## 4 Analytical and Numerical Solutions  
Applying analytical, numerical, and FE ANSYS(r) solutions, vast results of radial, tangential, Von-Mises stresses, and strain rates for all loadings and rotational speed conditions are obtained. However, as stated above, the depicted speeds used for calculations and result display were 300, 500, and 1000 rad/sec. Typical FE results obtained for radial stress \(\sigma_{r}\); tangential stress \(\sigma_{t}\); and equivalent Von Mises Stress \(\sigma_{e}\) in the radial direction are shown on part of the cylinder color contour graphs in Figure 4, for \(\omega=300\) rad/sec. Meanwhile, Figures 5-8 show the radial stress, tangential stress, axial stress, and equivalent Von-Mises stress obtained from both analytical and numerical solutions for the three rotational speeds 300, 500, and 1000 rad/sec, respectively. All results shown are in full agreement with similar published data and display excellent compliance and high accuracy between analytical and numerical solutions [31; 32; 33; 34; 40; 41; 42; 43]. Figure 4: _Cont._  
Figure 3: Analysis time versus rotational velocity \(\omega\). Figure 4: Variation of (**a**) Radial Stress \(\sigma_{r}\); (**b**) Tangential Stress \(\sigma_{t}\); (**c**) Equivalent Von Mises in the radial direction Stress \(\sigma_{e}\), for \(\omega=300\) rad/sec. Figure 5: Variation of Radial Stress \(\sigma_{r}\) in the radial direction Stress of cylinder. Figure 6: Variation of Radial Stress \(\sigma_{t}\) in the radial direction Stress of cylinder. In Figure 5, trends of radial stresses \(\sigma_{t}\) distribution obtained from analytical and numerical solutions are shown in the radial direction of the cylinder for \(\omega=300\), \(\omega=500\), and \(\omega=1000\) rad/sec. These stresses remain constant both at the outer and inner surfaces. However, they possess maximum values for radius \(r\cong(r_{in}+r_{out})/2\) for \(\omega=300,\ \omega=500,\ \text{and}\ \omega=1000\) rad/sec. But they increase gradually with an increase in the rotational velocity \(\omega\). Figure 6 depicts the trends in tangential stresses \(\sigma_{t}\) acquired through analytical and numerical approach radially through cylinder for \(\omega=300,\ \omega=500,\ \text{and}\ \omega=1000\) rad/sec. The tangential stresses decline gradually at the outer periphery of the cylinder while minutely increasing at the inner periphery of the cylinder as the rotational velocity \(\omega\) increases. The tangential stresses are higher at the inner periphery of the cylinder than at the outer periphery. As seen from Figure 6 these stresses become nearly constant radially through the cylinder for \(\omega=300\) rad/sec and low rotational velocities. Figure 7 depicts the trends in axial stresses \(\sigma_{z}\) acquired through analytical and numerical approach radially through cylinder for \(\omega=300,\ \omega=500,\ \text{and}\ \omega=1000\) rad/sec. The axial stresses decline gradually at the outer periphery of the cylinder while minutely increasing at the inner periphery of the cylinder as the rotational velocity \(\omega\) increases. The tangential stresses are higher at the inner periphery of the cylinder than at the outer periphery. As seen from Figure 7 these stresses become nearly constant radially through the cylinder for \(\omega=300\) rad/sec and low rotational velocities. Figure 10: Variation of Tangential Strain Rate \(\dot{\epsilon}_{t}\) in the radial direction of the cylinder. Figure 8 represents the trends of Von Mises stresses \(\sigma_{e}\) acquired through analytical and numerical approach radially through cylinder for \(\omega=300,\ \omega=500\), and \(\omega=1000\) rad/sec. The Von Mises stresses decline gradually at the outer periphery of the cylinder while minutely increasing at the inner periphery of the cylinder as the rotational velocity \(\omega\) increases. The Von Mises stresses are higher at the inner periphery of the cylinder than at the outer periphery.