On Free Vibrations of Elastodynamic Problem in Rotating Non-Homogeneous Orthotropic Hollow Sphere

<table>Variation of non-dimensional frequency (three modes) versus the radius <svg height="7.9499998" id="M125" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 7.3000002 7.9499998" width="7.3000002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg> of the hollow sphere at absent of rotation <svg height="11.6375" id="M126" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 52.612499 11.6375" width="52.612499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.362)"><path d="M719 146l-18 -146h-243l-5 18q2 4 17 25t24 36l25 42t26.5 52.5t21 56.5t16 65t5.5 68q0 121 -55.5 194.5t-149.5 73.5q-95 0 -150 -72t-55 -199q0 -54 18.5 -114t36.5 -94t48.5 -82.5t32.5 -51.5l-7 -18h-244l-17 146l27 1q17 -49 31.5 -62t45.5 -13h89l-21 21t-26 26.5
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</svg> and non-homogeneous <svg height="11.125" id="M127" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 53.150002 11.125" width="53.150002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
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q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g><g transform="matrix(.017,-0,0,-.017,41.046,10.862)"><path d="M113 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16z" id="x2E"></path></g><g transform="matrix(.017,-0,0,-.017,44.922,10.862)"><path d="M153 550l-26 -186q79 31 111 31q90 0 141.5 -51t51.5 -119q0 -93 -89 -166q-85 -69 -173 -71q-32 0 -61.5 11.5t-41.5 23.5q-18 17 -17 34q2 16 22 33q14 9 26 -1q61 -50 124 -50q60 0 93 43.5t33 104.5q0 69 -41.5 110t-121.5 41q-53 0 -102 -20l38 305h286l6 -8
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</svg> (inner fixed surfaces and outer free surfaces) and  <svg height="11.125" id="M128" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 36.237499 11.125" width="36.237499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg>.</table>

Mathematical Problems in Engineering

fig5

Figure 5

Figure 5: On Free Vibrations of Elastodynamic Problem in Rotating Non-Homogeneous Orthotropic Hollow Sphere 