Competitive time marching solution methods for systems with friction-induced nonlinearities

Finding efficient and accurate solution methods for nonlinear equilibrium equations is a challenging task. This is the case of systems with friction-induced nonlinearities, e.g., friction-damped turbomachinery assemblies and automotive applications such as brakes. In order to tackle this strategic task, several methods have been developed, both in the time and in the frequency domains. Time marching methods are regarded as the most accurate option, but their computational cost becomes prohibitive when friction nonlinearities are present. This poses a problem in all those cases where alternative frequency domain methods cannot be applied effectively, e.g., if transients, non-periodic excitation/solution, or highly nonlinear systems are of interest. The purpose of this paper is to propose three independent methods to make time-marching more competitive. Two of these methods can be applied to any existing direct integration scheme with minimal adjustments, but the computational time cut they introduce is significant. The last method is instead tailored for systems where the inertia force contribution is negligible. All methods are thoroughly validated numerically using a standard Newmark-β integration scheme as a reference

Forced response of rotating bladed disks: Blade Tip-Timing measurements

The Blade Tip-Timing is a well-known non-contact measurement technique currently employed for the identification of the dynamic behaviours of rotating bladed disks. Although the measurement system has become a typical industry equipment for bladed disks vibration surveys, the type of sensors, the positioning of the sensors around the bladed disk and the used algorithm for data post-processing are still not standard techniques, and their reliability has to be proved for different operation conditions by the comparison with other well-established measurement techniques used as reference like strain gauges. This paper aims at evaluating the accuracy of a latest generation Tip-Timing system on two dummy blisks characterized by different geometrical, structural and dynamical properties. Both disks are tested into a spin-rig where a fixed number of permanent magnets excite synchronous vibrations with respect to the rotor speed. A new positioning for the Blade Tip-Timing optical sensors is tested in the case of a shrouded bladed disk. Due to the presence of shrouds, the sensors cannot be positioned at the outer radius of the disk pointing radially toward the rotation axis as in the most common applications, since the displacements at the tips are very small and cannot be detected. For this reason a particular placement of optical laser sensors is studied in order to point at the leading and trailing edges' locations where the blades experience the largest vibration amplitudes with the aim of not interfering with the flow path. Besides the typical Blade Tip-Timing application aimed at identifying the dynamical properties of each blade, an original method is here proposed to identify the operative deflection shape of a bladed disk through the experimental determination of the nodal diameters. The method is applicable when a small mistuning pattern perturbs the ideal cyclic symmetry of the bladed disk

Vibration Parameters Estimation by Blade Tip-Timing in Mistuned Bladed Disks in Presence of Close Resonances

The present paper is focused on the post processing of the data coming from the Blade Tip-Timing (BTT) sensors in the case where two very close peaks are present in the frequency response of the vibrating system. This type of dynamic response with two very close peaks can occur quite often in bladed disks. It is related to the fact that the bladed disk is not perfectly cyclic symmetric and the so called “mistuning” is present. A method based on the fitting of the BTT sensors data by means of a 2 degrees of freedom (2DOF) dynamic model is proposed. Nonlinear least square optimization technique is employed for identification of the vibration characteristics. A numerical test case based on a lump parameter model of a bladed disk assembly is used to simulate different response curves and the corresponding sensors signals. The Frequency Response Function (FRF) constructed at the resonance region is compared with the traditional Sine fitting results, the resonance frequencies and damping values estimated by the fitting procedure are also reported. Accurate predictions are achieved and the results demonstrate the considerable capacity of the 2DOF method to be used as a standalone or as a complement to the standard Sine fitting method

on the choice of contact parameters for the forced response calculation of a bladed disk with underplatform dampers

Abstract Underplatform dampers (UPDs) are still in use to reduce the vibration amplitude of turbine blades and to shift the position of resonant frequencies. The dynamics of blades with UPDs is nonlinear and the analysis is challenging from both the experimental and the numerical point of view. A key point in obtaining a predictive numerical tool is the choice of the correct contact parameters (contact stiffness and friction coefficient) that are required as input to the contact model. The paper presents different approaches to choose these parameters: the contact stiffness in normal and tangential direction are both calculated and measured. The calculation is based on the analytical models in literature, the measurements are carried out on a dedicated test rig. The friction coefficient is also measured. Test results of the forced response of the same bladed disk with UPDs are available for each blade, they come from an experimental campaign under controlled excitation and centrifugal force. The forced response of the bladed disk is not used as a mean to tune the contact parameters, but rather as a validation tool: the effect of the different choices of contact parameters in the code is highlighted by the comparison of the calculated and experimental forced response of the bladed disk

Synchronous vibration parameters identification by tip timing measurements

The Blade Tip Timing (BTT) measurement system is a technique to measure vibration parameters of a rotating bladed disk. In particular for synchronous vibrations the BTT provides signals versus the rotation speed of the disk starting from the measurement of the time of arrival (TOA) of each blade under the tip timing probes. The signals must be post processed in order to obtain the interesting parameters of each blade vibration. The paper presents a method to extract the main parameters (amplitude and frequency) in resonance condition from the tip timing measurements. The proposed method is a revision of the already existing well known Two-Parameter Plot (2PP) method which requires a minimum of two probes. Improvements to the existing 2PP method are here suggested mainly in the part of engine order identification. The proposed method is then applied to the BTT measured signals coming from a rotating bladed disk excited at different engine orders. At the same time on the disk the vibration of one blade was detected by strain gauges. The strain gauges were calibrated and they provide the reference values of the vibration parameters. The vibration parameters derived by the proposed method are in agreement with those obtained by the strain gages methodolog

Forced response of rotating bladed disks: Blade Tip-Timing measurements

The Blade Tip-Timing is a well-known non-contact measurement technique currently employed for the identification of the dynamic behaviours of rotating bladed disks. Although the measurement system has become a typical industry equipment for bladed disks vibration surveys, the type of sensors, the positioning of the sensors around the bladed disk and the used algorithm for data post-processing are still not standard techniques, and their reliability has to be proved for different operation conditions by the comparison with other well-established measurement techniques used as reference like strain gauges. This paper aims at evaluating the accuracy of a latest generation Tip-Timing system on two dummy blisks characterized by different geometrical, structural and dynamical properties. Both disks are tested into a spin-rig where a fixed number of permanent magnets excite synchronous vibrations with respect to the rotor speed. A new positioning for the Blade Tip-Timing optical sensors is tested in the case of a shrouded bladed disk. Due to the presence of shrouds, the sensors cannot be positioned at the outer radius of the disk pointing radially toward the rotation axis as in the most common applications, since the displacements at the tips are very small and cannot be detected. For this reason a particular placement of optical laser sensors is studied in order to point at the leading and trailing edges' locations where the blades experience the largest vibration amplitudes with the aim of not interfering with the flow path. Besides the typical Blade Tip-Timing application aimed at identifying the dynamical properties of each blade, an original method is here proposed to identify the operative deflection shape of a bladed disk through the experimental determination of the nodal diameters. The method is applicable when a small mistuning pattern perturbs the ideal cyclic symmetry of the bladed disk

Substructuring for Contact Parameters Identification in Bladed-disks

Single stage bladed-disks are fundamental bricks of the rotating parts of a turbomachine. Although made of nominally identical sectors, the presence of imperfections or misalignment produces a large amplification of the forced response. Furthermore, due to their high modal density, friction dampers must be designed to mitigate resonance stresses, since a perfect detuning of the resonances from the excitation forces is impossible. Blade-root joints in these structures can provide the much-desired damping but the contact between the disk slot platform and blade-root lobes is characterized by uncertainty due to the actual locking position and machining tolerances. The cases of two simple beams and a bladed-disk test rig of an array of blades with dovetail root joints are studied to identify contact parameters. A dynamic Lagrange multiplier frequency based sub-structuring (LM-FBS) method is applied in a hybrid manner (experimental and numerical frequency response functions) to identify a parameter associated to each contact by mounting only one blade at a time. A sensitivity analysis is performed that will provide the basis for future work on non-linear frequency response prediction

Improved identification of a blade-disk coupling through a parametric study of the dynamic hybrid models

Joint identification of blade-root joints in typical bladed-disk assemblies is not possible with the classic decoupling methods due to inaccessibility of interface degrees-of-freedom. In a recent study, an attempt was made to identify such a joint by an expansion based decoupling strategy called System Equivalent Model Mixing (SEMM). The expanded sub-models of the connected substructures and their assembly can be influenced by the measurement errors and the discrepancies between the numerical and experimental sub-models. Therefore, the accuracy of the identified joint is compromised. In this work, we investigate some key factors to improve the expanded sub-models through a new measurement campaign on the unconstrained substructures and the assembly. These factors are i) expansion error, ii) interface type, and iii) singular value filtering. The resulting identified joint properties are validated by recoupling the joint with the respective substructures. It is shown that, by controlling these factors, the joint identification can be highly improved

Synchronous vibration parameters identification by tip timing measurements

The Blade Tip Timing (BTT) measurement system is a technique to measure vibration parameters of a rotating bladed disk. In particular for synchronous vibrations the BTT provides signals versus the rotation speed of the disk starting from the measurement of the time of arrival (TOA) of each blade under the tip timing probes. The signals must be post processed in order to obtain the interesting parameters of each blade vibration. The paper presents a method to extract the main parameters (amplitude and frequency) in resonance condition from the tip timing measurements. The proposed method is a revision of the already existing well known Two-Parameter Plot (2PP) method which requires a minimum of two probes. Improvements to the existing 2PP method are here suggested mainly in the part of engine order identification. The proposed method is then applied to the BTT measured signals coming from a rotating bladed disk excited at different engine orders. At the same time on the disk the vibration of one blade was detected by strain gauges. The strain gauges were calibrated and they provide the reference values of the vibration parameters. The vibration parameters derived by the proposed method are in agreement with those obtained by the strain gages methodolog

Tunable Vibration Absorber Design for a High-Precision Cartesian Robot

In metal sheet processing for automotive application, it is crucial to guarantee high robot dynamics for reduced cycle times and adequate components accuracy to be competitive in the market. Since the two aspects are closely and inversely related, the problem becomes challenging. After the first cutting tests, the Cartesian Robot prototype displayed insufficient dimensional accuracy when undergoing high accelerations. The solution hereby proposed is the design of a Tuned Mass Damper (TMD), working in shear mode, to reduce the robot vibration amplitude. To this end, an initial assessment of the robot frequency response and natural frequencies was performed both by using a Finite Element (FE) model of the machine and experimentally. Further, frequency response analyses were carried out to evaluate the TMD effectiveness and to highlight possible criticalities from the manufacturing point of view. On a numerical level, the proposed design can damp the machine resonant frequencies, also showing a certain grade of tunability before operation and in-plane orientation insensitiveness thanks to the use of cylindrically shaped spring