A new dynamic model of rotor–blade systems

Highlights

• A new dynamic model of rotor–blade systems is developed.

• A test rig of a rotor–blade system is established.

• The proposed model is validated using FE method and experiment.

• Effects of blade numbers on natural frequencies are analyzed.

Abstract

A new dynamic model of rotor–blade systems is developed in this paper considering the lateral and torsional deformations of the shaft, gyroscopic effects of the rotor which consists of shaft and disk, and the centrifugal stiffening, spin softening and Coriolis force of the blades. In this model, the rotating flexible blades are represented by Timoshenko beams. The shaft and rigid disk are described by multiple lumped mass points (LMPs), and these points are connected by massless springs which have both lateral and torsional stiffness. LMPs are represented by the corresponding masses and mass moments of inertia in lateral and rotational directions, where each point has five degrees of freedom (dofs) excluding axial dof. Equations of motion of the rotor–blade system are derived using Hamilton׳s principle in conjunction with the assumed modes method to describe blade deformation. The proposed model is compared with both finite element (FE) model and real experiments. The proposed model is first validated by comparing the model natural frequencies and vibration responses with those obtained from an FE model. A further verification of the model is then performed by comparing the model natural frequencies at zero rotational speed with those obtained from experimental studies. The results shown a good agreement between the model predicted system characteristics and those obtained from the FE model and experimental tests. Moreover, the following interesting phenomena have been revealed from the new model based analysis: The torsional natural frequency of the system decreases with the increase of rotational speed, and the frequency veering phenomenon has been observed at high rotational speed; The complicated coupling modes, such as the blade–blade coupling mode (BB), the coupling mode between the rotor lateral vibration and blade bending (RBL), and the coupling mode between the rotor torsional vibration and blade bending (RBT), have also been observed when the number of blades increases.

Introduction

Blades play an important role in turbomachinery, and the vibration analysis of elastic rotating blade is becoming a growing concern. The effects of centrifugal stiffening, Coriolis force, stagger angle, pre-twisted angle and elastic root constraints on the natural frequencies of a rotating beam have been studied by many researchers [1], [2], [3], [4], [5], [6], [7], [8]. Based on the thin shell theory, Sinha and Turner [1] derived the governing partial differential equations of motion for the transverse deflection of a rotating pre-twisted plate, where the effect of warping of the cross-section is included in the strain–displacement relationships. Hashemi and Richard [3] proposed a finite element (FE) method for vibration analysis of rotating assemblages composed of beams. In their model, Coriolis forces have been taken into consideration. Using Hamilton׳s principle, Lin et al. [4] derived the governing differential equations for the coupled bending-bending vibration of a rotating pre-twisted beam with an elastically restrained root and a tip mass, subjected to the external transverse forces and rotating at a constant angular velocity. Lee and Sheu [8] developed an accurate power-series solution to free vibration of a rotating inclined Timoshenko beam, and analyzed the effects of the extensional deformation and the Coriolis force on the natural frequencies of the rotating beam.

With the development of the research, many researchers found that blades are coupled with the outer circumference of the disk, and the disk–blades are coupled with the shaft; the coupling of disk and blades (or shaft and disk–blade) produces various vibration modes [9], [10], [11], [12], [13], [14], [15], [16], [17]. Santos et al. [9] conducted theoretical and experimental studies on the problem of rotor–blades dynamic interaction using a simple test rig, which is built with a mass-spring system attached to four flexible rotating blades. In their model, only lateral displacement of the rotor in the horizontal direction is taken into account. The blades are modeled as Euler–Bernoulli beams neglecting gyroscopic effect. Anegawa et al. [10] provided a detailed discussion on blade–shaft coupled resonance and excessive vibrations experienced in the experiment of in-plane blade and out-of-plane blade systems. Yang and Huang [11], [12] investigated the coupling vibrations among shaft-torsion, disk-transverse and blade-bending in a shaft–disk–blades unit, and derived the equations of motion for the shaft–disk–blades unit by using a energy approach in conjunction with the assumed modes method, and analyzed the effects of disk flexibility, blade׳s stagger angle and rotational speed on the natural frequencies and mode shapes. Al-Bedoor [13] developed a dynamic model considering the effects of the coupled shaft-torsional and blade-bending using the Lagrangian approach in conjunction with the assumed modes method, and analyzed the coupled natural frequencies for various combinations of system parameters. Based on lumped mass method and the Lagrangian approach, Cao et al. [14] developed a dynamic model of a bladed overhang rotor system with squeeze film dampers. Considering the coupling between shaft torsion and blade bending in a multi-disk rotor system, Chiu and Chen [15] analyzed the natural frequencies and the mode shapes of the system for one- to three-disk cases. Okabe et al. [16] developed a blade–shaft coupled bending vibration model in which the shaft is represented in an inertial coordinate system as usual, and the blade system is defined in a rotating coordinate system, and the coupled equations of motion are represented by a mix type of rotating and inertial coordinate systems. Lee et al. [17] developed a computational model of a rotor–blade system by using the Lagrangian equation in conjunction with the assumed modes method. Considering the effects of the rotary inertia and gyroscopic moments, Sinha [18] derived equations of motion for a multiple-blade flexible rotor (shaft and disk) supported by a set of bearings at multiple locations.

Many models of rotor–blade system consider the coupling of the shaft torsion and blade bending [13], [15], [17]; however, little attention has been paid to the coupling of rotor lateral motion and blade bending [11], [12]. In this study, a new dynamic model of rotor–blade systems is developed by considering the coupling action of many factors, including the bending and torsion motions of the shaft, gyroscopic effects of the rotor, and the centrifugal stiffening, spin softening and Coriolis force of the blades. In the new model, the equations of motion of the system are described by using the Hamilton׳s principle in conjunction with assumed modes method to approximate the blade deformation. The model is validated using the results obtained from FE model analysis and experimental studies, where the natural frequencies and vibration responses determined by the proposed model are compared with those obtained from FE model. In addition, the new model is also verified by comparing the model simulated natural frequencies with those measured in experiments. A flow chart of the new dynamic modeling procedure for rotor–blade systems is shown in Fig. 1.

The paper is organized as follows. After this introduction, a dynamic model of a rotor–blade system is established using the Hamilton׳s principle in conjunction with assumed modes method in Section 2. Kinetic and potential energy expressions and equations of motion of the rotor–blade system are given in 2.1 Kinetic energy expressions of the rotor–blade system, 2.2 Potential energy expressions of the rotor–blade system, 2.3 Equations of motion for the rotor–blade system, respectively, and the natural frequencies from the complex eigenvalue solution is introduced in Section 2.4. In Section 3, the model verification using an FE model and experiment studies is conducted in 3.1 Model verification based on FE model analysis, 3.2 Model verification by experimental studies, respectively. Finally, the conclusions are drawn in Section 4.