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<ul><li><p>CONSISTENT MATRICES IN ROTOR DYNAMICS Giancarlo Genta* </p><p>MECCANICA </p><p>20 (1985), 235~ </p><p>SOMMARIO. Nel presente lavoro vengono ricavate le espres- </p><p>sioni della matrice congruente delle masse e giroscopica per un elemento di albero a s ezione costante. Tali espressioni tengono conto sia della deformazione a taglio che dell'iner- zia trasversale. I risultati, in termini di velocitd, critiche e di diagramma di Campbell, per un albero a sezione costante vengono confrontati con soluzioni in forma chiusa di varia complessitd. Vengono discussi gli effetti della deformazione a taglio, dell'inerzia trasversale, dello smorzamento e dello </p><p>sforzo assiale sul comportamento dinamico. Viene inoltre presa in considerazione una distribuzione lineare nota di squilibrio. II modello cosi ottenuto pub venire usato anche per lo studio di sistemi smorzatL purchb lo smorzamento possa essere considerato di tipo o . Viene infine mostrata un'applicazione ad un modello in cui si tiene conto anche dello smorzamento e chefa uso </p><p>della condensazione delle matrici per ridurre l'ordine del problema dinamico. </p><p>SUMMAR Y. The expressions for the consistent mass and </p><p>gyroscopic matrices for a constant section shaft element </p><p>are obtained taking into account both sl, ear deformation and transversal inertia. The results are compared with closed </p><p>form solutions, which are available in simple cases. The </p><p>results obtained show that the study o f the dynamic behaviour of the rotor with a model which includes rotational inertia but not shear deformation is, at least in the case examined, misleading. Formulae for matrix condensation and for taking into account the effects o f axial load and of a linear unbalance distribution are given. Damped systems can be studied using the same model, provided that damping can be assumed to be o f either viscous or hysteretic type. Some formulations found in the literature are however not considered correct. An application o f consistent matrices to a model which includes damping and uses matrix con- densation is shown. </p><p>SYMBOLS </p><p>{f} vector of the forces due to the unbalance of the </p><p>element [g] gyroscopic matrix of the element </p><p>i vc-i" [k] stiffness matrix of the element l shaft length </p><p>l e length of the element </p><p>* Associate Professor, Dipartimento di Meccanica, Politecnieo di Torino. </p><p>[m ] mass matrix of the element n number of elements r/ inner radius </p><p>r 0 outer radius {q} generalized displacements at the nodes x y z system of reference t time A area of the cross section [C] viscous damping matrix E Young's modulus { F} force vector due to the unbalance </p><p>F a axial force </p><p>G shear modulus [G] gyroscopic matrix I area moment of inertia of the cross section [!] identity matrix [K'] real part of the stiffness matrix [K"] imaginary part of the stiffness matrix [M] mass matrix T kinetic energy </p><p>U potential energy </p><p>a slenderness (a = I X /~ </p><p>"r shear strain </p><p>e unbalance; strain ~" non dimensional coordinate (~" = z/l e) </p><p>~7 loss factor v a unbalance k whirl speed v Poisson's ratio ~r/~" rotating system of reference (fixed to the shaft </p><p>element) p density ~0 x, ~oy rotations about x and y axes X shear factor w spin speed </p><p>SUBSCRIPTS </p><p>n non rotating element r rotating element [ inertial section characteristics, imaginary part </p><p>R real part S structural section characteristics. </p><p>I. INTRODUCTION </p><p>The dynamic study of rotors is based on principles which </p><p>are well extablished from long time. The increasing complexity of the problems encountered </p><p>in many fields of the technology and the need of a more accurate prediction of the dynamic behaviour of rotating </p><p>20 (1985) 235 </p></li><li><p>machines, led to formulations based on the finite element method (e.g. [1], [2], [3], [4]), which are now replacing those based on transfer matrices. </p><p>It is well known that with appropriate formulations it is possible to overcome the limitations of both the conventional distributed parameter closed form solutions, which can be used only for very simple geometrical configurations, and the lumped parametel:s approach. </p><p>When considering rotor elements however the often used for structural dynamics can be not suited in order to obtain the required accuracy. </p><p>The aims of the present paper are those of stating a complete formulation of the problem and of comparing the results so obtained with those obtained using the distributed parameter closed form solution in a simple ease in which the latter is available. </p><p>2. CLOSED FORM SOLUTIONS - EFFECT OF ROTATIO- NAL INERTIA AND OF SHEAR DEFORMATIONS </p><p>2.1. First approximation solution </p><p>The most common formulation of the problem of finding the critical speeds of a distributed parameter rotor is based on the assumption that both rotational inertia and shear </p><p>deformations are negligible. Even with these assumptions, together with the others </p><p>usually accepted in rotor dynamics such as material linearity and axial symmetry, the problem yields simple solutions only for shafts with constant cross section and statically determined constraints. </p><p>The value of the i-th critical speed as can be found e.g. in [5], is: </p><p>l 1/< ('Ocri = 32 7 [ pA 1 (I) </p><p>where the suffixes I and S have been introduced in order to take into account that only a part of the cross section of the shaft can contribute to its stiffness. </p><p>The constants 3 i depend upon the constraints. In the case of a shaft which is simply supported at both ends their values are: </p><p>3i = i= (2) </p><p>As a consequence of having neglected the rotatory inertia of the cross section the critical speeds coincide with the natural frequencies of the non-rotating shaft. The Campbell diagram, i.e. the plot of the whirl speeds against the spin speed, consists of an infinity of straight lines parallel to </p><p>the spin axis. </p><p>2.2. Second approximation solution </p><p>A ~second approximation>> solution can be obtained by taking into account the rotational inertia of the shaft [6]. </p><p>The critical speeds can still be obtained from eq. (1) but with different values of the constants 3 i. </p><p>In the case of a simply supported uniform shaft they are: </p><p>3 i = i~ 1 - =2i2/Ct])-1/4 (3) </p><p>where the s t is defined as: </p><p>a / = ~ (4) </p><p>From eq. (3) the following considerations can be drown: i) The critical speeds computed by taking into account </p><p>the rotational inertia of the shaft are higher than the ones obtained from eq. (2). This effect is similar to that due to the presence of >. </p><p>ii) Only a limited number of critical speeds exists. The following relationship: </p><p>i </p></li><li><p>* is a function of three The value of the critical speed ~ocr </p><p>parameters only, namely the two values of the slenderness </p><p>~x and as and the ratio Q/r o between the inner and outer radii of the cross section. </p><p>The function coc,(a) , or better, *0X ~/~c*r/c*r/Tr(a) is plotted in fig. 1 for a solid shaft in which the two values of the </p><p>slenderness a! and a s coincide. The solutlon of eq. (10) is compared with the ones </p><p>obtained using the first and second approximation. The same function, only for the third approximation Solution, is reported in fig. 2 in which the solid shaft is compared with a limiting case of very thin tube (rt/r o -~ 1). </p><p>From figures 1 and 2 the following conclusions can </p><p>10 </p><p>0 </p><p>I . l - J - i I 'l ~, , , . . . . . . . 9 . . . . . . . . . . . . . . . . . . . . </p><p>; ; i ; , . . . . . . . . . . </p><p>lt ~v Z ~ 1.~ .. </p><p>a r ; I I f | O 2ql 40 60 </p><p>1 t Q~ al fiat IO0 </p><p>Fig. 1. Non dimensional critical speeds for a solid uniform shaft simply supported at the ends as a function of the slenderness ~ = a s = a I . . . . . . . . . . . . . . First approximation (eq. (1)) . . . . . . . Second approximation (eq. (3)) - - Third approximation (eq. (10)). </p><p>~.' i ' Jr. = 9 </p><p>it . . _ . . - ..__,,.___,.....__.._ </p><p>- - - - - - r, I f o - - I - - '7 . . . . . . </p><p>O ~ 0 20 40 66 a= %=a I IOO </p><p>Fig. 2. Same graph as fig. 1 in which a solid shaft (ri/ro= O) is compa- red with a thin wall tubular shaft (ri/r 0 -* 1). Only third approximation shown. </p><p>be drawn: </p><p>i) The effect of shear deformation is that of lowering the value of the critical speeds. At least in the case of </p><p>simply supported constant section shafts the effect due to shear deformations is stronger than the one due to rotational </p><p>inertia and consequently the values obtained from the third approximation are lower than the ones from the first </p><p>approximation. ii) An infinity of critical speeds exists in any case. iii) The effects of shear deformation and of rotational </p><p>inertia vanish with increasing values of a, particularly in the case of low order critical speeds. </p><p>From i) it follows that the error linked with the second approximation is higher than the one linked with the first approximation, at least in the case studied. </p><p>This amounts to the statement that if the shear deforma- tion is neglected it is of no use, and can indeed be misleading, </p><p>to take into account the rotational inertia of the .cross section of the shaft. This goes against the practice, suggested e.g. in [ 1 ], of including in an F.E.M. model the consistent mass and gyroscopic matrices without taking into account shear deformation. </p><p>The equation needed for plotting the Campbell diagram can be easily solved in the spin speed 6o*. In non dimensional form it yields: </p><p>= {x 'X* ' - + + 4x* ) ] + (13) </p><p>+ - 2i 4) </p><p>The Campbell diagram for a solid shaft with a t = as=C(= = 15 is shown in fig. 3. From the figure it is "clear that the differences between the second and the third approximation, which are small for what the first critical speed is concerned, are increasingly important for higher order critical speeds and for high speed supercritical whirl motions. " </p><p>3. CONSISTENT MASS AND GYROSCOPIC MATRICES FOR CONSTANT SECTION AXlSYMMETRICAL SHAFT </p><p>ELEMENT </p><p>The shaft element can be considered as a element. The element here considered is what in [8] is defined as a one as only four degrees of freedom are considered in each plane; namely the displace- ments and rotations at the two ends. </p><p>Following the sketch in fig. 4 and the detailed definition </p><p>of the systems of reference in [3], each node has 4 degree~ of freedom, the displacements x and y and the rotation, </p><p>~x and ~y. The same expression obtained in [3] for the evaluation </p><p>of the kinetic energy of a thin disc can be used for the </p><p>length dz of the element (fig. 4): </p><p>[;_ dT = pAz(~2 + p2) + ~o201 e + (14) </p><p>1 Ie ] </p><p>2o (tgss) 237 </p></li><li><p>0 </p><p>-50 </p><p>-lOG </p><p>-150 </p><p>-200 </p><p>-250 </p><p>1" V "4/ t. . " " / - </p><p>I 150 I~. - </p><p>50 ~ . . . . . . . - </p><p>/ " s "~ ; . , </p><p>0 50 lnO 150 ra ~ 250 </p><p>Fig. 3. Campbell diagram for a short solid shaft (ct --- a s =~1 = 15) in non dimensional form. Only the first four modes are shown. . . . . . . Second approximation (eq, (6)) </p><p>Third approximation (eel. (13)), </p><p>(. I x </p><p>k k k l ) - le ,_]2 </p><p>Fig. 4. ~Timoshenko beam~ shaft element, System of reference. </p><p>As the length dz is vanishingly small, it follows that </p><p>The expression for the potential energy is simply: </p><p>2/ SL/, + +- - x (15) </p><p>where the shear deformations 7 are linked to the displa- cements and rotations by the relationships: </p><p>dx </p><p>dy (16) </p><p>The above relationships hold only for a perfectly balanced shaft element on which no axial force is acting. The effects of unbalance and axial forces can however be easily taken into account. </p><p>The displacements x, y and the rotations 9x, 9y of the internal points of the element can be expressed as functions of the displacements at the nodes, i.e. of the degrees of freedom of the element </p><p>t X I </p><p>{qx}= ~'y, X 2 </p><p>by the expressions: </p><p>(17) </p><p>x -- [N1] {qx } Y = [Nt] { qy } (18) </p><p>,a~, = [N2l{q x ~ ,ax =- [N~l{qT} </p><p>The usual expressions of the shape functions for Timo- shenko beam are: </p><p>,[ 1 +~(1 - ~') - 3~'2 + 21"3; [3/1] = (1 + r tfl +- - r162 ; </p><p>2 </p><p>]] ~(3~-2~ 2+r l~ ~- r (19) </p><p>1 IN21 = - - [6~'g'- 1); t~[1-- 4~" + 3~'2+ r -D] ; </p><p>le(1 + r </p><p>- 6~'(~'- 1); tel- 2~" + 3~ "~ + 4,~'ll (20) </p><p>where ~" is the nondimensional z coordinate (~" = z/l e) and parameter $ is defined as:, </p><p>= 12gZsx/~ast2 ~ = ]2x*/,~s 2, (21) </p><p>The integration of.equations (14) and (15) yields directly the kinetic and potential energy of the element: </p><p>2 {qx}r[Nllr[N1]{dlx} dr + </p><p>f' ] + {#/T[Ndr[N ~] {#y} d~ + ~0 </p><p>(22) </p><p>+ . {~lxI[N2lr[Nzl{cix} d~" + {~y} [N2lr[N2] 9 2 </p><p>el </p><p>238 MECCAN I(~A </p></li><li><p> ,,II U = 2le {qx} r ~-~ [Na]r ~-~ [g2]{qx} d[ + </p><p>f l d d + {qyir "0 d'~ [N2IT ~'~ [N2I{qY} d~ + ,2II + - - {qx}r[N3]r[N3l{qx} d~ + r </p><p>(23) </p><p>11 1 + {qy}r[Nslr[N3l{qy } dt where the shape function [N3] is: </p><p>d [N 3 ] = [iV 2 ] - ~ [N 1 ] (24) </p><p>l e d~ </p><p>The equation of motion of the insulated element can be directly obtained from Lagrange equation: </p><p>- - - ~ (T - U) = 0 (25) dt ~ ~ Oqi </p><p>Introducing the complex coordinates: </p><p>{q-}= {qx} + i{qy} </p><p>and performing all derivatives in eq. (25), the following equation of motion can be obtained: </p><p>11 s pAxl e [Nxlr[Nlld[{~}+ P~le [N2lr[N2] d~{~}+ (27) </p><p>' el s (d -2icop/fl. [N21T[N2I df{~-}+ ~ [ ~-~ [N21 r. </p><p>9 d--:" IN2] d~ + - - [N3lr[N3] d {7}= 0 </p><p>The same equation can be written in a more compact notation as: </p><p>([mr] + [m R l) {}- i oJ[g]{~'} + [kl{g} = 0. (28) </p><p>The expressions of the matrices appearing in eq. (28) can easily be obtained by performing the integrals of eq. (27). This computation yields: </p><p>- Translatory inertia matrix [m r ]: </p><p>PAlle ] 12rnslerr l412em6[ [mr] = 420(1 +~)2 ] ml lem2 I (29) </p><p>L symmetrical / </p><p>where: </p><p>ml= 156+294 ~+ 140 ~2 </p><p>m2= 22+ 38,5~+ 17.5~ 2 </p><p>m3= 54+ 126 ~+ 70 ~2 </p><p>m4= 13+ 31.5~+ 17.5~ 2 </p><p>m s = 4 + 7 ~ + 3.5# 2 </p><p>m6= 3 + 7 ~ + 3.5~ 2 </p><p>- Rotational inertia matrix [m R ]: </p><p>m lem 8 - m 7 </p><p>p I I 2 - l e m8 l~m 9 [mR] = 30/e(1 + r m7 </p><p>symmetrical </p><p>where: </p><p>m 7 =36 </p><p>m s = 3 - 15~ </p><p>m 9 = 4+ 5q~+10~b 2 </p><p>m]0 = 1+ 5r 5~ 2 </p><p>- Gyroscopic matrix [g]: </p><p>[g] = 2[m R ] </p><p>- Stiffness matrix [k]: </p><p>[k] = le3(1 + r </p><p>12 6 l , -12 6t e ] </p><p>(4+~)l~ -61 e (2-~)./2 L 12 - 6l e [ </p><p>symmetrical (4 + r J </p><p>] - ml0] -lm / </p><p>l m, ] </p><p>(30) </p><p>(31) </p><p>(32) </p><p>The mass and stiffness matrices so computed are the same obtained in [7], [9] for a non rotating Timoshenko beam. Following [9] there is no difficulty in obtaining the consistent matrices for a tapered shaft element. </p><p>Once that the mass, gyroscopic and stiffness.matrices ot aI1 elements have been computed there is no difficulty to assemble them and to add the masses and moments ot inertia of all those bodies which can be modeled as .lumped elements, the stiffnesses of components such as joints, bearings etc. and the constraints. When assembling non-ro- tating elements their gyroscopic matrix must be set equal tc zero . </p><p>The equation of motion of the whole system is then: </p><p>[m]{~}- i~ [G]{~} + [X]{~-}= 0 (33; </p><p>As the solution ofeq. (33) is of the type: </p><p>{q'} ={ q'o } ~t (34: </p><p>equation (33) yields the eigenproblem: </p><p>det [ - X2[M] + reX[G] + [K]] = 0 (351 </p><p>The shape functions (19) and (20) are approximated, a, they correspond to the shape taken by a beam loaded onl., at its ends. </p><p>zo (1985) 239 </p></li><li><p>It is possible to introduce an expression of the shape functions which corresponds to the solution of the equation of motion of the beam element, obtaining an ~...</p></li></ul>

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