The equation of motion of the shaft is given by:

where, and are density, shear modulus and twist angle of the shaft, respectively. The solution of Equation (1) may be found as follows [22] :

where is the frequency of vibration and A, B, C and D are unknown coefficients. The characteristic equation

is obtained by substituting the boundary conditions as:

where and and J are polar moment of inertia of upper shaft, polar moment of inertia of lower shaft and polar mass moment of inertia of the disk, respectively, and, and are unknown coefficients. For non-trivial solutions, the determinant of the matrix should be set to zero which will yield the natural frequencies.

In the Rayleigh-Ritz solution, displacement field of a structure is defined as linear combination of admissible functions. In this study, the deflection shape is considered as:

where are admissible functions satisfying at least the geometrical boundary conditions and are arbitrary coefficients. For a shaft with distributed mass and elasticity, the kinetic and potential energy expressions are given by:

where is the frequency of vibration. For harmonic vibrations, we have:

The conditions for the stationary of the natural frequencies with respect to the arbitrary coefficients in the assumed deflection expression formulate the eigenvalue problem of the structure. It is well-known that the natural frequencies obtained by the Rayleigh-Ritz method are the upper bound. In this study, the following formulation is employed to obtain orthogonal admissible functions [23] .

where

It should be noted that, increasing the number of admissible functions improves the convergence of the results.

Discretization technique may be regarded as the simplest, and the least accurate method that is used to find the fundamental frequency of the structure quickly. In this solution, the stiffness constants of the upper and lower shafts are found, individually. Total stiffness of the stepped shaft is obtained as a series combination of these two shaft segments. It should be noted that, solving the problem using this approach necessitates assuming linear torsional deflection through the stepped shaft, while the exact solution reveals trigonometric functions for the shaft deflection. The stiffness constants of the upper and lower shafts are

where and are shear modulus, polar moment of inertia and length of each shaft, respectively. Total stiffness of the stepped shaft is given by:

Finally, fundamental frequency of the structure is obtained as:

where is polar mass moment of inertia of the disk.

The exact solution for the fundamental frequency of the structure is obtained as 121.022 rad/s. The results obtained by the Rayleigh-Ritz solution reveal that, using one term of admissible function gives fundamental frequency of the structure as 256.88 rad/s. It should be noted that, the admissible functions satisfy the geometric boundary condition, along with continuity of the angular displacement and torque at the point of step change in the shaft cross section. Table 1 shows convergence of the fundamental frequency when the admissible functions satisfy only the geometric boundary conditions. The results given in Table 1 indicate that, they converge to the exact value, although the rate of convergence is poor.

Just using two elements in the finite element model yields the same result as obtained by the exact solution. It is interesting to note that the discretization technique gives fundamental frequency of the structure equal to 121.056 rad/s. The discretization technique considered the displacement field of the structure as a linear function, while the exact solution uses trigonometric functions to describe the displacement field. This is attributed to the magnitude of λ in the exact solution, which has a very small value. In fact, in this order of λ, may be assumed to be equal to λ; as a result, the frequency obtained by linear deflection assumption is the same as that of the exact solution. In order to clarify this behavior, another example is presented. In this example, all parameter values of the structure remain unchanged except the density, which is assumed to be. In this case, does not have a small value and. The fundamental frequency obtained by the exact solution is 40.8408 rad/s, while the frequency obtained using discretization technique is equal to 121.056 rad/s. Moreover, the Rayleigh-Ritz method―using one admissible functions satisfying geometric boundary condition and continuity of torque and displacement―yields. The difference between these results derives from the fact that, for higher values of, the assumption of linear deflection through the shaft is incorrect. In this case, more terms in the Rayleigh-Ritz method is required to obtain accurate results.

When the variation of natural frequencies against length ratio is plotted, the curves change smoothly everywhere except in some regions, where they show sudden changes. As mentioned earlier, Leissa [2] called these regions “transition zones”. This behavior was reported as a result of approximate solution method used to solve the problem, but in the present study curve veering is present whether the results are obtained by the exact method or the approximate methods. Hence, the curve veering may be regarded as an inherent behavior of the system. Figure 2 shows the variation of the first five natural frequencies against length ratio, when, , and M = 100 kg. It is observed that, the variation of the fundamental frequency of the structure against length ratio is not significant. In fact, the curve veering does not occur, when the structure vibrates in the first vibrational mode. The curve veering may be observed in other modes of vibration. The graph

shows that the number of transition zones increases when mode number increases. For instance, there is just one transition zone in the second mode of vibration, while three, five and seven transition zones may be observed in the third, fourth and fifth vibrational modes, respectively. In order to understand the curve veering phenomenon, transition zone corresponding to the fourth and fifth modes has been magnified. It can be seen that, natural frequencies approach each other and veer away in this region. This behavior is of great significance for the designers.

Table 2 shows the variation of the first five resonant frequencies of the structure, when d₁ = 0.1 m, d₂ = 0.01 m, d = 0.5 m and M = 100 kg and. For the range of considered, the shaft behaves, approximately, as a slender bar of d₂ = 0.01 m; as a result, the fundamental natural frequency of the structure becomes very low, in the range of 3 - 4 rad/s. The bold frequencies indicate the curve veering point in the transition zones. In Figure 3 the variation of the first five natural frequencies versus length ratio of the structure is shown when d₁ = 0.1 m, d₂ = 0.05 m, d = 0.5 m and M = 100 kg. In this case, the curve veering is observed. Figure 4 shows an analogous study on the structure, when d₂ = 0.095 m. It may be seen that variation of length ratio does not show a drastic change in the frequencies. It is attributed to the diameter of upper and lower shafts, which have almost the same magnitudes.

Frequency (rad/s)

Frequency (rad/s)