Free and Forced Whirling Analyses of a Single-Disk Rotor Subject to Axial Force

Abstract

Keywords: Axial load; lateral vibration; whirling; campbell diagram.

1. Introduction

Rotor systems are widely used assemblies for power transmission in various kinds of machinery. The lateral vibrations of the rotors must be limited because excessive vibrations will affect the running status of it and may lead to catastrophic accidents. Due to several factors, which contribute to the energy transfer, the rotor rotations generate different modes of vibrations: lateral, torsional and axial modes. Among these modes, the lateral is of the greatest concern [1]. The simplest model that can be adopted to study the flexible behavior of rotors consists of a point mass attached to a massless shaft, often referred to as Jeffcott rotor. One of the most important vibration analysis parameter is a natural frequency. Equality of the system rotation speed and *

Corresponding Author: e-mail: zeinolabedini_mr@yahoo.com, tel: 00983531232498 21

system frequency creates a critical state in the system. Also critical speed is an important parameter in the design of systems and their vibration analysis. The operation of rotating systems in the frequency range of the resonator can be very risky and cause serious damage to it. Parameters like mass and stiffness affect the natural frequency, and any change in the system that changes these two parameters will change the natural frequency of the system. In general, axial forces are indeed small therefore are not considered. In aircraft engines, the thrust generates axial force in the rotor system. In compressors stall and surge phenomena can generate horizontal force on disks. These forces may be constant, while in rotating machinery is specific, they may be harmonic or random. This is the main motivation of the current research. Grybos [2] studied the effect of shear deformations on the critical speeds of rotors. Jei and Lee [3] derived critical speeds of an un-symmetric rotor with rigid disks. Free and forced whirling analysis of viscoelastic rotors was investigated by Sturla and Argento [4]. Jun and Kim [5] studied the effect of torsional torque on whirling of rotors. Tiwari and Bhaduri [6] assessed the effect of boundary conditions on the natural frequencies of the rotor bearing system. Nelson et al. [7] studied the vibration analysis of the Timoshenko rotor with internal damping under axial load. Edney et al. [8] presented the dynamic analysis of the tapered Timoshenko rotor. Chen et al. [9] analyzed an exact and direct modeling for rotor bearing system subject to axial load. Choi et al. [10] presented the consistent derivation of a set of governing differential equations describing the vibration in two orthogonal planes and the torsional vibration of a straight rotor with dissimilar lateral principle moments of inertia, subjected to a constant compressive axial load. Huajian gouyang et al. [11] presented a dynamic model for the vibration of a rotating Timoshenko beam subjected to a three- directional load moving in the axial direction. Askarian et al. [12] assessed the effect of various parameters such as axial force, unbalance, and coupling misalignment on vibration of a rotor. Nawal H.Al et al. [13] derived the equation of motion that governs the transvers vibration of a beam loaded axially and compared the natural frequencies thereof. Motallebi et al. [14] adopted homotopy analysis based method to assess the vibration of a nonlinear beam subject to axial force. Torabi et al. [15] presented an analytical solution for whirling analysis of axial-loaded Timoshenko rotor and corresponding basic function. In this study the dynamics of a rotating shaft, modeled as a Timoshenko rotor, subject to distributed harmonic load acting on the surface of a disk on the rotor are investigated. For this, the dynamic equations of lateral vibrations of the rotor are derived using Timoshenko beam model and Newton’s second law. Then, these governing equations are solved by Galerkin and Newmark method and effect of the symmetric axial force acting on the disk on the Campbell diagram and the rotor vibration response are analyzed.The most advantages of the presented paper are consideration of time-dependency of rotating speed in forced whirling analysis.

2. Governing Equations And Solution Procedure

A simply supported Timoshenko rotor subject to distributed axial force acted on disk is shown in Figure 1. The rotational speed increases from zero to its nominal value ( ) in time as: Ω Ω

1 An element of rotor with forces and bending moment is shown in Figure 2.

(b) Figure 2. An element of rotor a) shear forces and bending moments, b) axial forces

By applying the equilibrium of forces and momentums, the set of equations of motion are extracted as follows: ,

Ω where, , , are displacement and rotation components in and directions, respectively; and ρ is the mass density. A, , are the cross- sectional area, moment of inertia around the x and y axes and polar moment of inertia, respectively, and and are the forces per unit length in x and y directions and axial force and the disk position. According to Timoshenko beam theory, components of bending moment (M) and shear force (F) in x and y directions are presented as follows [16]: ∂ EI ∂z ∂ (3) EI ∂z where E and G are the modulus of elasticity and shear modulus, respectively; k is the shear correction factor depending on the shape of the section and Poisson's material ratio [17]. and applying the following equation for a circular section: By neglecting the term of 2

where, and are the reaction of force in the support and axial force exerted on disk. By re-arranging the equations of motion Eq.5 is yield: ,

By introducing the following complex variables: (8) f and The external forces that consisting weight and unbalanced force written as: Ω

where, is the disk mass, is the unbalanced mass, is the unbalanced radius and θ is the angle of unbalanced mass with respect to horizontal axis, the following set of equations of motion is yield:

2 0 For a rotor with simple support the following equation must be met: 0∶

Inserting Eq.12 into Eq.10 and then running the required simplifications the following equations are obtained:

3. Numerical Results And Discussion

In order to present and analyze the results obtained in this study, some specifications of the subject rotor are selected from Table 1.

Table 1. Considered rotor characteristics characteristics Rotor diameter

4. Conclusions

In this study, the effect of applied horizontal distributed force on a disk of a single disk rotor was studied using Campbell's diagram. The rotor was assumed to be uniform and Timoshenko theory was used. Partial differential equations of motion were derived by considering equilibrium equations for an element of the rotor. The rotor was considered simply supported and GalerkinNewmark method was applied directly to the partial differential equations of motion.The results show that the application of horizontal force in a rotor causes a change in the critical speed of the system. This is due to the effect of horizontal force on the stiffness of the system. The greatest effect is seen in the first mode of the system and increases with the increase of the applied horizontal force. However, with the increase of the force of the second mode, the second mode also changes the critical velocities, which is less than the first mode and decreases. The results of this study show that the effect of this force should be considered in designing rotary systems that are observing the presence of a horizontal force. Also¸ the effects of axial symmetrically force acted on disk on lateral vibration of a single disk rotor was analysed. Responses of lateral vibration for without and with axial symmetrically force were a analysed. From the results it is observed that by increasing the amplitude of force the amplitude of direction shear force increases.

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