Abstract
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.
1. 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.
In this era of machines, tapered shafts are widely used for rotating systems. Using the approach of Whalley and Abdul-Ameer [6], the dynamic analysis of hollow tapered shaft-rotor has been done. Later the results obtained were compared with that obtained from finite element method. The effect of length and speed on the dynamic analysis of the hollow tapered shaft-rotor system is also clearly shown. Frequency response of the rotor system for an impulse of unit force at the free end is determined in terms of critical speeds for various rotor speeds and shaft lengths.
2. Transfer Matrix Method
2.1. Shaft Model
The shaft model is derived in matrix form Whalley and Abdul-Ameer [6] as where
The complete derivation is present in [6].
2.2. 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 , , , and 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.
3. 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.
3.2. 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 The rotational shape function is given by The element matrices are assembled together to get the equation of motion for the complete system.
4. 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 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.
The stiffness matrix for hollow tapered shaft element with uniform thickness is given by is given bywhere elements of the stiffness matrices are Translational mass matrix is given bywhere elements of translational mass matrix are given by
Rotational mass matrix is given bywhere elements of rotational mass matrix are given by
Gyroscopic matrix is given bywhere 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 .
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.
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 bywhere elements of translational mass matrix are given by
Rotational mass matrix is given bywhere the elements of the rotational mass matrix are given by The gyroscopic matrix is given bywhere the elements of the gyroscopic matrix are As proceeded in Example 1, bode plots are obtained for various shaft lengths and rotor speeds as shown in Figures 13 and 14 and are found to be in good agreement with bode plots found using TMM as shown in Figures 10 and 11.
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 Figure 15 with TMM.
Applying FEM, then for zero rpm we get the bode plot as shown in Figure 16.
5. 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 shaft-rotor 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 theory. Mass, stiffness, and gyroscopic matrices are found and values of all these elements are stated in a systematic manner for ease of understanding. The results obtained from both methods are compared and are found to be in good agreement. However, the above procedures show that the method of TMM is simpler in calculations. Two types of hollow tapered shafts have been analyzed, that is, one with uniform thickness and another with uniform bore.
The length of the shaft is a vital parameter that affects the frequency response of the shaft-rotor system. As shown in Figures 3, 7, 10, and 13 there is increase in amplitude of vibration for increased value of shaft length, while reducing the whirling speed of shaft. The system exhibits this behaviour due to the bending effect of the shaft and stiffness change. For changing lengths, there are large differences in frequencies even for small increase in lengths. Bode plots obtained in Figures 4, 8, 11, and 14 show that rotating speeds have very little effect on the critical frequencies; however with increasing speed, the amplitude is lowered due to gyroscopic couple. The phase angle changes abruptly for lower value of shaft speeds than higher speeds. This is due to the fact that there is reduction in gyroscopic couple as the rotational speed of the shaft decreases thereby giving larger amplitude of vibration. Nonrotating conditions are also shown in Figures 15 and 16 as bode plots which show that any rotating system at a very low speed vibrates with high amplitudes due to lack of gyroscopic couple.
Effects of bearing may be included in the problem. Geared systems and other rotary elements can be mounted instead of discs and further calculations can be made. Multidiscs and other complex problems can be solved using these methods.
Notations
| Compliance per unit of length (function) | |
| Inertia per unit of length (function) | |
| Length of shaft | |
| Modulus of elasticity | |
| System model matrix | |
| Bending moment in plane (function) | |
| Shear force (function) | |
| Vertical deflection of shaft (function) | |
| Slope of the shaft (function) | |
| Mass moment of inertia (function) | |
| Material density | |
| Shaft polar moment of inertia | |
| Whirling frequency | |
| Wave propagation factor (function) | |
| Shaft-rotor rotational speed | |
| Rigid rotor model matrix | |
| Polar moment of inertia of disc | |
| Beginning radius of the shaft element | |
| Inner radius of the shaft element | |
| End radius of the shaft element | |
| Mass of the disc attached at free end | |
| Translational mass matrix for disc | |
| Rotational mass matrix for disc | |
| Gyroscopic matrix for disc | |
| Translational mass matrix | |
| Rotational mass matrix | |
| Gyroscopic matrix for element | |
| Stiffness matrix for shaft element | |
| External force matrix | |
| Cross-sectional area (function) | |
| Diametral inertia (function) | |
| Polar inertia (function) | |
| Rotational speed in rad/s | |
| Translational shape function matrix | |
| Rotational shape function matrix. |
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.