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Optimal linearization method for dynamic analysis of non-linear rotors.


The rotor vibrations are strongly dependent on the rotor geometry, on the bearings type and on the excitation forces type. Usually the performances of the mechanical systems are improved by methods, which lead to the increasing of the dynamic stiffness. In the case of linearized model of the bearings we can build the optimization problems for rotor-bearing systems by coupling the finite element method (FEM) to the methods of non-linear programming with constraints (Nicoara et al, 2000). The goal is the diminishing the vibrations by the maximizing of the dynamic stiffness. To do this we need to find out the design parameters: the position of the bearings and the diameters of the shaft (different diameters for several segments of the shaft). In the case of non-linear rotors the optimization problem is formulated by the "energy" of the response as the objective function.


The model consists of a rotor treated as a continuous elastic shaft with several rigid disks, supported on the bearings with a non-linear behavior. In the study of the lateral motion of the rotor, the displacement of any point is defined by two translations (v, w) and two rotations ([[phi].sub.y], [[phi].sub.z]). The model use the beam C1 finite element type based on Timoshenko beam model. The beam finite element has two nodes. In the case of the dynamic analysis four degrees of freedom (DOF) per node are considered: two displacements and two slopes measured in two perpendicular planes containing the beam, (Kramer, 1993).

In the case of linearized model of rotor bearing-systems, when the fluid film bearings are analyzed, the physical model of fluid film bearing may be simplified as a linear element.

It is well known that the behavior of both lubricated journal bearings and rolling element bearings is strongly non-linear and can cause rotors to behave in a non-linear way. In this paper the nonlinearities are involved only in the "elastic" part of the system. The non-linear bearings have a cubic non-linear term (Genta 1993), where the force-displacement relation for a nonlinear spring element can be written as a function of the complex displacement z by the law

f(z) = k(1 + [[absolute value of 2].sup.2])z (1)

This law is particularly well suited for modeling some rolling element bearings, in particular preloaded angular contact bearings but has a more general application.

The linear part of the equation (1) can be introduced directly into the stiffness matrix.

The global mass matrix and the damping matrix are the same as in the linearized model of the bearings, but the global stiffness matrix contains non-linear terms k + [[??]u.sup.2] in nodal displacements, due to the stiffness matrix of the bearings. Thus, the global stiffness matrix can be written as the sum of two matrices by isolating the non-linear part

[[K({x})].sup.n] = [K] + [[[??]({x})]] (2)

where the first matrix is the stiffness matrix of the structure which refers to the shaft and to the constant terms of the bearing stiffness and the second matrix appears due to the non-linearity of the bearings.

The equations of motions of anisotropic rotor-bearing systems which consist of a flexible non-uniform axisymmetric shaft, rigid disk and anisotropic bearings are obtained in second order form, by assembling the element matrices and may be written as


Where {x} is the global displacement vector, whose upper half contains the nodal displacements in the y-x plane, while the lower half contains those in z-y plane, and where the global stiffness matrix is defined by the equations (2)? The positive definite matrix [M] is mass (inertia) matrix, the skew symmetric matrix [G] is gyroscopic matrix, and the non-symmetric matrices [C] and [[K]] are called the damping and the stiffness matrices, respectively.


The numerical time response solution for the non-linear system is calculated using the Wilson-9 method in conjunction with an iteration procedure. In the non-linear rotor's case, the stiffness matrix


have non-linear terms, which depend on the values of the elements in vector [{x}.sub.t+[theta][DELTA]t].

Equation (4) is a set of non-linear algebraic equations now. Therefore an iteration procedure is utilized in conjunction with the Wilson-[theta] method to find [{x}.sub.t+[theta][DELTA]t] and then the displacements [{x}.sub.t+[DELTA]t].


In this section we propose a passive optimization model for non-linear rotor-bearing systems in dynamic range, i.e. to minimize the amplitudes of unbalanced response (synchronous harmonic). The goal is the diminishing the vibrations by the maximizing of the dynamic stiffness. To do this we need to find out the design parameters in order to minimize certain coast functional. The design parameters are the geometrical building elements as the distances between the bearings, [s.sub.i], and the diameters of the shaft, [d.sub.i]. In the case of non-linear rotors the optimization problem is formulated by the "energy" of the response as the objective function:


Numerical example. For the rotor model represented in Figure 1, we propose a numerical example with the value [??] = [10.sup.14]N/[m.sup.3], and the rotor data given in (Nicoara et al., 2000).

The variation of the cost functional with distance between bearings, both for the linear model ([??] = 0) and non-linear model, is represented in Figure 2. The optimization has been done for spin speed of the shaft [ohms] = 8000 rpm and nstep = 4096.

The numerical solution of the non-linear system (3) is calculated using the method described in section 3.

The optimal value of the distance between bearings obtained from the solving of the optimization problem, with the cost functional the energy, is [s.sub.opt] = 0.25 m.


The linearization of the non-linear model using the optimization principle means the replacing of the non-linear system (3) by an equivalent linear system.

[M]{[??]} + [C]{[??]} + [[K].sup.lin.sub.eq]{x} = {F} (6)

The equivalent linear system is obtained by finding of an equivalent stiffness matrix [[K].sup.lin.sub.eq] which replaces the non-linear r stiffness matrix [[K]]. The elements of the equivalent stiffness matrix are obtained by minimizing the cost functional defined as an error between the numerical solution of the nonlinear system and the linear system solution i.e., the equivalent linear system is found by solving the following optimization problem




The numerical solution of the non-linear equation (3) was found using the Wilson-9 method in conjunction with an iteration procedure, shown in section 3. The author elaborated several computer codes in MATLAB programming language. The optimization problem is solved by using the method BFGS, Broyden-Fletcher-Goldfarb-Shanno.

Numerical example. We consider again the rotor-bearing system from Fig. 1. We shall determine an equivalent linear system for the optimized non-linear system that means for the system with a determined distance between bearings by resolving the optimization problem (5). Thus, numerical solution [{x(t)}.sup.num] is calculated for the optimal configuration of the bearings, [s.sub.opt] = 0,25 m. The equivalent stiffness matrices, for station 1 and 2, obtained by solving of the optimization problem (7) are, respectively



The method of the optimal linearization or equivalent linearization allows the determination of some elastic stiffness matrices by which a linear model can be built. Its dynamic behavior is equivalent to that of the non-linear system.

The optimal linearized model is easier to be used at the balancing of the multi-disk-rotor systems supported on bearings with non-linear flexibility.

The optimization program written by the authors in MATLAB language allows the optimal determination of the design parameters both by the direct study of the non-linear model and by the obtaining of a linearized model equivalent with the non-linear model.


Genta, G., (1993). Vibration of Structures and Machines: Practical Aspects. Springer-Verlag, New-York, Berlin

Kramer, E., (1993) Dynamics of Rotors and Foundations. Springer Verlag, Berlin

Nicoara, D.; Cotoros, D. & Munteanu, M. (2000). External optimization of the rotor-bearing systems, Proceedings of the 11th International DAAAM Symposium, Katalinic, B. (Ed.), pp. 331-332, ISBN 3-901509-13-5, Opatjia , Croatia, October 2000, DAAAM International Vienna
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Author:Nicoara, Dumitru
Publication:Annals of DAAAM & Proceedings
Date:Jan 1, 2008
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