Vibration Analysis of Hollow Tapered Shaft Rotor

Shafts or circular cross-section beams are important parts of rotating systems and their geometries play important role in rotor dynamics. Hollow tapered shaft rotors with uniform thickness and uniform bore are considered. Critical speeds or whirling frequency conditions are computed using transfer matrix method and then the results were compared using finite element method. For particular shaft lengths and rotating speeds, response of the hollow tapered shaft-rotor system is determined for the establishment of dynamic characteristics. Nonrotating conditions are also considered and results obtained are plotted.


Introduction
Shaft is a major component of any rotating system, used to transmit torque and rotation.Hence the study shaft-rotor systems has been the concern of researchers for more than a century and will continue to persist as an active area of research and analysis in near future.Geometry of shaft is of the main concern during the study of any rotating system.Most papers related to shaft-rotor systems consider cylindrical shaft elements for study and analysis of rotating systems.The first idea of transfer matrix method (TMM) was compiled by Holzer for finding natural frequencies of torsional systems and later adapted by Myklestad [1,2] for computing natural frequencies of airplane wings, coupled in bending and torsion.Gyroscopic moments were first introduced by Prohl [3] for rotor-bearing system analysis.Lund [4] used complex variables as the next significant advancement in the method.An improved method for calculating critical speeds and rotor stability of turbo machinery was investigated by Murphy and Vance [5].Whalley and Abdul-Ameer [6] used frequency response analysis for particular profiled shafts to study dynamic response of distributed-lumped shaft rotor system.They studied the system behavior in terms of frequency response for the shafts with diameters which are functions of their lengths.They derived an analytical method which uses Euler-Bernoulli beam theory in combination with TMM.On the other hand, there are large numbers of numerical applications of finite element techniques for the calculation of whirling and the computation of maximum dynamic magnitude.In this regard, Ruhl and Booker [7] modeled the distributed parameter turbo rotor systems using finite element method (FEM).Nelson and McVaugh [8] reduced large number of eigenvalues and eigenvectors identified, following finite element analysis, and the erroneous modes of vibration predicted were eliminated.Nelson [9] again formulated the equations of motion for a uniform rotating shaft element using deformation shape functions developed from Timoshenko beam theory including the effects of translational and rotational inertia, gyroscopic moments, bending and shear deformation, and axial load.Greenhill et al. [10] derived equation of motion for a conical beam finite element form Timoshenko beam theory and include effects of translational and rotational inertia, gyroscopic moments, bending and shear deformation, axial load, and internal damping.Genta and Gugliotta [11] also analyzed element with annular cross-section based on Timoshenko beam theory having two degrees of freedom at each node.Mohiuddin and Khulief [12] derived a finite element model of a tapered rotating cracked shaft for modal analysis and dynamic modeling of a rotor-bearing system, based on Timoshenko beam theory, that is, included shear deformation and rotary inertia.Rouch and Kao [13] presented numerically integrated formulation of a tapered beam element for rotor dynamics.

Shaft Model. The shaft model is derived in matrix form
Whalley and Abdul-Ameer [6] as where ] . ( The complete derivation is present in [6].

Rigid Disk.
The output vector from the shaft will become the input for the rigid disk model, as shown in Figure 1; that is, for disk model, we have where ( 3 (),  1 ()), ( 3 (),  1 ()), ( 3 (),  1 ()), and ( 3 (),  1 ()) are the deflections, slopes, bending moments, and shear forces at the free and fixed end, respectively.Hence, writing in matrix form, we have where Form transfer matrix method [6]  () =  ()  () , where and input-output vectors relationship is given by After applying the boundary conditions for cantilever beam, deflection at the free end is obtained and hence leads to transfer function.

Finite Element Method
The vector of nodal displacements is given by So, each element is having eight degrees of freedom.
3.1.Rigid Disc.Rigid disk is having two translations and two rotations in  and  direction, respectively (considering  coordinate in axial direction).For constant spin condition, the Lagrangian equation of motion is given by where The forcing term may include mass unbalance and other external forces.

Finite Shaft-Rotor Element.
The rotor-shaft element considered here has eight degrees of freedom, that is, four degrees of freedom per node as in Nelson and McVaugh [8].For constant spin condition, the Lagrangian equation of motion is given by where Except skew-symmetric gyroscopic matrix [  ], others are symmetric matrices.Since the element is linearly tapered, area and inertias are the function of the shaft-rotor length.
The translational shape function is given by where (15) The rotational shape function is given by The element matrices are assembled together to get the equation of motion for the complete system.

Numerical Results
4.1.Rotating Condition.By using the transfer matrix approach as in the paper of Whalley and Abdul-Ameer, we will ultimately get the transfer function which will be plotted.The FEM will be applied and then is compared with the TMM approach of Whalley and Abdul-Ameer [6].
Example 1.Let us consider a cantilever tubular shaft with uniform thickness and a disc at the free end with downward unit force, , on the disc as shown in Figure 2. The default values of various parameters are tabulated in Table 1.
Transfer function for tubular shaft with constant thickness as shown in Figure 2 for default values is given by 1.344 + 1.016 × 10 4  3 + 7564 2 + 8.401 × 10 5  + 1.589 × 10 9 . (17 Bode plots for different lengths and rotating speed have been plotted using MATLAB software, as shown in Figures 3 and  4.
Applying FEM on the same system, we get mass, gyroscopic, and stiffness matrices.A finite hollow tapered shaft element is shown in Figure 5.
Table 1: Various parameters of the system shown in Figure 2  The stiffness matrix for hollow tapered shaft element with uniform thickness is given by is given by where elements of the stiffness matrices are Translational mass matrix is given by Advances in Acoustics and Vibration 5 where elements of translational mass matrix are given by Rotational mass matrix is given by where elements of rotational mass matrix are given by Advances in Acoustics and Vibration Gyroscopic matrix is given by where the elements of the gyroscopic matrix are given by  Discretizing the tapered shaft into six elements as shown in Figure 6 and then assembling, we get the assembled equation of motion where [   ] is the assembled mass matrix containing both the translational and rotational mass matrices.
The assembled equation of motion is arranged in the first order state vector form where The shaft rotor has been discretized into six elements of equal length.Hence the order of assembled matrices, after applying the fixed-free boundary condition, is 7 × 4 − 4 = 24.MATLAB program is used to find the bode plot for different values of shaft length and rotor speed as shown in Figures 7 and 8 and are found to be in good agreement with bode plots found using TMM as shown in Figures 3 and 4.
Example 2. Let us consider a cantilever hollow shaft with uniform bore and a disc at the free end, as shown in Figure 9.The values of various parameters are tabulated in Table 2.
Figure 5: Hollow tapered shaft finite element with uniform thickness.The transfer function for hollow shaft with constant thickness for default values is given by The bode plots for different lengths and rotating speeds have been plotted using MATLAB software, as shown in Figures 10  and 11.
Applying FEM in the same system, we get mass, gyroscopic, and stiffness matrices.A finite hollow tapered shaft element with uniform bore is shown in Figure 12.
The stiffness matrix for hollow tapered shaft with uniform bore is given by where elements of the stiffness matrices are Translational mass matrix is given by where elements of translational mass matrix are given by Rotational mass matrix is given by where the elements of the rotational mass matrix are given by Advances in Acoustics and Vibration 11 The gyroscopic matrix is given by where the elements of the gyroscopic matrix are    Applying FEM, then for zero rpm we get the bode plot as shown in Figure 16.

Conclusions
Shaft geometry plays one of the important roles in dynamic characteristics of rotating systems.Vibration analysis with the help of bode plots has been done for hollow tapered shaftrotor system.Both TMM and FEM have been used for the purpose.The equation of motion for a tapered beam finite element has been developed using Euler-Bernoulli beam

Figure 9 :
Figure 9: Hollow tapered shaft disc with uniform bore and vertically downward force  on the disc.

4. 2 .
Nonrotating Conditions.Bode plot for nonrotating (1 or 2 rpm) tapered shaft-rotor system is slightly different from rotating conditions in terms of amplitude.Hollow shaft with uniform thickness is considered.Bode plots are obtained for zero rpm as shown in Figure15with TMM.

Table 2 :
Various parameters of the system shown in Figure9and their default values.