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ELSEVIER Finite Elements in Analysis and Design 26 1997 41 55 FINITE ELEMENTS IN ANALYSIS AND DESIGN On the dynamic analysis of rotors using modal reduction Y A Khulief M A Mohiuddin Department of Me chanical Engineering King Fahd University of Petroleum and Minerals KFUPM Box 1767 Dhahran 31261 Saudi Arabia Abstract A finite element dyr amic model for a rotor bearing system is developed The model accounts for the gyroscopic moments and anisotropic bearings For dynamic response analysis a reduced order model using modal truncation is obtained Two modal truncation schemes are invoked one with planar undamped modes and the other with complex damped modes Both the modal characteristics and the dynamic responses of two rotor systems are evaluated using the two reduction schemes Simulation results are presented to demonstrate the validity and accuracy of the reduced order form The question of whether to use planar or complex modes in establishing the modal transformation is addressed 1 Introduction Rotating machines are extensively used in diverse engineering applications such as power stations marine propulsion systems aircraft engines machine tools automobiles and household appliances The dynamic modeling of rotors is essential to the dynamic analysis and control of vibrations in such systems Early dynamic models of rotor systems were formulated either analytically 1 or using the transfer matrix approach 2 However the potential of the powerful finite element technique for modeling rotor systems has been recognized at a very early stage see e g 3 Nelson and McVaugh 4 1 and Zorzi and Nelson 5 1 improved upon the model presented in 3 1 by including secondary effects such as rotary inertia internal damping and axial torque Several finite element formulations using a uniform shaft element 6 7 1 or tapered shaft element 8 11 were introduced to evaluate the modal characteristics of rotor bearing systems For dynamic response analysis the transient response of a flexible rotor subjected to a sudden imbalance was evaluated using Galerkin s method 12 A finite element model was employed by Ozguven and Ozkan 13 to study the unbalance response of a rotor bearing system Firoozian and Stanway 14 1 derived a linearized finite element model in order to establish a control scheme Corresponding author 0168 874X 97 17 00 1997 Elsevier Science B V All rights reserved PII S0 168 8 74X 96 00070 4 42 Y A Khulief M A Mohiuddin Finite Elements in Analysis and Design 26 1997 41 55 for rotor systems The finite element method was also applied to a complex rotor system to evaluate its vibration response due to fluid forces 15 and gyroscopic moments 16 Rouch et al 17 utilized general purpose finite element codes to simulate the dynamics of some complex rotor systems All of the previously cited finite element analysis studies were based on the utilization of the nodal or physical coordinates The use of nodal coordinates however results in a large dimen sionality thus inhibiting the efficiency of the finite element solution Moreover the use of nodal coordinates results in a dynamic model of widely spread eigenspectrum that includes many insignificant modes Consequently a numerically stiff system is often created which causes the numerical integration scheme to search inefficiently for a solution or may even fail to find one In order to alleviate this problem reduced order models using modal coordinates were introduc ed Likins 18 introduced modal reduction using complex modes in his early formulation of the elastic appendage equations Gunter et al 19 utilized modal transformations to obtain a reduced order modal form of the equations of motion Although they recognized that the actual vibration modes of a rotor system are complex they have used planar modes in their evaluation of the unbalance response of the rotor They have also referred to the numerical difficulties associated with the use of complex modes Nevertheless they recommended that complex modes damped modes be considered in the final analysis Laurenson 20 addressed the issue of complex mode shapes in rotating flexible structures It was suggested however that planar modes be used in modal reduction of complex geometric configurations by employing the technique presented in 21 for converting the complex eigenvalue problem to one defined by real matrices Stephenson and Rouch 22 invoked modal reduction using planar modes wherein the mass matrix was modified to include the gyroscopic effects Modal transformations using planar modes were also utilized in evaluating the unbalance response 23 24 stability 25 and the gyroscopic effect in rotor systems 26 In this regard one can also refer to the general area of flexible multibody applications 27 30 where planar modes were employed to obtain reduced order models inspite of the existence of damping and gyroscopic forces Kane and Torby 31 referred to the different methods for reducing the size of the finite element model while preserving the lower significant frequencies It was stated that the static reduction usually results in poor accuracy at higher modes and therefore cannot be applied to general rotor systems because it is derived for systems having symmetric mass and stiffness matrices Therefore they introduced a modal transformation based on complex modes that resulted in reduced mass and stiffness matrices and demonstrated how the reduced model preserved the same modal characteristics of the original finite element model Their work however was not carried out to the dynamic response analysis stage Having examined the previously cited investigations one recognizes a strong view in support of using complex modes in modal transformations of systems with gyroscopic matrices though acknowledging the associated numerical complexities To avoid such numerical difficulties an other view suggested the use of planar modes obtained after modifying the mass matrix to include account for the gyroscopic effects Nevertheless other investigators especially those concerned with dynamic response analysis have consistently employed planar modal transformations It is noteworthy to mention that no dynamic response analysis study that invoked complex modal transformations was reported in the available literature In this paper a finite element elastodynamic model of a rotor bearing system is presented The model accounts for the gyroscopic effects as well as the coupling between flexural and torsional Y A Khulief M A Mohiuddin Finite Elements in Analysis and Design 26 1997 41 55 43 deformations In addition the formulation permits the use of either a uniform or tapered shaft element and a general anisotopic bearing Two modal transformations were established using planar modes and complex modes respectively In either case a reduced order model is obtained the modal characte dstics are evaluated and the transient dynamic responses are simulated for two rotor bearing systems Numerical results are presented to demonstrate the validity of such modal reductions 2 The elastodynamic model Two reference frames are employed to describe the rotor system One is the fixed reference X YZ and the other is the rotating reference xyz The angular displacement between the two reference frames is O t and the rotational speed of the shaft is defined by O t Referring to Fig 1 let X Y iZi be a Cartesian coordinate system with its origin fixed to the undeformed beam element The xiylz is the Cartesian coordinate system after the deformation of the beam and is rotated through a set of rotations b fl and 7 with respect to the xiy Z coordinate system as shown in Fig 2 Now the i I J I 1 k i P t Jl x Fig 1 Generalized coordinates of element i i V 44 Y A Khulief M A Mohiuddin Finite Elements in Analysis and Design 26 1997 41 55 i i Ya Y2 yi yi e i x 3 X i i x 2 P i i i Z Z 3 Fig 2 Cross section rotation angles instantaneous angular velocity of vector c5 of the xiyiz i is given by 6 6 f 2 1 where I j2 and are unit vectors along the X y and z axes One can express the global position fp as f R o a 2 where a represents the deformation vector of point pi The time derivative of fp can be expressed in the form dfp a c x p 3 Y A Khulief M A M ohiuddin Finite Elements in Analysis and Design 26 1997 41 55 45 Now the kinetic energy expressions can be written as Substituting Eq 3 into Eq 4 and putting if Nv e we get l v r r Nv T Nv e T Nv T 03 fV N e e dV 5 where is the mass density of the ith beam element and o3 is a skew symmetric matrix 11 Here the superscript i has been dropped out for notational simplicity In Eq 5 the first term represents the translational kinetic energy the last term represents the kinetic energy due to rotational effects that include gyroscopic moments while the second and third terms are identically zero if moments of inertia are calculated with respect to the elemental center of mass The kinetic energy in Eq 5 can be expressed as T T M 12C0 2 0 e T Go e 6 where M Mt Mr I M 2 Me is the composite mass matrix The constituent matrices and other invariants of Eq 6 are given by M fl N t A No dx 7 m fl N ID N dx 8 M fl N TIp N dx 9 rt M Jo Ip I N T Nar e Na N T Na e Nat dx I0 Go fi Nor XIe Naa dx Co flledx II where A x is the cross sectional area l is the length ID is the diametral mass moment of inertia and Ie is the polar moment of inertia of the shaft element Here IN Np represents the rotational shape function and N is the torsional shape function Explicit expressions of such shape functions are given in Ref 11 46 Y A Khulief M A Mohiuddin Finite Elements in Analysis and Design 26 1997 41 55 2 1 Strain energy expression The strain energy of the finite shaft element can be written in the form 2 1 1 dx 12 where v and w are flexural deformations in Y and Z directions while b fl and 7 represent small rotations about the X J2 and k axes respectively The other parameters represent modulus of elasticity E the shear modulus G the second moment of the cross sectional area I the polar moment of inertia J and the shear correction factor x Now Eq 12 can be written in matrix form as U I e T K e 13 The matrix K Ko Ks K is the element composite stiffness matrix where Ke represents the elastic stiffness Ks accounts for the shear stiffness and K represents the torsional stiffness These can be expressed as K fi a N T ax GJ a N dx ax 14 ax N N Ke fl GA a Nvw N w dx L ax N l N 15 Here the different shape functions are defined by the following relationships Nvv b N e 18 Using the Lagrangian approach the equation of motion can be derived as M 0 G O K e Q 19 where M is the composite element mass matrix G Go Go T is the element gyroscopic matrix K1 is the composite element stiffness matrix and Q is a vector of generalized forces Y A KJ ulief M A Mohiuddin Finite Elements in Analysis and Design 26 1997 41 55 47 A generalized rotor system comprises shaft elements disks and bearings If one considers a rigid disk attached to the shaft the equation of motion of such a rigid disk may be written as O G O 20 where M d is the disk mass matrix G d is the gyroscopic matrix and Qd is the vector of generalized forces of the disk In addition the generalized forces acting on the bearings can be written as ECb K e Qb 21 where C b is the bearing damping matrix K b is the bearing stiffness matrix and Qb is the vector of generalized forces of the bearings Now the equation of motion of a rotor bearing system can be written in the assembled general form as C I O 22 where t is the associated mass matrix of the system taking into account the disk mass matrix at the corresponding locations Similarly C is the assembled gyroscopic matrix is the assembled stiffness matrix and is the assembled generalized force vector The vector comprises nodal coordinates of the whole system 3 Modal reduction Two modal reduction schemes are established The first scheme utilizes planar modes obtained by solving the self adjoint eigenvalue while the second scheme invokes the complex modes of the non self adjoint eigenvalue In each case a reduced order modal form of the equations of motion is obtained 3 1 Planar modal transformation In order to obtain the real eigenvalues and the associated planar modes one must ignore the damping matrix in Eq 22 To this end the associated homogenous adjoint equation can be written as Eg o o 23 Upon solving the self adjoint eigenvalue problem associated with Eq 23 one obtains a set of real eigenvalues and eigenvectors Let H denote the modal matrix that comprises a selected subset of the resulting real eigenvectors planar modes Now a modal transformation can be defined as H v 24 48 Y A Khulief M A Mohiuddin Finite Elements in Analysis and Design 26 1997 41 55 where v is the vector of modal coordinates If only a truncated set of significant modes are retained the corresponding truncated form of Eq 22 can be written as H T r H i H T 12 H HIT K H v H T T 25 or simply as r 1 Cr 1 r 11 H 26 where Mr S r and K r are the reduced modal mass gyroscopic and stiffness matrices respectively Eq 26 represents the reduced order model using planar mode truncation 3 2 Complex model transformation The elastodynamic model of Eq 22 can be represented in the state space form as o o 27 or simply as A 4 BJ q f 28 where q T g T T One can write the following two homogeneous adjoint equations A 4 B q 0 29 A T BIT q 0 30 Let R and L denote the complex modal matrices of the differential operators of Eqs 29 and 30 respectively 32 Introducing the transformation q JR u 31 where u is the vector of modal coordinates If only a set of significant modes are to be retained the truncated modal form of the equations of motion can be written as L T A R fi L T B R u L T f 32 or simply as ar i 4 Br U LIT f 33 where R and L contain only those complex eigenvectors that represent a set of selected modes Now Eqs 26 and 33 represent truncated models using planar modal transformation and complex modal transformation respectively In general a subset of eigenvectors which spans the frequency spectrum of the forcing function are retained as significant models 4 Numerical results and discussions In this dynamic simulation two rotor systems are considered The first is a simple shaft rotor while the second is a stepped complex rotor system with disks and overhung In general planar Y A Khulief M A Mohiuddin Finite Elements in Analysis and Design 26 1997 41 55 49 modes are generated by solving the eigenvalue problem of Eq 23 while the actual complex modes are produced by Eq 27 The complex modal characteristics based on reduced order matrices are obtained by solving either Eq 26 or Eq 33 4 1 A simple shaft rotor system As a first examp ie a uniform steel shaft rotating at 400 rad s and supported at the two ends by flexible bearings is considered Fig 3 The stiffness coefficients of the bearings are Kyy Kzz 1 7513 x 107 N m Kyz Kzy 2 917 x 106 N m and the damping coefficients are Cyy C 1 752 103 N s m and Cyz Czy 0 0 The shaft is of diameter d 10 16 cm and length l 127 cm The density and elastic modulus of the shaft material are p 7833 kg m 3 and E 0 2608 x 1012 N m 2 respectively The shaft is divided into six equal finite elements In this case planar modes are obtained by solving the eigenvalue problem associated with Eq 23 and actual modes by utilizing the full dimension 50 x 50 matrices of Eq 27 The 25 x 25 full dimensional system was reduced using planar modal transformation to one of dimension 6 x 6 The modal characteristics of the reduced model are compared to the first six modes of the actual system and the first six planar modes as shown in Table 1 The comparison shows that the reduced matrices using planar transformations produce modal characteristics almost within the same level of accuracy of the actual full dimension matrices Here the error in the sixth mode is in the order of 0 02 which is in the same order of magnitude of the computational error It must be noted however that the reduced matrices of Eq 33 produce the exact values as those produced by the full dimensional system Both the step and impulse responses produced by solving either Eq 26 or Eq 27 are presented in Figs 4 and 5 respectively The figures show almost identical time responses of the reduced form Eqs 26 and 27 or Eq 33 4 2 A complex rotor system As a second example the rotor bearing system studied by Nelson and McVaugh 4 1 is used to illustrate the merits of the present mathematical formulation for the determination of natural frequencies and time responses The configuration of the rotor system is shown in Fig 6 and the corresponding data are given in Ref 331 The rotor bearing system is rotating at a constant speed ZCb K b Fig 3 Rotating shaft supported by flexible bearings 50 Y A Khulief M A Mohiuddin Finite Elements in Analysis and Design 26 1997 41 55 Table 1 Natural frequencies rad s of the simple rotor No Planar Actual Reduced by planar modes 1 490 8132 9 3819 490 9265 9 382 490 9164 2 543 3553 7 0884 543 4634 7 0885 543 4548 3 1005 137 56 606 1004 006 56 554 1003 529 4 1173 259 53498 1172 594 53 453 1172 059 5 2166 952 93 947 2164 179 93 893 2163 715 6 2307 3862 94 695 2305 706 94 653 2305 198 x 10 r Step Response Due to Unit Force Original System t 09 1 t 6 Q g 1 t o 6 3 Reduced System 011 012 013 014 0 5 0 6 Time Fig 4 of I2 2000 rad s Here the 95 x 95 was reduced using planar transformations to one of dimen sions 10 x 10 The actual complex modes are obtained by solving the 190 x 190 full dimension state space form of Eq 27 Table 2 presents similar comparisons where reduced matrices using planar modal transformations resulted in modal characteristics very close to the actual ones In this case the error in the tenth frequency is about 1 6 The step and impulse responses of the reduced model of Eq 26 is almost identical to the one produced by Eq 33 or Eq 27 as shown in Figs 7 and 8 Y A K lzulief M A Mohiuddin Finite Elements in Analysis and Design 26 1997 41 55 51 x 10 5 Impulse Response 4 2 r 51 C 0 0 I1 e e o 6 2 a Original System Reduc