华侨大学陈振川全球求助翻译英语论文_(10).pdf

69 The Fifth International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY Huazhong University of Science and Technology Wuhan P R CHINA 27 28 August 2009 2009 Huazhong Universiti of Science and Technology Press INFLUENCE OF CENTRIFUGAL STIFFENING ON A ROTOR SYSTEM WITH A FLEXIBLE DIAPHRAGM COUPLING S Ganesan C Padmanabhan Machine Design Section Department of Mechanical Engineering Indian Institute of Technology Madras Chennai India Email mouli iitm ac in ABSTRACT In this paper an investigation to estimate the influence of rotational speed on the flexible coupling rotor using a time invariant model has been carried out A rigid hollow shaft with a flexible diaphragm coupling as a rotor system is studied for its dynamic behavior by deriving a finite element rotor model To estimate stiffness parameters of flexible coupling to include in the finite element model a solid modeling of the coupling has been generated and a quasi static analysis has been carried out at different speeds using ABAQUS software The estimated stiffness in all six degrees of freedom has been lumped as a stiffness matrix for the coupling in the time invariant rotor model Eigen analysis performed on the discretized flexible coupling rotor model clearly indicates that the bending critical speeds of the model vary depending on angular velocity due to centrifugal stiffening and gyroscopic couple For speeds up to 20 000 rpm the increase in the first critical of the flexible rotor is 15 Critical speed obtained is verified with simplified analytical calculation Also a limited experiment has been done to validate the stiffness value input in the FEM model KEYWORDS fexible coupling rotor centrifugal stiffening I INTRODUCTION In general two mechanical systems get connected in a driveline by a coupling that tolerates a certain misalignment between them in static and dynamics conditions Misalignment is due to various operating conditions such as thermal expansion of mechanical parts sinking and relative movement of mounting beds There are many types of couplings used in practice such as universal joint coupling gear coupling flector type coupling and diaphragm type disk coupling For high speed application with large misalignment diaphragm type disk couplings are preferred since they are constant velocity coupling The dynamics of a coupling rotor system with such a coupling needs to be accurately predicted as the system may have to operate beyond its first critical speed Then the influence of angular velocity on the modal properties of rotor system has to be understood clearly The influence of rotational speed on the critical speeds of the flexible rotor is the main focus of the study To carry out this objective a time invariant model based on finite element FE method is developed A rigid hollow shaft with a flexible diaphragm coupling as a rotor system is studied for its dynamic behavior by deriving a rotor finite element model which is an extension of Nelson and McVaugh rotor model 1 A FEM rotor model with 2D beam element has been derived by Nelson and McVaugh 1 to find the influence of gyroscopic couple on a simply supported disk They have clearly shown effect of 70 gyroscopic couple in splitting the critical speeds They also showed the natural mode variation with speed due to the gyroscopic couple Modal controllability and observability of a bladed disk and their dependency on the angular velocity was studied by Christensen and Santos 2 Their model presented parametric vibration and blade natural frequency change depending on the angular velocity due to centrifugal stiffening Xi 3 has carried out modeling and analysis for active control of circular saw vibrations used in wooden saw mills In his study influence of the rotation speed on the natural frequencies was considered The dynamic response of a spinning CD disk was studied by Heo et al 4 when the axis of the disk is misaligned with the centre of rotation In their study effect of rotation speed on the in plane mode was reported From the literature review it is clear that effect of rotational speed on modal characteristics is studied on many mechanical systems like bladed disk circular saw and CD disk Unlike the systems investigated the coupling rotor system investigated here involves complexities due to the fact that are assembled to form one piece that takes four thin disks to misalignment To account for this an equivalent stiffness model of the coupling is derived using a quasi static analysis at different speeds A rigid hollow shaft with a flexible coupling as a rotor system is shown in Fig 1 The rigid shaft connected by a flexible coupling forms a rotor system capable of handling large misalignments Since the coupling is made out of four thin circular plates it can deflect to provide flexibility Fig 1 Flexible coupling rotor schematic II TIME INVARIANT ROTOR MODEL WITH CENTRIFUGAL STIFFENING 2 1 Finite Shaft Element of Rotor The entire rotor coupling system is modeled using FEM In this section a brief description of the model developed for the coupling rotor based on Nelson and McVaugh 1 is presented To include axial and torsional modes the finite element model has been derived including transverse bending axial and torsion effects as an extension of the Nelson and McVaugh 1 rotor model Axial and torsional terms are introduced into the mass and stiffness matrices at the appropriate places In deriving the bending element the shaft is assumed to be elastic and flexible and the disk rigid The nodal variables are u v which are bending displacements in the x and y directions while x and y are the bending rotations The mass matrices are given by d T T Ms I 2 1 M T PR ds 1 q v u q y x Where is mass of the shaft element per unit length Ip is polar moment of inertia and q is nodal degrees of freedom The gyroscopic matrices are obtained as T s nn 2 1 g ba nnn 2 Z Y X 1 2 3 4 5 6 7 71 d T aPxy nIs d T bPyx n Is 3 Where is rotational speed d d x x v s d d y y u s 4 The stiffness matrix is given by 1 0 d T b KEIs 5 Where E is Young s modulus and I is area moment of inertia The axial stiffness Ka and torsional stiffness KT terms are obtained by 1 0 d T a KEAN N s 1 0 d T T KCIpN N s 6 Where A is the cross sectional area C is modulus of rigidity and N is the standard axial torsional shape function matrix The equations of motion representing all DOF are 0000000 000000 0000000 s ds ds x s ds ds y aa z mxgxkxf mygykyf mzzkzf 7 where ms d represents sum of mass matrices of shaft and disk in transverse bending gs d represents sum of gyroscopic matrices of shaft and disk in transverse bending ks represents shaft stiffness in bending direction in x and y ma represents mass matrices in axial and torsional direction and ka represents stiffness of shaft in axial and torsional direction In the above equation the vectors x y and z represent T 1122 iiiii yy xuu T 1122 iiiii xx yvv T 1122 iiiii zz zww Equation of motion with all DOFs given by FQKQGQM 8 Where Q represent the total system DOFs in the model 2 2 Characterisation of Flexible Coupling Fig 2 Flexible coupling Fig 3 Discretised coupling model This flexible diaphragm coupling shown can handle large axial misalignment and lateral misalignments In the axial direction z flexible rotor system can expand or compress by 2 5mm In the Z Y XRigid links 72 lateral bending directions it can take angular misalignment of 1 50 or a parallel offset up to 1 5mm When the rotor is subjected to axial misalignment either expansion or compression circular plates in the flexible coupling are subjected to symmetric bending But during angular misalignment parallel offset circular plates are subjected to asymmetric bending The flexible coupling stiffness characteristics are required in the system to understand the behavior of the rotor system under the influence of speed The flexible coupling shown in Fig 2 is made out of 4 thin circular plates which are welded to form an integral piece The thin circular plate is supported at the inner and outer surface by the hub and rim which are comparatively thicker The thickness of the circular plate varies from 1 mm at the hub to 0 5 mm at the rim To obtain the stiffness characteristics of the flexible coupling a finite element model is generated in the commercial software ABAQUS 5 The model representing the diaphragm coupling is meshed using quadrilateral shell elements with appropriate aspect ratio as shown in Fig 3 Since a shell element is used to model the geometry a single layer is used to represent the thickness of the thin circular plates A four noded shell element with six degrees of freedom DOF u v w x y z at each node can represent the entire thickness as one layer There are 6266 elements and 6600 of nodes representing the geometry The total number of DOF in the model is 39564 Mesh refinement has been done to ensure the convergence of the results The h type mesh optimization starting from the finer mesh to the present level has been carried out without affecting the convergence to represent the geometry Non linear analysis has been performed to estimate the stiffness of the coupling under the influence of rotational speed At both ends rigid links are used to connect the nodes at the hub to the center of the diaphragm as shown in Fig 3 These two centre points of the model have been used to impose the boundary conditions Displacement of one of the centre nodes with respect to the other centre node is used The stiffnesses in all six directions are obtained by displacing the coupling at one end by a unit value and fixing the other end to find the reaction forces under the influence of rotational speed From the initial position the unit displacement in all directions one at a time is given to get a set of reaction forces for a speed Then speed is varied and the above steps were repeated to find the reaction forces and stiffnesses are obtained and tabulated in Table 1 The stiffness obtained is then used in the rotor finite element model as a part of a stiffness matrix Table 1 Stiffness variation due to centrifugal stiffening Stiffness in X axis Stiffness in Y axis Stiffness in Z axis Rotational Stiffness about X axis Rotational Stiffness about Y axis Rotational Stiffness about Z axis Speed r m Nm 1 Nm 1 Nm 1 Nm rad 1 Nm rad 1 Nm rad 1 0 4 85E 06 4 85E 06112800 676 66 676 09 6 07E 04 2000 4 85E 06 4 85E 06112800 676 66 676 09 6 07E 04 4000 4 89E 06 4 90E 06113000 684 11 683 59 6 07E 04 6000 4 96E 06 4 96E 06113400 696 14 695 6 07E 04 8000 5 05E 06 5 06E 06113900 713 33 711 04 6 07E 04 10000 5 16E 06 5 18E 06114600 734 53 731 09 6 07E 04 20000 6 06E 06 6 11E 06119200 901 26 886 93 6 07E 04 30000 7 36E 06 7 46E 06124100 1147 63 1116 7 6 07E 04 40000 8 85E 06 9 02E 06127600 1455 31 1402 02 6 07E 04 50000 1 03E 07 1 06E 07129100 1814 56 1734 91 6 07E 04 100000 1 45E 07 1 55E 07127500 4269 11 3976 9 6 07E 04 2 3 Model with Centrifugal Stiffening The stiffnesses of coupling element in all DOFs with different rotational speed are obtained by quasi static analysis The stiffness values obtained are then used to represent the coupling element in FEM model of the flexible rotor system in a stiffness matrix 12 12 This matrix is generated at each running speed Stiffness variation with different rotational speed is represented in the equation of motion as mean co efficient stiffness now variation with speed due to centrifugal stiffening 73 MQGQKQF 9 The coupling rotor shown in Fig 1 is discretized with two noded element of general rotor beam element having six degrees of freedom to represent the entire rotor There are six element with 42 DOFs The discretization of the flexible coupling rotor is shown in Fig 1 Mass of the coupling is represented as a single disk with equivalent thickness instead of four circular plates Eigen analysis was performed and the critical speeds in all the modes of vibrations are found The critical speeds of rotor in bending are plotted in a graph as frequency versus speed which is shown in Fig 4 From the graph plotted it is observed that the bending critical speeds are varying with speed due to gyroscopic couple and centrifugal stiffening When compared to the gyroscopic effect the centrifugal stiffening is predominant Lower modes of bending participate in the centrifugal stiffening effect in a greater way when compared with higher modes of bending This is because the lower modes of bending are influenced by the coupling element while higher bending modes correspond to the rigid hollow shaft In the speed range up to 20 000 rpm the increase in first critical of the flexible rotor is 15 But the axial and torsional resonances are not affected due to the centrifugal stiffening effect of the coupling The gyroscopic effect is less predominant since the mass of the flexible coupling is small Fig 4 Bending critical speed variation with speed 2 4 Verification of Centrifugal Stiffening Model To verify the critical speed found by the centrifugal stiffening rotor model estimation has been carried out using stiffness and mass parameters of flexible coupling The flexible coupling has four thin plates which act like a disk and are supported by two adaptor flanges on either side A rigid tube is connected to flexible coupling by the adaptor on one side to form flexible coupling rotor The mass of the disk is 0 2 kg and accounting for one half of the rigid tube and adaptor to contribute to disk mass the total mass of the disk is 1 12 kg The flexible coupling contributes to the stiffness in the lower modes The stiffness of the flexible coupling corresponding to zero rpm is 4 853e6 Nm 1 as seen from Table 1 To check the modal participation the mode shape of the first mode has been plotted for translational DOFs u and v in Fig 5 From the plot the modal participation factor is seen to be 0 9 Then critical speed of the rotor is simply calculated using a SDOF and its value is 2194 2 rad sec The centrifugal stiffening model of FEM has predicted the critical speed for zero rpm as 2194 6 rad sec These two values agree well indicating consistency of the ABAQUS model The stiffness of the coupling obtained from ABAQUS is validated in a limited way An experiment has been conducted to find the axial stiffness Z direction The experimental setup shown in Fig 6 has 74 one fixed mounting bracket and one moving bracket The assembly of the diaphragm coupling is mounted between these brackets and a unit displacement is given to find the resistance The load sensor mounted in line with displacement measures the load due to the coupling axial misalignment The stiffness of the coupling is obtained and compared with the results obtained from the quasi static analysis model in ABAQUS which is specified in Table 1 against stiffness in Z direction for static condition zero rpm While the finite element model predicts the axial stiffness to be 112 8 Nmm 1 the experimental value is about 15 higher at 130 2 Nmm 1 The thin plates forming the coupling are very sensitive to small changes in thickness Due to machining inaccuracies the plates in the experiment are slightly thicker leading to an increase in stiffness Fig 5 Mode shape of displacement u and v Ist mode Fig 6 Static Thrust Test Rig III CONCLUSIONS An investigation made on the flexible coupling rotor to study the influence of rotational speed using a time invariant finite element model revealed that critical speed of the