Using a cosine function of z so as to approximate a mode shape with a single nodal point along the shaft, approximate from the Rayleigh quotient the fundamental frequency of the system presented in Problems 7.8 and 7.11. Show that if I1, I2, p (recall I1 and I2 are polar moments of inertia of the added rigid cylinders and Ip is the polar moment of inertia of the connecting shaft) that
Problems 7.8:
Problem 7.8:
Do the first part of Problem 7.7 this time including the effects of two rigid cylinders, respectively of uniform mass per unit length ρ1 and ρ2 and polar moments of inertia of the cross section I1 and I2, placed at the ends z = 0 and z = L of the bar.
Problem 7.7:
Derive the differential equation and boundary conditions for the torsional oscillation of a bar in the absence of warping in terms of the rotation angle θ(z,t) and using the polar moment of inertia of the cross-vectors . Also discuss under what conditions the effect of the geometric cross section of the bar will not be a factor in the solution.
Problems 7.11:
Find the Rayleigh quotient for torsional vibration of Problem 7.8. What is the effect of the added masses on the free vibration frequency?
Problem 7.8:
Do the first part of Problem 7.7 this time including the effects of two rigid cylinders, respectively of uniform mass per unit length ρ1 and ρ2 and polar moments of inertia of the cross section I1 and I2, placed at the ends z = 0 and z = L of the bar.
Problem 7.7:
Derive the differential equation and boundary conditions for the torsional oscillation of a bar in the absence of warping in terms of the rotation angle θ(z,t) and using the polar moment of inertia of the cross-vectors . Also discuss under what conditions the effect of the geometric cross section of the bar will not be a factor in the solution.
Note that this corresponds to weak (or no) coupling with the added masses at the boundary conditions, which are ordinarily inhomogeneous.