^{1}

^{*}

^{1}

In this study, the influence of geometrical parameters on the curve veering phenomenon in a tor-sional system with stepped shaft is investigated. Three approximate solutions including finite el-ement, Rayleigh-Ritz and discretization methods, along with an exact solution are employed to obtain the natural frequencies of the structure. The study reveals that, under specific circumstances, the results obtained by approximate methods are very close to the exact solution. The curve veering behavior is manifested irrespective of the method employed. It is concluded that for the structure studied the curve veering behavior is not because of the approximate techniques used to compute the natural frequencies, and is an inherent behavior of the structure.

Curve veering is defined as an abrupt veering of the natural frequency plots, when plotted against some system parameters [

In high speed rotating machinery, a considerable number of studies have been carried out on the natural frequencies and mode shapes. Most often, in view of the complex geometry of the rotor systems, they are treated as lumped rotors mounted on shafts. In many practical situations, the shafts may have different cross sections and may have stepped configuration. Accurate determination of the natural frequencies is imperative in order to ensure that the system does not operate near resonant frequencies and particularly in the vicinity of curve veering ranges. Exact solutions are possible only in the case of well-defined uniform shaft geometries, and for practical rotors with many cross sectional changes, approximate techniques such as the discretization method, the Rayleigh Ritz method, finite element method are used. The first general theory for free vibration analysis of torsional systems was reported by Beddoe [

In the present study, a stepped shaft supporting a rotating disk at the tip is analyzed for its curve veering behavior by computing the natural frequencies by different methods. The effect of geometric parameters of the stepped shaft disk system on the curve veering phenomenon is investigated. Although approximate solutions exhibit curve veering in the structure, an exact method is also employed to confirm this phenomenon as an inherent property of the structure.

An isotropic, homogeneous torsional system composed of a stepped shaft with a lumped disk at the tip, as depicted in

The equation of motion of the shaft is given by:

where

where

is obtained by substituting the boundary conditions as:

where

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

where

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 [

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

Finally, fundamental frequency of the structure is obtained as:

where

In this study, the following base line values are assumed in the analysis: shear modulus of the structure is 79.3 Gpa, density is 7800 kg/m^{3},

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.

Number of admissible functions | First frequency |
---|---|

1 2 3 4 5 6 7 8 | 256.868 165.885 137.485 136.992 130.562 130.562 127.832 127.790 |

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,

When the variation of natural frequencies against length ratio

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.

_{1} = 0.1 m, d_{2} = 0.01 m, d = 0.5 m and M = 100 kg and_{2} = 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 _{1} = 0.1 m, d_{2} = 0.05 m, d = 0.5 m and M = 100 kg. In this case, the curve veering is observed. _{2} = 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.

L_{1}/L | First | Second | Third | Fourth | Fifth |
---|---|---|---|---|---|

0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 | 3.640 3.660 3.679 3.700 3.720 3.741 3.762 3.783 3.806 3.828 3.851 3.874 3.897 3.897 3.945 3.970 3.996 4.022 4.048 | 5329 5386 5444 5504 5565 5628 5692 5757 5824 5893 5963 6035 6108 6184 6261 6340 6421 6505 6590 | 10657 10771 10888 11008 11130 11255 11383 11514 11648 11785 11925 12069 12215 12365 12482 11924 11383 10888 10435 | 15985 16157 16332 16512 16695 16883 17074 17270 17467 16694 15652 14732 13914 13183 12562 12682 12844 13010 13181 | 21313 21542 21776 22015 22260 22501 20869 19265 17680 17889 17893 18104 18325 18551 18873 19020 19264 19514 19771 |

Frequency (rad/s)

Frequency (rad/s)

This study deals with an analysis on the curve veering phenomenon in a torsional structure, which consists of a stepped shaft and a rotating disk. Different approximate techniques including the Rayleigh-Ritz, finite element and discretization methods, along with the exact solution were employed to extract natural frequencies of the structure. The results reveal that curve veering in this structure is not due to application of approximate solution, and it appears even if an exact solution is employed. As a result, the curve veering may be regarded as an inherent behavior of the structure. The geometric parameters affect the curve veering, noticeably. Moreover, a comparison of the results obtained by approximate solutions and those of the exact one was carried out. It was realized that, under some specific geometries and material properties, the frequencies obtained from approximate solutions are as accurate as the exact solution. Under such conditions, the trigonometric functions which describe angular displacement field can be replaced by a linear function.