Reduced Order Modeling Based on Complex Nonlinear Modal Analysis and its Application to Bladed Disks With Shroud Contact

Abstract

The design of bladed disks with contact interfaces typically requires analyses of the resonant forced response and flutter-induced limit cycle oscillations. The steady-state vibration behavior can efficiently be calculated using the Multi-Harmonic Balance method. The dimension of the arising algebraic systems of equations is essentially proportional to the number of harmonics and the number of degrees of freedom (DOFs) retained in the model. Extensive parametric studies necessary e.g. for robust design optimization are often not possible in practice due to the resulting computational effort. In this paper, a two-step nonlinear reduced order modeling approach is proposed. First, the autonomous nonlinear system is analyzed using a Complex Nonlinear Modal Analysis technique based on the work of Laxalde and Thouverez [1]. The methodology in [1] was refined by an exact condensation approach as well as analytical calculation of gradients in order to efficiently study localized nonlinearities in large-scale systems. Moreover, a continuation method was employed in order to predict nonlinear modal interactions. Modal properties such as eigenfrequency and modal damping are directly calculated with respect to the kinetic energy in the system. In a second step, a reduced order model is built based on the Single Nonlinear Resonant Mode theory. It is shown that linear damping and harmonic forcing can be superimposed. Moreover, similarity properties can be exploited to vary normal preload or gap values in contact interfaces. Thus, a large parameter space can be covered without the need for re-computation of nonlinear modal properties. The computational effort for evaluating the reduced order model is almost negligible since it contains a single DOF only, independent of the original system. The methodology is applied to both a simplified and a large-scale model of a bladed disk with shroud contact interfaces. In contrast to [1], the contact constraints account for variable normal load and lift-off in addition to dry friction. Forced response functions, backbone curves for varying normal preload and excitation level as well as flutter-induced limit cycle oscillations are analysed and compared to conventional methods. The limits of the proposed methodology are indicated and discussed.

On the computation of the slow dynamics of nonlinear modes of mechanical systems

“…Krack et al [16,17] applied this approach to various nonlinear mechanical systems and significantly improved the numerical performance of the analysis.…”

“…This frequency-domain modal analysis concept allows for the direct calculation of iso-energy orbits on the invariant manifold as well as eigenfrequency and modal damping ratio of the nonlinear system. The modal properties have also been exploited in a ROM formulation for the prediction of steady-state vibrations [15,16,17]. Krack et al [16,17] also investigated the limitations of the ROM and demonstrated that the validity of this approach is restricted to those regimes in which the energy is confined to a single nonlinear mode.…”

“…The modal properties have also been exploited in a ROM formulation for the prediction of steady-state vibrations [15,16,17]. Krack et al [16,17] also investigated the limitations of the ROM and demonstrated that the validity of this approach is restricted to those regimes in which the energy is confined to a single nonlinear mode. This finding is completely in line with the work of Blanc et al [18], who concluded that the absence of nonlinear modal interactions represents an intrinsic limitation to ROMs based on the invariant manifold concept.…”

Reliability optimization of friction-damped systems using nonlinear modes

“…The concept of nonlinear modes can generally be employed to extract the essential signature of oscillatory nonlinear systems [19,20] and therefore gives rise to order reduction opportunities for nonlinear systems [21,22]. In Krack et al [23,24], an approach was proposed that reduces the computational effort by several orders of magnitude within reasonable ranges of validity. However, it cannot directly be employed in a sophisticated design process since the ROM lacks of an appropriate parameter space.…”

“…It should be stressed that preload, damping and forcing parameters are often regarded as uncertain and/or relevant design variables in the field of friction damping [39,14,24]. Hence, the proposed parameter treatment strategy can In the solution step, the constructed ROM can then be utilized for the efficient prediction of the vibration behavior.…”