Linearly coupled shaft-torsional and blade-bending vibrations in multi-stage rotor-blade systems (original) (raw)
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A geometrically exact mechanical model for the overall dynamics of elastic isotropic rotating blades is proposed. The mechanical formulation is based on the special Cosserat theory of rods which includes all geometric terms in the kinematics and in the balance laws without any restriction on the geometry of deformation besides the enforcement of the local rigidity of the blade cross sections. All apparent forces acting on the blade moving in a rotating frame are accounted for in exact form. The role of internal kinematic constraints such as the unshearability of the slender blades is discussed. The Taylor expansion of the governing equations obtained via an Updated Lagrangian formulation is then employed to obtain the linearized perturbed form about the prestressed configuration under the centrifugal forces. By applying the Galerkin approach to the linearized equations of motion, the linear eigenvalue problem is solved to yield the frequencies and mode shapes. In particular, the natural frequencies of unshearable blades including coupling between flapping, lagging, axial and torsional components are investigated. The angular speeds at W. Lacarbonara ( ) which internal resonances may arise due to specific ratios between the frequencies of different modes are determined thus shedding light onto the overall modal couplings in rotating beam structures depending on the angular speed regime. The companion paper (part 2) discusses the nonlinear modes of vibration away from internal resonances.
Study of the vibration behaviour of rotatory blades using the finite element method
2019
The physics lying behind rotordynamics is complex to model, so that in many cases numerical processing is the only feasible approach. Being rotordynamics a field of great interest in the aerospace industry, the efforts devoted to its understanding are increasing day by day. Following this tendency, the aim of the present study is to develop a simplified elastodynamic model for the case of rotating structures such that can be addressed through numerical tools, built using the finite element method. For the purpose of analysing the vibration phenomena, modal decomposition and numerical integration have been taken advantage from. In this context, it has been found that the singular value decomposition could be applied in structural analysis to extract dominant displacement fluctuations, allowing the unfolding of global properties of the dynamic response. In the present report, the singular value decomposition has been applied to cantilever beams undergoing a single rotation, giving ris...
GT2009-59471 ROTOR-BLADE COUPLED VIBRATION ANALYSIS BY MEASURING MODAL PARAMETERS OF ACTUAL ROTOR
The designers of rotor shafts and blades for a traditional turbinegenerator set typically employed their own models and process by neglecting the coupled torsional effect. The torsional coupled umbrella mode of recent longer blades systems designed for higher output and efficiency tends to have nearly doubled the frequency of electric disturbance (i.e., 100 or 120 Hz). In order to precisely estimate the rotor-blade coupled vibration of rotating shafts, the analysis must include a process to identify the parameters of a mathematical model by using a real model. In this paper we propose the use of a unique quasi-modal technique based on a concept similar to that of the modal synthesis method, but which represents a unique method to provide a visually reduced model. An equivalent mass-spring system is produced for uncoupled umbrella mode and modal parameters are measured in an actual turbine rotor system. These parameters are used to estimate the rotor-blade coupled torsional frequencies of a 700-MW turbine-generator set, with the accuracy of estimation being verified through field testing. NOMENCLATURE a coupling factor kij spring constant [K1], [K3] Stiffness matrix of shaft and blade k23 Coupling stiffness between shaft and blade [M1], [M3] Mass matrix of shaft and blade * m Modal mass of blade * c m Coupled modal mass of blade eq m , eq k Equivalent mass and equivalent spring constant p t I M m = = δ δ δ 3
A new dynamic model of rotor–blade systems
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.
The Influence of Shaft’s Bending on the Coupling Vibration of a Flexible Blade-Rotor System
Mathematical Problems in Engineering, 2017
The influence of shaft bending on the coupling vibration of rotor-blades system is nonignorable. Therefore, this paper analyzed the influence of shaft bending on the coupling vibration of rotor-blades system. The vibration mode function of shaft under elastic supporting condition was also derived to ensure accuracy of the model as well. The influence of the number of blades, the position of disk, and the support stiffness of shaft on critical speed of system was analyzed. The numerical results show that there were two categories of coupling mode shapes which belong to a set where the blade's first two modes predominate in the system: shaft-blade (SB) mode and interblade (BB) mode due to the coupling between blade and shaft. The BB mode was of repeated frequencies of (− 2) multiplicity for number blades, and the SB mode was of repeated frequencies of (2) multiplicity for number blades. What is more, with the increase of the number of blades, natural frequency of rotor was decreasing linearly, that of BB mode was constant, and that of SB mode was increasing linearly. Natural frequency of BB mode was not affected while that of rotor and SB mode was affected (changed symmetrically with the center of shaft) by the position of disk. In the end, vibration characteristics of coupling mode shapes were analyzed.
2014
Turbo-generator trains are susceptible to torsional vibration which can lead to fatigue cracking and failure. Methods are available for the measurement and calculation of the torsional natural frequencies of these systems for the purpose of design, monitoring and life prediction. Calculation methods are conventionally based on one dimensional (1D) finite element (FE) methodologies which require the simplification of a number of aspects including the participation of flexible blades in torsional vibration modes. The accuracy of 1D, three dimensional (3D) and three dimensional cyclic symmetric (3DCS) FE methods was investigated by the application thereof on a small test rotor. Experimental measurements of static and dynamic vibration responses were conducted with rotation and torsional forcing accomplished through the use of a DC motor and a digital control system optimised for fast transient and stable steady state response. Blade stagger angle was demonstrated to have a significant effect on torsional frequencies although no stress stiffening effects were noted in the speed range considered. Similarly, damping was measured to decrease with blade stagger angle but not with rotational speed. Step changes in torsional frequencies due to the activation of the motor field and armature currents required optimisation of the motor models for static and dynamic conditions. © © U Un ni iv ve er rs si it ty y o of f P Pr re et to or ri ia a accuracy is required, direct field measurements should be used for model calibration or model updating.
The Assumed Mode Method in Structural Dynamics of Bladed-Disk-Shaft Systems
Volume 5: Manufacturing Materials and Metallurgy; Ceramics; Structures and Dynamics; Controls, Diagnostics and Instrumentation; General, 1990
Analytical investigation into the effect of transverse bending of continuous flexible shafts is presented. While the blades are allowed both in-plane and out-ofplane deformations, the considered disk is rigid, and the shaft is allowed to bend in two planes. The assumed mode method is used to express flexible blade and shaft deformations, and the Lagrangian approach is used to derive the governing equations of motion for the considered structure. Stiffness and inertia properties of an actual experimental rotor, typical of a fan stage, are used in the analysis. Calculations are performed for three different disk-shaft configurations, and results are presented for different shaft stiffness and inertia parameters, as well as for a wide range of rotational speed. NOMENCLATURE a n (t),b n (t) Generalized coordinates used to describe shaft's flexible deformation. (EI) x ,(EI) y Bending rigidity of the shaft along the X and Y directions. I bd Mass moment of inertia for the bladed-disk. Kb Modal stiffness of the blade, associated with the fundamental blade bending mode. KM Ratio of the shaft's mass to the mass of the bladed-disk. KS Ratio of the shaft's length to the radius of blade tip. Mbd Mass of the bladed-disk. M O P) Modal mass of the shaft.
Stability Analysis of Rotating Blade Vibration due to Torsional Excitation
Journal of Vibration and Control, 2007
This paper presents an approximate analysis of the vibration stability of a rotating blade due to shaft torsional vibration excitation. The governing equation adopted in the study is a Hill's type linear second order ordinary differential equation with multiple harmonically variable coefficient terms. The strained parameters method, a perturbation technique, is utilized in developing the stability transition curves in the plane of parameters related to the rotor speed, the torsional vibration excitation frequency and the blade natural frequency. The stable and unstable regions obtained by perturbations are contrasted to those obtained by numerical stability analysis performed using Floquet theory and an excellent match is observed for small torsional vibration amplitudes. Numerical integration of the original equation at selected points in the predicted stable and unstable regions showed that the predicted behavior of the responses is correct, wherein the unstable regions growing blade vibration is exhibited.
Journal of Sound and Vibration, 1997
A new approximate method is presented for the analysis of the modal characteristics of straight, pretwisted non-uniform blades corresponding to the coupled flapwise bending, chordwise bending and torsion of both rotating and non-rotating blades. An integral approach is described based on the use of Green functions (structural influence functions), which are used to develop the equations of motion. A clamped-free blade is analyzed and comparisons are made with numerical results from the literature. Several examples regarding specific aspects of the flapwise bending, coupled bending-bending, coupled bending-torsion and coupled bending-bending-torsion vibration analysis are presented. The method presented gives good results and can be used for modelling of turbomachine blades, aircraft propellers or helicopter rotor blades which may be considered as straight non-uniform beams with built-in pretwist.
Study on the Coupled Vibration Characteristics of a Two-Stage Bladed Disk Rotor System
Applied Sciences, 2021
This paper conducts a coupled vibration analysis of a two-stage bladed disk rotor system. According to the finite element method, the bladed disk rotor system is established. The substructure modal synthesis super-element method (SMSM) with a fixed interface and free interface is presented to obtain the vibration behaviors of the rotor system. Then, the free vibration results are compared with the ones calculated by the cyclic symmetry analysis method to validate the analysis in this paper. The results show that the modes of the two-stage bladed disk not only include the modes of the first- and second-stage bladed disk, but also the coupled modes of the two-stage bladed disk.