Raghavendra Murthy

Nonparametric Stochastic Modeling of Structural Uncertainty in Rotordynamics: Unbalance and Balancing Aspects

This paper focuses on extending an earlier investigation on the systematic and rational consideration of uncertainty in reduced order models of rotordynamics systems. The current effort concentrates on the consistent introduction of uncertainty in mass properties on the modal mass and gyroscopic matrices as well as on the unbalance force vector. The uncertainty in mass is separated into uncertainty that maintains the rotor symmetry and the one which disrupts it. Both types of uncertainties lead to variations in the system modal matrices but only the latter induces an unbalance. Accordingly, the approach permits the selection of separate levels on the uncertainty on the system properties (e.g. natural frequencies) and on the unbalance.It was first found that the unbalance response is increased by considering the uncertainty in the rotor modal mass matrices. It was next noted that the approach presented not only permits the analysis of uncertain rotors but it also provides a computational framework for the assessment of various balancing strategies. To demonstrate this unique feature, a numerical experiment was conducted in which a population of rotors were balanced at low speed and their responses were predicted at their first critical speed. These response predictions were carried with the uncertainty in the system modal mass matrices but with or without the balancing weights effects on these matrices. It was found that the balancing at low speed may in fact lead to an increase in both the mean and 95th percentile of the response at critical speed.Copyright

Nonparametric Stochastic Modeling of Uncertainty in Rotordynamics—Part I: Formulation

In the first part of this series, a comprehensive methodology was proposed for the consideration of uncertainty in rotordynamic systems. This second part focuses on the application of this approach to a simple, yet representative, symmetric rotor supported by two journal bearings exhibiting linear, asymmetric properties. The effects of uncertainty in rotor properties (i.e., mass, gyroscopic, and stiffness matrices) that maintain the symmetry of the rotor are first considered. The parameter λ that specifies the level of uncertainty in the simulation of stiffness and mass uncertain properties (the latter with algorithm I) is obtained by imposing a standard deviation of the first nonzero natural frequency of the free nonrotating rotor. Then, the effects of these uncertainties on the Campbell diagram, eigenvalues and eigenvectors of the rotating rotor on its bearings, forced unbalance response, and oil whip instability threshold are predicted and discussed. A similar effort is also carried out for uncertainties in the bearing stiffness and damping matrices. Next, uncertainties that violate the asymmetry of the present rotor are considered to exemplify the simulation of uncertain asymmetric rotors. A comparison of the effects of symmetric and asymmetric uncertainties on the eigenvalues and eigenvectors of the rotating rotor on symmetric bearings is finally performed to provide a first perspective on the importance of uncertainty-born asymmetry in the response of rotordynamic systems.

Optimal Representation of a Varying Temperature Field for Coupling with a Structural Reduced Order Model

This investigation focuses on determining how to optimally represent the temperature distribution of a structure to capture at best its effects on the nonlinear geometric response of the structure expressed in a given modal expansion format. More specifically, with the temperature assumed in an expansion form, it is desired to find thermal basis functions most adapted for the ensuing structural computations. Under the assumptions that the tensor of elasticity and coefficient of thermal expansion are independent of temperature, it is justified that these thermal basis functions should be proportional to linear and nonlinear stress distributions induced by each structural mode and linear combinations of two such modes in the absence of temperature. The implementation of this finding in the context of structural and thermal finite element models is described and validated on a hypersonic panel under strong coupling between structural, thermal, and aerodynamic analyses. It is observed that the effects of the temperature on the structural response are indeed accurately captured without requiring a full modeling of the temperature field.

On the Benefits of Intentional Mistuning of Friction Dampers to Reduce the Response of Tuned and Mistuned Bladed Disks

This paper investigates the potential benefits resulting from using two different types of friction dampers on a bladed disk. In this scenario, every blade or platform is equipped with a damper of either type A or type B with these two types differing by their friction forces. The variations in friction force are assumed here to be induced by similar variations in the damper mass. The benefit of this strategy is measured in comparison with using identical dampers of optimized mass on every blade/platform and is dependent on the pattern of A/B dampers around the disk as well as the damper masses.It is accordingly desired to optimize both the pattern and the damper masses to obtain the largest benefit. As a discovery effort, this optimization is accomplished here through an exhaustive search for all patterns and on a large grid of values of the two damper masses. Owing to the large computational cost of this effort, only single degree of freedom per blade models are assumed with both blade-blade and blade-ground dampers and only small blade counts are considered (6 and 12).Three particular situations are considered: disks with tuned blades except for the arrangement of A/B dampers, disks that also exhibit random mistuning of the blade stiffnesses, and, finally, disks exhibiting random mistuning of both blade stiffness and of the friction forces of the dampers. This latter situation is considered to include the variability induced by manufacturing and wear on the friction dampers.In all cases considered, the benefit of this intentional mistuning of friction dampers is either zero or small, of the order of a few percent, consistently with a single data point reported in the literature.Copyright

Nonparametric stochastic modeling of structural uncertainty in rotordynamics: Unbalance and balancing aspects

This paper focuses on extending an earlier investigation on the systematic and rational consideration of uncertainty in reduced order models of rotordynamics systems. The current effort concentrates on the consistent introduction of uncertainty in mass properties on the modal mass and gyroscopic matrices as well as on the unbalance force vector. The uncertainty in mass is separated into uncertainty that maintains the rotor symmetry and the one which disrupts it. Both types of uncertainties lead to variations in the system modal matrices but only the latter induces an unbalance. Accordingly, the approach permits the selection of separate levels on the uncertainty on the system properties (e.g. natural frequencies) and on the unbalance.It was first found that the unbalance response is increased by considering the uncertainty in the rotor modal mass matrices. It was next noted that the approach presented not only permits the analysis of uncertain rotors but it also provides a computational framework for the assessment of various balancing strategies. To demonstrate this unique feature, a numerical experiment was conducted in which a population of rotors were balanced at low speed and their responses were predicted at their first critical speed. These response predictions were carried with the uncertainty in the system modal mass matrices but with or without the balancing weights effects on these matrices. It was found that the balancing at low speed may in fact lead to an increase in both the mean and 95th percentile of the response at critical speed.Copyright

On the Effects of Blade-Disk Interface Mistuning on the Response of Integrated Bladed Rotors

The response of blades in bladed disks can be represented as a sum of modal contributions from their cantilevered modes and a component induced by the motion of the disk and its interface with the blades. This last contribution referred to here as the disk-induced blade motions is generally considered to be tuned when performing mistuning analysis of bladed disks. Yet, as most of the blade properties, its structural coupling to the disk is likely to be uncertain, for example due to variations in thickness at the blade filet. One thus expects a mistuning of the interface stiffness and mass matrices in particular. The effect of this mistuning on the blade response, which does not appear to have received significant attention, is the focus of the present investigation. A Craig-Bampton methodology is introduced to highlight the disk-blade interface and a mistuning modeling of its stiffness matrix is introduced following the nonparametric modeling method. The analysis with various mistuning models is carried out on a 15-blade impeller finite element model at several resonances. It is found that a small mistuning of the disk-induced blade does not alter notably the mistuned response of the blades.Copyright

Decreasing Bladed Disk Response With Dampers on a Few Blades: Part II—Nonlinear and Blade-Blade Dampers Applications

This two part paper focuses on the optimum placement of a limited number of dampers, i.e. fewer than the number of blades, on a bladed disk to induce the smallest possible amplitude of blade response with or without involuntary, random mistuning. Intentional mistuning is also considered as an option to reduce the amplitude of blade response and the pattern of two blade types (referred to as A and B blades) is then part of the optimization effort in addition to the location of the dampers on the disk.This second part of the investigation focuses on the application of the optimization algorithms developed in Part I to nonlinear dampers, more specifically friction dampers, as well as to the consideration of blade-blade dampers, linear or nonlinear (underplatform dampers). Additionally, the optimization of blade-only and blade-blade linear dampers will be extended to include uncertainty/variability in the damper properties that arise during the manufacturing and/or inservice. It is found that the optimum achieved without considering such uncertainty/variability is robust with respect to it.Copyright

Decreasing Bladed Disk Response With Dampers on a Few Blades: Part I—Optimization Algorithms and Blade-Only Dampers Applications

This two part paper focuses on the optimum placement of a limited number of dampers, i.e. fewer than the number of blades, on a bladed disk to induce the smallest possible amplitude of blade response with or without involuntary, random mistuning. Intentional mistuning is also considered as an option to reduce the amplitude of blade response and the pattern of two blade types (referred to as A and B blades) is then part of the optimization effort in addition to the location of the dampers on the disk.This first part focuses on the formulation and validation of dedicated algorithms for the selection of the damper locations and, when appropriate, of the intentional mistuning pattern. Given the limited number of dampers, there is a concern that the failure of one or several of them could lead to a sharp rise in blade response and this issue is addressed by including the possibility of damper failure in the optimization process to yield a fail-safe optimized solution. The high efficiency and accuracy of the optimization algorithms is assessed in comparison with computationally very demanding exhaustive search results.Copyright

Nonparametric Stochastic Modeling of Uncertainty in Rotordynamics

A systematic and rational approach is presented for the consideration of uncertainty in rotordynamics systems, i.e. in rotor mass and gyroscopic matrices, stiffness matrix, and bearing coefficients. The approach is based on the nonparametric stochastic modeling technique which permits the consideration of both data and modeling uncertainty. The former is induced by a lack of exact knowledge of properties such as density, Young’s modulus, etc. The latter occurs in the generation of the computational model from the physical structure as some of its features are invariably ignored, e.g. small anisotropies, or approximately represented, e.g. detailed meshing of gears. The nonparametric stochastic modeling approach, which is briefly reviewed first, introduces uncertainty in reduced order models through the randomization of their system matrices (e.g. stiffness, mass, and damping matrices of nonrotating structural dynamic systems). Here, this methodology is first extended to permit the consideration of uncertainty in symmetric and asymmetric rotor dynamic systems. Its application is next demonstrated on a symmetric rotor on linear bearings and uncertainties on the rotor stiffness (stiffness matrix) and/or mass properties (mass and gyroscopic matrices) are introduced that maintain the symmetry of the rotor. The effects of these uncertainties on the Campbell diagram, damping ratios, mode shapes, forced unbalance response, and oil whip instability threshold are analyzed. The generalization of these concepts to uncertainty in the bearing coefficients is achieved next. Finally, the consideration of uncertainty in asymmetric rotors is addressed and exemplified.Copyright