined shaft with no undue friction or loss of speed or drive output potential. Typical applications of universal coupling include aircraft appliances, control mechanisms, electronics, instrumentation, medical and optical devices, ordnance, radio, sewing machines, textile machinery and tool drives. Fig. \hyperref[fig_0]{1} shows a commercially available universal coupling and Fig.  {\ref 2} shows CAD design of a coupling.\par
There are some available literature on universal coupling \hyperref[b2]{[3]}. The novelty of the present literature is that it gives emphasis on yoke and hub separately. The main stress zone of a coupling is the yoke slot-hub interface and the hub corners. In the yoke slot-hub interface, the stress acts is circumferential. At the same time, friction at the interface generates heat. This heat generates thermal stress. So during the operation of a coupling, mainly two types of stresses works, namely circumferential stress and thermal stress. During our investigation, we applied circumferential stress at the yoke slot-hub interface to see how it affects the stress propagation in both the yoke and the hub. In addition, we applied thermal stress and increased it gradually to show how it influences generation of stresses. To show the strain rate and the displacement in the yoke, the strain contour and the displacement contour are also plotted. The simulation was carried out in SolidWorks 2010, the validity and acceptability of which is well established.  The angles ? 1 and ? 2 in a rotating joint will be function of time. Differentiating the equation of motion with respect to time and using the equation of motion itself to eliminate a variable yields the relationship between the angular velocities ? 1 =d? 1 /dtand ? 2 =d? 2 /dt? 2 =? 1 cos?/(1-sin 2 ? cos 2 ? 1 )\par
The angular velocities are not linearly related but rather are periodic with a period twice that of the rotating shafts. The angular velocity relation can again be differentiated to get the relation between the angular acceleration ? 1 and ? 2 ,? 2 =? 1 cos?/(1-sin 2 ? cos 2 ?1) -(? 1 2 cos? sin 2 ? sin2? 1 )/(1-sin 2 ? cos 2 ? 1 ) 2 b) Solution Methodology\par
The CAD drawing is carried out in SolidWorks 2010. At first yoke, hub and disc are drawn in part drawing option separately. Then these three things are assembled in assembly drawing option. The main parameters used during the drawing are given below. No. of nodes (yoke) 12059\par
Fig. \hyperref[fig_3]{3} shows SolidWorks drawing of yoke, hub and disc before being assembled. Fig.  {\ref 4} shows different parts of the yoke and the hub.\par
The main features used during drawing are extrude, extrude cut, mirror and mate. For simplicity, details description of the drawing has been avoided.  
\section[{Results and Discussion}]{Results and Discussion}\par
In our specimen, the material considered is Al 1060 alloy. Modulus of elasticity of the material is E=69\par
GPa and Poisson's ratio is v =0.33. The simulation has been carried out in room temperature, which is considered to be 25 o C. To find out the most critical condition in terms of stress and strain, the clearance between the hub and the slot of yoke is kept zero.\par
Fig. \hyperref[fig_6]{5} shows the generation of strain across the yoke. It is found to be maximum along the edge of the yoke extension. Besides there is also an abrupt rise of strain at the extension-base intersection. The maximum value of strain is found to be 6.2x10 -5 and the minimum value is found to be 1.93x10 -8 . The value of strain around the slot is found to be almost 3x10 -5 . Fig.  {\ref 6} is a demonstration of displacement, takes place during the operation of a universal coupling. The displacement is found to be maximum at the free end of the yoke extension. It is in conformity with the physical condition because the extension works as a cantilever and a cantilever with a load at the free end displays maximum displacement at that end. On the other hand the displacement is negligible at the base. It is also valid because the base is considered to be rigidly fixed. The maximum value of the displacement is found to be almost 0.02mm. Fig. \hyperref[fig_7]{7} shows the demonstration of von Mises stress generated in the yoke. Like the generation of strain, maximum stress is found along the edge and at the extension-base intersection of the yoke. The maximum value is found to be 6.03 MPa and the minimum value is found to be almost 0.00099 MPa. Stress around the slot is about 3 MPa, which is half of the maximum stress. So in terms of von Misesstress, the most critical zone of a yoke is the base-extension intersection and the edge of the yoke extension having the maximum probability to fail. But under the given load at room temperature, the yoke would not fail because the maximum stress is 6.03 MPa which is much smaller than the yield strength of Al 1060 alloy, which again is 27.57 MPa.\par
Fig. \hyperref[fig_8]{8} shows relationship between temperature at the slot of the yoke and generation of maximum stress in the yoke. With increase of temperature in the slot surface, stress increases across the yoke. The    In Fig. \hyperref[fig_10]{10}, a relationship between temperature rise and displacement in the yoke has been showed. The relationship is not linear. The displacement at generation will be. The friction can be reduced significantly using bearing and lubricant. From the figure it is evident that under given loading and restrained relationship is linear in nature. That means the more friction between the hub and the slot of the yoke, the more temperature rise will be, hence the more stress is linear in the temperature range between 305K and 320K.\par
Fig. \hyperref[fig_0]{11} shows the distribution of von Mises stress in the hub. At the two free ends of the hub, circumferential pressure is applied at the slot-hub interface. The other two ends are assumed to be fixed. From the figure it is evident that, for the same loading condition as like in the yoke, generation of stress in the hub is larger. In case of yoke, the maximum stress generation is 6.03 MPa, where as in case of hub it is 7.577 MPa, which is about 20.4\% larger than the previous one. That means between the yoke and the hub, the hub will fail first, provided that both of them are facing same loading conditions. The extreme failure regions are found at the corners of the hub.\par
IV. 
\section[{Conclusions}]{Conclusions}\par
Stress and strain generated in a universal coupling is discussed elaborately. Attention is mainly given to the yoke slot and the hub because they are the main frictional zones. Effect of thermal stress has also been demonstrated in case of the yoke. It is showed that friction can cause significant thermal effect which eventually can increase the stress intensity of the yoke. For example, if the temperature at the slot of the yoke increases up to 315K (42 o C), the material may yield, because the generated stress will cross the yield strength of Al 1060 alloy.  \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Figure 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2212}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Figure 2 : 2 U 1 ? 2}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Figure 3 (}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_4}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-7.png}
\caption{\label{fig_5}Figure 3 (}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Figure 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{7}\includegraphics[]{image-9.png}
\caption{\label{fig_7}Figure 7 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{8}\includegraphics[]{image-10.png}
\caption{\label{fig_8}Figure 8 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{9}\includegraphics[]{image-11.png}
\caption{\label{fig_9}Figure 9 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{10}\includegraphics[]{image-12.png}
\caption{\label{fig_10}Figure 10 :}\end{figure}
 			\footnote{© 2013 Global Journals Inc. (US)} 			\footnote{© 2013 Global Journals Inc. (US) Failure Analysis of a Universal Coupling} 			\footnote{using Finite Element Method} 			\footnote{using Finite Element Method} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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\bibitem[Krauthamer ()]{b2}\label{b2} 	 		‘Universal Swivel Coupling or’.  		 			J Krauthamer 		.  	 	 		\textit{Joint. NY}  		1948.  	 
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\end{document}
