186 Mechanics of Materials §8.12 8.12.Variation of data along shaft length-torsion of tapered shafts This section illustrates the procedure which may be adopted when any of the quantities normally used in the torsion equations vary along the length of,the shaft.Provided the variation is known in terms of x,the distance along the shaft,then a solution can be obtained. 8 d: Fig.8.9.Torsion of a tapered shaft. Consider,therefore,the tapered shaft shown in Fig.8.9 with its diameter changing linearly from d to de over a length L.The diameter at any section x from end A is then given by d=d+d。-d2 Provided that the angle of the taper is not too great,the simple torsion theory may be applied to an element at section XX in order to determine the angle of twist of the shaft,i.e.for the element shown, Gde T dxJxx Therefore the total angle of twist of the shaft is given by 0= GJxx 0 Now nd4 Jxx=32=32 a+a,-d克 Substituting and integrating, da+dad8+di dida the standard esult for a paralle shaf 32TL When d=da=d this reduces to 0= 8.13.Power transmitted by shafts If a shaft carries a torque T Newton metres and rotates at w rad/s it will do work at the rate of To Nm/s (or joule/s).186 Mechanics of Materials $8.12 8.12. Variation of data along shaft leogth-torsion of tapered shafts This section illustrates the procedure which may be adopted when any of the quantities normally used in the torsion equations vary along the length of,the shaft. Provided the variation is known in terms of x, the distance along the shaft, then a solution can be obtained. I-L-' Fig. 8.9. Torsion of a tapered shaft. Consider, therefore, the tapered shaft shown in Fig. 8.9 with its diameter changing linearly from d, to dB over a length L. The diameter at any section x from end A is then given by Provided that the angle of the taper is not too great, the simple torsion theory may be applied to an element at section XX in order to determine the angle of twist of the shaft, i.e. for the element shown, Gd8 T dx Jxx - Therefore the total angle of twist of the shaft is given by Now Substituting and integrating, the standard result for a parallel shaft. 32 TL When dA = dB = d this reduces to 8 = - nGd 8.13. Power transmitted by shafts If a shaft carries a torque T Newton metres and rotates at o rad/s it will do work at the rate of Tw Nm/s (or joule/s)