Malte Krack

A method for nonlinear modal analysis and synthesis: Application to harmonically forced and self-excited mechanical systems

Abstract The recently developed generalized Fourier–Galerkin method is complemented by a numerical continuation with respect to the kinetic energy, which extends the framework to the investigation of modal interactions resulting in folds of the nonlinear modes. In order to enhance the practicability regarding the investigation of complex large-scale systems, it is proposed to provide analytical gradients and exploit sparsity of the nonlinear part of the governing algebraic equations. A novel reduced order model (ROM) is developed for those regimes where internal resonances are absent. The approach allows for an accurate approximation of the multi-harmonic content of the resonant mode and accounts for the contributions of the off-resonant modes in their linearized forms. The ROM facilitates the efficient analysis of self-excited limit cycle oscillations, frequency response functions and the direct tracing of forced resonances. The ROM is equipped with a large parameter space including parameters associated with linear damping and near-resonant harmonic forcing terms. An important objective of this paper is to demonstrate the broad applicability of the proposed overall methodology. This is achieved by selected numerical examples including finite element models of structures with strongly nonlinear, non-conservative contact constraints.

A HIGH-ORDER HARMONIC BALANCE METHOD FOR SYSTEMS WITH DISTINCT STATES

Abstract A pure frequency domain method for the computation of periodic solutions of nonlinear ordinary differential equations (ODEs) is proposed in this study. The method is particularly suitable for the analysis of systems that feature distinct states, i.e. where the ODEs involve piecewise defined functions. An event-driven scheme is used which is based on the direct calculation of the state transition time instants between these states. An analytical formulation of the governing nonlinear algebraic system of equations is developed for the case of piecewise polynomial systems. Moreover, it is shown that derivatives of the solution of up to second order can be calculated analytically, making the method especially attractive for design studies. The methodology is applied to several structural dynamical systems with conservative and dissipative nonlinearities in externally excited and autonomous configurations. Great performance and robustness of the proposed procedure was ascertained.

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

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.Copyright

Parametric studies on the harvested energy of piezoelectric switching techniques

Piezoelectric energy harvesting techniques have experienced increasing research effort during the last few years. Possible applications including wireless, fully autonomous electronic devices, such as sensors, have attracted great interest. The key aspect of harvesting techniques is the amount of converted and stored energy, because the energy source and the conversion rate is limited. In particular, switching techniques offer many parameters that can be optimized. It is therefore crucial to examine the influence of these parameters in a precise manner. This paper addresses an accurate analytical modeling approach, facilitating the calculation of standard-DC and parallel SSHI-DC energy harvesting circuits. In particular the influence of the frequency ratio between the excitation and the electrical resonance of the switching LR-branch, and the voltage gaps across the rectifier diodes are studied in detail. Additionally a comparison with the SSDI damping network is performed. The relationship between energy harvesting and damping is indicated in this paper.

Multiharmonic Analysis and Design of Shroud Friction Joints of Bladed Disks Subject to Microslip

Vibration reduction of turbine blades by means of friction damping in shroud joints is a well-established technology in the field of turbomachinery dynamics. Three-dimensional contact constraints in the shroud coupling can induce highly nonlinear dynamics in the bladed disk assembly. Moreover, large normal contact stresses, which are typical for this application, necessitate the consideration of microslip effects.This study focuses on the accurate prediction of the forced response of tuned bladed disks subject to friction joints. In order to account for extended friction interfaces, the contact area is discretized into several contact points. Microslip behavior is explicitly enforced by a non-uniform normal pressure distribution. Local elastic properties of the contact area are accurately captured in the reduced order model of the structure by employing a component mode synthesis method. The steady-state forced response is efficiently computed using a Multi-Harmonic Balance ansatz. Thus, it is possible to study and explain the occurrence of internal resonances. Planar Coulomb friction and unilateral normal contact conditions are considered in terms of the Dynamic Lagrangian formulation. The normal preload of the shroud interface is varied in order to study the effect on vibration amplitude and resonance frequency.Copyright

Robust Design of Friction Interfaces of Bladed Disks With Respect to Parameter Uncertainties

Friction damping is a well-known technology in the field of turbomachinery. The design of friction contacts is subject to various uncertainties in the contact parameters and operating conditions. In order to obtain a robust design, it is thus necessary not only to optimize the design for a specific set of parameters but also to assess the performance of the design regarding sensitivities with respect to changes in the parameters. An optimization method for the design of friction interfaces for bladed disks subject to uncertainties has been developed. The nonlinear forced vibrations are computed by efficiently solving the equation of motion using the Multi-Harmonic Balance Method. Coulomb friction and unilateral normal contact constraints are enforced employing an analytical formulation of the Dynamic Lagrangian method. Resonance response levels and frequencies are directly computed with respect to design parameters. Analytically derived sensitivities are then used to obtain the probability for that a certain response level is not exceeded. The method is applied to a tuned blisk in order to obtain the optimum normal preload in the nonlinear shroud coupling subject to a given uncertainty in the level of excitation, for example.Copyright

On the efficacy of friction damping in the presence of nonlinear modal interactions

Abstract This work addresses friction-induced modal interactions in jointed structures, and their effects on the passive mitigation of vibrations by means of friction damping. Under the condition of (nearly) commensurable natural frequencies, the nonlinear character of friction can cause so-called nonlinear modal interactions. If harmonic forcing near the natural frequency of a specific mode is applied, for instance, another mode may be excited due to nonlinear energy transfer and thus contribute considerably to the vibration response. We investigate how this phenomenon affects the performance of friction damping. To this end, we study the steady-state, periodic forced vibrations of a system of two beams connected via a local mechanical friction joint. The system can be tuned to continuously adjust the ratio between the first two natural frequencies in the range around the 1:3 internal resonance, in order to trigger or suppress the emergence of modal interactions. Due to the re-distribution of the vibration energy, the vibration level can in fact be reduced in certain situations. However, in other situations, the multi-harmonic character of the vibration has detrimental effects on the effective damping provided by the friction joint. The resulting response level can be significantly larger than in the absence of modal interactions. Moreover, it is shown that the vibration behavior is highly sensitive in the neighborhood of internal resonances. It is thus concluded that the condition of internal resonance should be avoided in the design of friction-damped systems.

Motion complexity in a non-classically damped system with closely spaced modes: From standing to traveling waves

This article addresses the phenomenon of motion complexity in a periodically oscillating system, i.e. the occurrence of non-trivial phase lags among the systems coordinates. Specifically, the steady-state forced response of a linear, weakly damped, self-adjoint system is studied, for which the extent of motion complexity is typically expected to be small. Yet, it is shown that under the condition of closely spaced modes, weak non-classical damping may lead to a transition from standing waves to traveling waves. A system of two oscillators weakly coupled via a linear spring-damper element is considered. The emergence of these motions is related to the distribution of the applied forces and the effect of adding nonlinearity in the form of a cubic spring to the system is investigated. Moreover, it is demonstrated that under certain conditions, the traveling wave response is critical, i.e. it is associated with a resonance or anti-resonance.

Global complexity effects due to local damping in a nonlinear system in 1:3 internal resonance

It is well known that nonlinearity may lead to localization effects and coupling of internally resonant modes. However, research focused primarily on conservative systems commonly assumes that the near-resonant forced response closely follows the autonomous dynamics. Our results for even a simple system of two coupled oscillators with a cubic spring clearly contradict this common belief. We demonstrate analytically and numerically global effects of a weak local damping source in a harmonically forced nonlinear system under condition of 1:3 internal resonance: the global motion becomes asynchronous, i.e., mode complexity is introduced with a non-trivial phase difference between the modal oscillations. In particular, we show that a maximum mode complexity with a phase difference of