Andreas Hartung

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

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

An Approach for Estimating the Effect of Transient Sweep Through a Resonance

The difference of a stationary forced response situation of a turbine or compressor blade relative to a transient resonance sweep is well known and documented in the literature. Different approaches have been used to understand the effect on transient amplitude in comparison with forced response. The dependencies on damping levels and resonance passage speed have been noted. Estimates for a critical or/and maximum sweep velocity have been given.The understanding of transient response during resonance sweep is of practical importance for instance when running a certification stress test for an aircraft engine, where it needs to be decided upfront which acceleration rate (increase in rpm per second) to use to ensure that the maximum airfoil response that could be attained under stationary condition is being measured with sufficient precision. A second reason for understanding of transient response is the verification of correct, if relevant lower, component life usage during transient regimes in operation.This paper gives a proposal for a simple 1D method based on one degree of freedom (1-dof) system considerations for estimating the transient response dependency on the sweep velocity, damping levels and resonance frequency. The method is verified with 3D analyses of more complex blade-disc structures, which have been validated with air jet excitation rig and aero engine tests. Using the results of the 1-dof analysis, an estimate of the expected stationary resonance response increase can be formulated even in cases where the measured data are based on a significant deviation from the desired sweep velocity, where transient effects would be significant.Copyright

A Numerical Approach for the Resonance Passage Computation

This paper describes a numerical approach to compute the passage through the resonance with one of the natural frequencies and provides analytical proof of this. The numerical approach was used to compute the resonance passage of a turbine blade. The results are compared with complete resonance passages of a simplified multi body model and showed a very good agreement with the test results. In addition some parameter studies, such as structure damping resonance passage time and excitation amplitude, are presented.Copyright

Insert Damping of Hollow Airfoils

This paper describes a new mechanical damping device for hollow airfoils. The concept combines cooling and damping systems in one single element. Damping is achieved by frictional contact between an insert and specially selected contact locations on internal airfoil surfaces. The mechanical damping system is optimized for the first torsion mode of a shroud-less turbine blade and leads to very high damping effectiveness. In parallel an acceptable damping of the first flap mode was observed. In addition studies indicating the role of main parameters such as friction coefficient, excitation amplitude, geometry and location of damper elements are presented.Copyright

A Practical Approach for Evaluation of Equivalent Linear Damping From Measurements of Mistuned and/or Non-Linear Stages and Forced Response Validation

The equivalent linear damping is an important parameter for the design of blades and vanes. This parameter will normally be used based on experience and worst case considerations. In this paper, a practical approach for the evaluation of the equivalent linear damping of blade and vane stages from measurements is proposed. The method can especially be used for mistuned and/or non-linear systems as well as in case of non-satisfying quality of the measurement signals. Based on the approach developed, a way for the validation of the forced response analysis is presented.Copyright

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

Multi-Body Damping of a Vane Cluster

Spring-damper systems are standard for reducing blade vibration amplitude at vane clusters. Spring-dampers can only be used with an altered geometry of the inner shrouds. In most cases a separation of the inner shrouds is inevitable. In this paper an alternative damping system without changes of the outer inner shroud geometry is developed and analyzed. Two analytical models — a simplified Rigid Body Model and a 3D Finite Element Model show, based on similar results, a good comparison. The analytical results were validated by shaker tests. A high level of agreement between simulation and test was achieved.Copyright