The discretisation of rotordynamic systems usually results in a high number of coordinates, so the computation of the solution of the equations of motion is very time consuming. An efficient semianalytic time-integration method combined with a substructure technique is given, which accounts for nonsymmetric matrices and local nonlinearities. The partitioning of the equation of motion into two substructures is performed. Symmetric and linear background systems are defined for each substructure. The excitation of the substructure comes from the given excitation force, the nonlinear restoring force, the induced force due to the gyroscopic and circulatory effects of the substructure under consideration and the coupling force of the substructures. The high effort for the analysis with complex numbers, which is necessary for nonsymmetric systems, is omitted. The solution is computed by means of an integral formulation. A suitable approximation for the unknown coordinates, which are involved in the coupling forces, has to be introduced and the integration results in Green's functions of the considered substructures. Modal analysis is performed for each linear and symmetric background system of the substructure. Modal reduction can be easily incorporated and the solution is calculated iteratively. The numerical behaviour of the algorithm is discussed and compared to other approximate methods of nonlinear structural dynamics for a benchmark problem and a representative example.

The presence of skew-symmetric matrices is typical for rotordynamic systems. Hence the computation of rotordynamic systems differs from ordinary structural problems due to gyroscopic and circulatory terms, represented by skew-symmetric matrices in the equations of motion; see, for example, Krämer [

In contrast a state-space formulation in connection with a Duhamel-type time integration is given in this contribution. Modal analysis is performed in the confi-guration space considering linear symmetric operators, in order to derive a simple closed-form representation of the two transition matrices of the substructures in the state space. This avoids the more costly analysis with complex numbers due to nonsymmetric matrices and relates the state-space formulation to modal analysis with real eigenvectors of the configuration space. The time integration for both substructures involves the Duhamel-type convolution integral, where the transition matrices are used as kernels. The nonlinearities, the circulatory terms, and the coupling forces are treated as induced forces of the linear system with symmetric matrices. The solution of the second substructure is formulated in the same way. The derivation of an algorithm with a minimum number of nonlinear equations relates the present paper to the component mode analysis, which has been developed by Tongue and Dowell [

A damped gyroscopic nonlinear system with circulatory forces is considered in the configuration space in the following global form, see, for example, Meirovitch [

It is assumed that the nonlinearity is restricted to the second substructure with the index 2, so that we get

With the global configuration vector

Equation (

The above mentioned conditions

The present method takes advantage of the well-known solutions of linear vibration problems with symmetric matrices and proportional damping. A reformulation of (

The mass matrix is positive definite and the inverse

The eigenvectors

The integrals of (

Based on these approximations the integral in (

The

In (

The form of the used dummy matrices depends on the applied approximation of the time evolution of the coordinate, and for the given case of (

The matrix

The equations of motion of substructure 2 again are represented as a linear system with symmetric matrices which are excited by external, induced, and nonlinear forces:

The increment of the state-vector

Finally the increment of the displacement due to the nonlinearity is computed by

Again the dummy matrices are computed from (

When including an adaptive time stepping procedure, see, for example, Crisfield [

The analysis of the numerical behaviour of this substructure method involves the numerical dissipation, numerical dispersion, and stability, see, for example, Hughes [

Spectral Radius for the Present Method.

The presented method is applied to the rotordynamic system sketched in Figure

Geometry of the Rotordynamic System.

Horizontal Acceleration at the Bearings.

Results of the numerical computations with the above semianalytic procedure are shown in Figures

Motion at the Left Disk of the Rotor.

Motion of the Right Disk of the Rotor.

Motion at the Right Bearing.

A comparison to the unconditionally stable version of the Newmark method shows that the effort with the present substructure algorithm is about 20% less than that with the Newmark method, when converged results are considered.

The application of the substructure technique in the presented procedure results in a semi-analytic method, which allows for efficient computation of nonlinear rotordynamic systems. Only a few restrictions to the parameters of the system have been introduced. The system matrices can be very general, as for example, the nonproportional damping is considered in the generalized gyroscopic matrix. The stability of the resulting method is shown for the case of a benchmark system. Modal reduction can be implemented easily in the resulting formulation as the nonlinearity is treated in a suitable manner

Global mass matrix

Global damping matrix

Global gyroscopic matrix

Global stiffness matrix

Global stiffness and circulatory matrix

Global configuration vector

Global velocity vector

Global acceleration vector

Nonlinear restoring force

Externally imposed force

Coupling forces of substructure 1

Coupling matrix of substructure 1

Generalized load vector of substructure 1

Global state space vector of substr. 1

Increment of the state space vector

Initial value of

Modal state space vector

Vector of modal coordinates

Global transition matrix

Transition matrix of substructure 1

Approximate global transition matrix

Modal transition matrix of substructure 1

Transition matrix for step load

Transition matrix for ramp load

Transition matrix for parabolic load

Dummy matrix

Dummy matrix

Dummy matrix

Dummy matrix

Dummy system matrix

Total modal matrix of the substructure 1

Modal matrix of the substructure 1

Identity matrix

Diagonal matrix of the step response

Diagonal matrix of the impulse response

Degrees of freedom of substructure 1

Support of the author by the Linz Center in Mechatronics (LCM) and the Austrian Center of Competence in Mechatronics (ACCM) is greatly acknowledged.