Effects of non-unique friction forces on the dynamic behavior of turbine bladed disks with contact interfaces

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Determination of periodic response limits among multiple solutions for mechanical systems with wedge dampers

Wedge dampers are commonly used to utilize the frictional behavior in many engineering fields such as vehicle dynamics and turbo-machinery. However, the presence of non- unique contact forces in the damper interfaces creates an uncertainty that provides different dynamic response amplitudes even for the same input parameters. The maximum limits of the variability range always take the core attention in most of the damper design processes. In this paper, determination of an upper and a lower boundary among multiple steady-state solutions is presented by using a numerical approach. The method is specifically suitable for the mechanical systems with wedge dampers modeled by macro-slip frictional contact elements in the joint interfaces. In the approach proposed, a criterion that determines the periodic response boundaries according to the limit tangential force values is utilized. The method is demonstrated by illustrating several case studies on a lumped parameter system which represents a turbo-machinery application with a symmetric wedge damper pressed against two vibrating adjacent blades. A point-to-point 1D friction model with varying normal force is used in both contact sides. A parametric investigation on the variability range and response limits is performed for different damper configurations. Harmonic Balance Method with Newton’s iteration scheme is used in the numerical solution of the governing equations. The results show that a large variability exists for damper geometries where a strong coupling is present between tangential and normal contact forces. The method proposed successfully captures the limits of the variability range in all cases

A Consistency Analysis of Phase-Locked-Loop Testing and Control-Based Continuation for a Geometrically Nonlinear Frictional System

Two of the most popular vibration testing methods for nonlinear structures are control-based continuation and phase-locked-loop testing. In this paper, they are directly compared on the same benchmark system, for the first time, to demonstrate their general capabilities and to discuss practical implementation aspects. The considered system, which is specifically designed for this study, is a slightly arched beam clamped at both ends via bolted joints. It exhibits a pronounced softening-hardening behavior as well as an increasing damping characteristic due to the frictional clamping. Both methods are implemented to identify periodic responses at steady-state constituting the phase-resonant backbone curve and nonlinear frequency response curves. To ensure coherent results, the repetition variability is thoroughly assessed via an uncertainty analysis. It is concluded that the methods are in excellent agreement, taking into account the inherent repetition variability of the system

On the non-uniqueness of friction forces and the systematic computation of dynamic response boundaries for turbine bladed disks with contacts

Turbine bladed disks with friction contacts may have a large scattering of dynamic response amplitudes in laboratory conditions even for two consecutive tests. The non-repeatability of experimental studies might directly be related to a physical phenomenon associated with an uncertainty in contact forces. This observation has also been computationally shown in many studies with non-unique contact forces and multiple responses obtained for the same set of inputs. This study presents a numerical aspect and a deeper insight for understanding the variability observed in the periodic vibration analysis of turbine bladed disks with friction damping. A novel method based on an optimization algorithm is proposed to systematically detect the nonlinear dynamic response boundaries. The main idea of the developed approach is to minimize the system loss factor which ultimately determines the damping ability of the structure. In the meanwhile, algebraic set of dynamic balance equations are simultaneously imposed as the nonlinear constraints to be satisfied. In this way, two cases with the minimum values of the positive and negative loss factor determine the upper and the lower boundaries, respectively. The method is validated and demonstrated on a realistic turbine bladed disk with friction interfaces on the shrouds and on the blade-disk interface. Several case studies are performed on different cases by using the state of the art 2D friction model with varying normal load. The results show that the limits of the variability range can be successfully captured by utilizing the offered optimization algorithm. The great contribution of the study is also discussed with some accompanying numerical drawbacks

An Experimental Investigation on the Dynamic Response Variability of a Turbine Blade With Midspan Dampers

This paper addresses two main subjects. First, a novel test setup is described to experi-mentally study the nonlinear dynamic behavior of a turbine blade coupled with two mid-span dampers (MSDs). To this end, a representative turbine blade and midspan friction dampers are originally designed, and they are assembled to a special test rig which has been previously developed at Politecnico di Torino. Second, the variability of the dynamic response is intensively investigated with a purposely defined loading/unloading strategy. To better understand the inherent kinematics of the blade–damper interaction, contact forces are measured through the novel design of the experimental campaign. It is shown that multiple responses, which are obtained in different tests while keeping all user-controlled inputs nominally same, are due to nonunique contact forces that provide different static force equilibria on the damper. This outcome is further supported by the qualitative illustration of hysteresis cycles. This study contributes to the understanding of the response repeatability linked to the nonuniqueness of friction forces

A new modal superposition method for nonlinear vibration analysis of structures using hybrid mode shapes

In this paper, a new modal superposition method based on a hybrid mode shape concept is developed for the determination of steady state vibration response of nonlinear structures. The method is developed specifically for systems having nonlinearities where the stiffness of the system may take different limiting values. Stiffness variation of these nonlinear systems enables one to define different linear systems corresponding to each value of the limiting equivalent stiffness. Moreover, the response of the nonlinear system is bounded by the confinement of these linear systems. In this study, a modal superposition method utilizing novel hybrid mode shapes which are defined as linear combinations of the modal vectors of the limiting linear systems is proposed to determine periodic response of nonlinear systems. In this method the response of the nonlinear system is written in terms of hybrid modes instead of the modes of the underlying linear system. This provides decrease of the number of modes that should be retained for an accurate solution, which in turn reduces the number of nonlinear equations to be solved. In this way, computational time for response calculation is directly curtailed. In the solution, the equations of motion are converted to a set of nonlinear algebraic equations by using describing function approach, and the numerical solution is obtained by using Newton's method with arc length continuation. The method developed is applied on two different systems: a lumped parameter model and a finite element model. Several case studies are performed and the accuracy and computational efficiency of the proposed modal superposition method with hybrid mode shapes are compared with those of the classical modal superposition method which utilizes the mode shapes of the underlying linear system

A novel modal superposition method with response dependent nonlinear modes for periodic vibration analysis of large MDOF nonlinear systems

Design of complex mechanical structures requires to predict nonlinearities that affect the dynamic behavior considerably. However, finding the forced response of nonlinear structures is computationally expensive, especially for large ordered realistic finite element models. In this paper, a novel approach is proposed to reduce computational time significantly utilizing Response Dependent Nonlinear Mode (RDNM) concept in determining the steady state periodic response of nonlinear structures. The method is applicable to all type of nonlinearities. It is based on the use of RDNM which is defined as a varying modal vector with changing vibration amplitude. At steady-state, due to periodic motion, it is possible to define equivalent stiffness due to nonlinear elements as a function of response level which enables one to create new linear systems at each response level by modifying original stiffness matrix of the underlying linear system. In this method, a new linear system is defined at each response level corresponding to each excitation frequency step, and modal information of these equivalent linear systems is used to construct RDNMs which forms a very efficient basis for the nonlinear response space. The response of the nonlinear system is then written in terms of these RDNMs instead of the modes of the underlying linear system. This reduces the number of modes that should be retained in modal superposition method for accurate representation of solution of the nonlinear system, which decreases the number of nonlinear equations, hence the computational effort, significantly. Dual Modal Space method is employed to decrease the computational effort in the calculation of RDNMs for realistic finite element models, i.e. for large MDOF systems. In the solution, nonlinear differential equations of motion are converted into a set of nonlinear algebraic equations by using Describing Function Method, and the numerical solution is obtained by using Newton’s method with arc-length continuation. The method is demonstrated on two different systems. Accuracy and computational time comparisons are performed by applying different case studies which include several different nonlinear elements such as gap, cubic spring and dry friction. Results show that the proposed method is very effective in determining periodic response of nonlinear structures accurately reducing the computational time considerably compared to classical modal superposition method that uses the modes of the underlying linear system. It is also observed that the variation of natural frequency with energy level in a nonlinear system can be approximately obtained by using RDNM concept

A Modal Superposition Method for the Analysis of Nonlinear Systems

In the determination of response of nonlinear structures, computational burden is always a major problem even if frequency domain methods are used. One of the methods used to decrease the computational effort is the modal superposition method for nonlinear systems where the modes of the linear system are used in the calculation. However, depending on the type of the nonlinearity, in order to obtain an accurate response, the number of modes retained in the response calculations needs to be increased, which increases the number of nonlinear equations to be solved. In this study, a method is proposed to decrease the number of modes used for systems having nonlinearities where the equivalent stiffness varies between two limiting values. For such systems, one can define different linear systems for each value of the limiting equivalent stiffness. In this study, it is proposed to use a combination of these linear mode shapes in the modal superposition method. It is shown that proper combination of mode shapes of different linear systems provides satisfactory results by keeping the number of modes used at a minimum. The method is demonstrated on case studies where describing function method is used in the analysis of the nonlinear system