Consistent matrices in rotor dynamic

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    20 (1985), 235~

    SOMMARIO. Nel presente lavoro vengono ricavate le espres-

    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

    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

    della condensazione delle matrici per ridurre l'ordine del problema dinamico.

    SUMMAR Y. The expressions for the consistent mass and

    gyroscopic matrices for a constant section shaft element

    are obtained taking into account both sl, ear deformation and transversal inertia. The results are compared with closed

    form solutions, which are available in simple cases. The

    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.


    {f} vector of the forces due to the unbalance of the

    element [g] gyroscopic matrix of the element

    i vc-i" [k] stiffness matrix of the element l shaft length

    l e length of the element

    * Associate Professor, Dipartimento di Meccanica, Politecnieo di Torino.

    [m ] mass matrix of the element n number of elements r/ inner radius

    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

    F a axial force

    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

    U potential energy

    a slenderness (a = I X /~

    "r shear strain

    e unbalance; strain ~" non dimensional coordinate (~" = z/l e)

    ~7 loss factor v a unbalance k whirl speed v Poisson's ratio ~r/~" rotating system of reference (fixed to the shaft

    element) p density ~0 x, ~oy rotations about x and y axes X shear factor w spin speed


    n non rotating element r rotating element [ inertial section characteristics, imaginary part

    R real part S structural section characteristics.


    The dynamic study of rotors is based on principles which

    are well extablished from long time. The increasing complexity of the problems encountered

    in many fields of the technology and the need of a more accurate prediction of the dynamic behaviour of rotating

    20 (1985) 235

  • 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.

    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.

    When considering rotor elements however the often used for structural dynamics can be not suited in order to obtain the required accuracy.

    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.


    2.1. First approximation solution

    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

    deformations are negligible. Even with these assumptions, together with the others

    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.

    The value of the i-th critical speed as can be found e.g. in [5], is:

    l 1/< ('Ocri = 32 7 [ pA 1 (I)

    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.

    The constants 3 i depend upon the constraints. In the case of a shaft which is simply supported at both ends their values are:

    3i = i= (2)

    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

    the spin axis.

    2.2. Second approximation solution

    A ~second approximation>> solution can be obtained by taking into account the rotational inertia of the shaft [6].

    The critical speeds can still be obtained from eq. (1) but with different values of the constants 3 i.

    In the case of a simply supported uniform shaft they are:

    3 i = i~ 1 - =2i2/Ct])-1/4 (3)

    where the s t is defined as:

    a / = ~ (4)

    From eq. (3) the following considerations can be drown: i) The critical speeds computed by taking into account

    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 >.

    ii) Only a limited number of critical speeds exists. The following relationship:


  • * is a function of three The value of the critical speed ~ocr

    parameters only, namely the two values of the slenderness

    ~x and as and the ratio Q/r o between the inner and outer radii of the cross section.

    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

    slenderness a! and a s coincide. The solutlon of eq. (10) is compared with the ones

    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).

    From figures 1 and 2 the following conclusions can



    I . l - J - i I 'l ~, , , . . . . . . . 9 . . . . . . . . . . . . . . . . . . . .

    ; ; i ; , . . . . . . . . . .

    lt ~v Z ~ 1.~ ..

    a r ; I I f | O 2ql 40 60

    1 t Q~ al fiat IO0

    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)).

    ~.' i ' Jr. = 9

    it . . _ . . - ..__,,.___,.....__.._

    - - - - - - r, I f o - - I - - '7 . . . . . .

    O ~ 0 20 40 66 a= %=a I IOO

    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.

    be drawn:

    i) The effect of shear deformation is that of lowering the value of the critical speeds. At least in the case of

    simply supported constant section shafts the effect due to shear deformations is stronger than the one due to rotational

    inertia and consequently the values obtained from the third approximation are lower than the ones from the first

    approximation. ii) An infinity of critical speeds exists in any case. iii) The effects of shear deformation and of rotational

    inertia vanish with increasing values of a, particularly in the case of low order critical speeds.

    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.

    This amounts to the statement that if the shear deforma- tion is neglected it is of no use, and can indeed be misleading,

    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.

    The equation needed for plotting the Campbell diagram can be easily solved in the spin speed 6o*. In non dimensional form it yields:

    = {x 'X* ' - + + 4x* ) ] + (13)

    + - 2i 4)

    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 differ