Torsional Vibrations of a Conic Shaft with Opposite Tapers Carrying Arbitrary Concentrated Elements

Jia-Jang Wu

doi:10.1155/2013/491062

Abstract

The purpose of this paper is to present the exact solution for free torsional vibrations of a linearly tapered circular shaft carrying a number of concentrated elements. First of all, the equation of motion for free torsional vibration of a conic shaft is transformed into a Bessel equation, and, based on which, the exact displacement function in terms of Bessel functions is obtained. Next, the equations for compatibility of deformations and equilibrium of torsional moments at each attaching point (including the shaft ends) between the concentrated elements and the conic shaft with positive and negative tapers are derived. From the last equations, a characteristic equation of the form <svg style="vertical-align:-2.30685pt;width:78.125px;" id="M1" height="15.05" version="1.1" viewBox="0 0 78.125 15.05" width="78.125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.112)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,5.927,12.112)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28
h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g><g transform="matrix(.017,-0,0,-.017,20.835,12.112)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g><g transform="matrix(.017,-0,0,-.017,26.7,12.112)"><path id="x7B" d="M293 -169q-145 0 -145 128q0 35 4 106t3 104q0 17 -2 28t-10 25.5t-28 23t-51 10.5v29q31 2 51 10.5t28 23t10 25.5t2 28q0 31 -4 101t-3 104q0 132 145 132v-28q-51 -7 -67.5 -30t-16.5 -63q0 -32 6.5 -99.5t6.5 -100.5q0 -97 -83 -115v-4q83 -20 83 -117
q0 -34 -6.5 -99.5t-6.5 -96.5q0 -41 16.5 -65.5t67.5 -31.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,32.598,12.112)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.017,-0,0,-.017,44.582,12.112)"><path id="x7D" d="M283 255q-31 -2 -51 -10.5t-28 -23.5t-10 -27t-2 -30q0 -28 4 -97.5t4 -107.5q0 -128 -146 -128v28q50 7 67.5 31t17.5 63q0 33 -7 100t-7 98q0 96 84 117v4q-84 20 -84 115q0 34 7 102.5t7 98.5q0 40 -17.5 63t-67.5 30v28q28 0 50 -4t45.5 -16.5t37 -40t13.5 -68.5
q0 -37 -4 -106.5t-4 -98.5q0 -18 2 -30t10 -27t28 -23.5t51 -10.5v-29z" /></g><g transform="matrix(.017,-0,0,-.017,55.189,12.112)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,69.893,12.112)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
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" /></g> </svg> is obtained. Then, the natural frequencies of the torsional shaft are determined from the determinant equation <svg style="vertical-align:-2.22495pt;width:51.549999px;" id="M2" height="14.85" version="1.1" viewBox="0 0 51.549999 14.85" width="51.549999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path id="x7C" d="M162 -163h-61v866h61v-866z" /></g><g transform="matrix(.017,-0,0,-.017,4.533,12.012)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,19.441,12.012)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,28.62,12.012)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,43.324,12.012)"><use xlink:href="#x30"/></g> </svg>, and, corresponding to each natural frequency, the column vector for the integration constants, <svg style="vertical-align:-2.30685pt;width:23.9px;" id="M3" height="15.05" version="1.1" viewBox="0 0 23.9 15.05" width="23.9" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.112)"><use xlink:href="#x7B"/></g><g transform="matrix(.017,-0,0,-.017,5.961,12.112)"><use xlink:href="#x1D436"/></g><g transform="matrix(.017,-0,0,-.017,17.945,12.112)"><use xlink:href="#x7D"/></g> </svg>, is obtained from the equation <svg style="vertical-align:-2.30685pt;width:78.125px;" id="M4" height="15.05" version="1.1" viewBox="0 0 78.125 15.05" width="78.125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.112)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,5.927,12.112)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,20.835,12.112)"><use xlink:href="#x5D"/></g><g transform="matrix(.017,-0,0,-.017,26.7,12.112)"><use xlink:href="#x7B"/></g><g transform="matrix(.017,-0,0,-.017,32.598,12.112)"><use xlink:href="#x1D436"/></g><g transform="matrix(.017,-0,0,-.017,44.582,12.112)"><use xlink:href="#x7D"/></g><g transform="matrix(.017,-0,0,-.017,55.189,12.112)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,69.893,12.112)"><use xlink:href="#x30"/></g> </svg>. Substitution of the last integration constants into the associated displacement functions gives the corresponding mode shape of the entire conic shaft. To confirm the reliability of the presented theory, all numerical results obtained from the exact method are compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved.

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