The Vibrational Behavior of Bladed Disks in Consideration of Friction Damping and Contact Elasticity

Rotating turbine blading is subjected to fluctuating gas forces during operation that cause blade vibrations. One of the main tasks in the design of turbomachinery blading is the reduction of the vibration amplitudes of the blades to avoid high resonance stresses that could damage the blading. The vibration amplitudes of the blades can be reduced significantly to a reasonable amount by means of friction damping devices such as underplatform dampers. In the case of blade vibrations, relative displacements between the friction damping devices and the neighboring blades occur and friction forces are generated that provide additional damping to the structure due to the dry friction energy dissipation. In real turbomachinery applications, spatial blade vibrations caused by a complex blade geometry and distributed excitation forces acting on the airfoil accur. Therefore, a three dimensional model including an appropriate spatial contact model to predict the generalized contact forces is necessary to describe the vibrational behavior of the blading with sufficient accuracy, see [1] and [2]. In this paper the contact model presented in [2] is extended to include also local deformations in the contacts between underplatform dampers and the contact surfaces of the adjacent blades. The additional elasticity in the contact influences the resonance frequency of the coupled bladed disk assembly. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear vibrations of a mechanical system with non-regular nonlinearities and uncertainties

This paper presents the concept of the Stochastic Multi-dimensional Harmonic Balance Method (Stochastic-MHBM) in order to solve dynamical problems with non-regular non linearities in presence of uncertainties. To treat the nonlinearity in the stochastic and frequency domains, the Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) is proposed. The approach is demonstrated using nonlinear two-degree-of-freedom model with different types of nonlinearities (cubic nonlinearity, contact/no contact, friction). The quasi-periodic stochastic dynamic response is evaluated considering uncertainties in linear and nonlinear parts of the mechanical system. The results are compared with those obtained from the classical Monte Carlo Simulation (MCS). For various numerical tests, it is found that the results agreed very well whilst requiring significantly less computation.     

Parameter identification for nonlinear dynamical systems using Multi Harmonic Balance techniques

An usual approach to investigate nonlinear systems in the frequency domain is the application of the Harmonic Balance Method (HBM), assuming that a harmonic excitation of the system leads to a harmonic response. However, for systems where the steady state response is not just harmonic but periodic or for systems which are excited periodically, this assumption does not longer lead to satisfying results. Therefore, the Multi Harmonic Balance Method (MHBM) is utilized. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

This study presents a direct comparison of measured and predicted nonlinear vibrations of a clamped–clamped steel beam with non-ideal boundary conditions. A multi-harmonic comparison of simulations with measurements is performed in the vicinity of the primary resonance. First of all, a nonlinear analytical model of the beam is developed taking into account non-ideal boundary conditions. Three simulation methods are implemented to investigate the nonlinear behavior of the clamped–clamped beam. The method of multiple scales is used to compute an analytical expression of the frequency response which enables an easy updating of the model. Then, two numerical methods, the Harmonic Balance Method and a time-integration method with shooting algorithm, are employed and compared one with each other. The Harmonic Balance Method enables to simulate the vibrational stationary response of a nonlinear system projected on several harmonics. This study then proposes a method to compare numerical simulations with measurements of all these harmonics. A signal analysis tool is developed to extract the system harmonics’ frequency responses from the temporal signal of a swept sine experiment. An evolutionary updating algorithm (Covariance Matrix Adaptation Evolution Strategy), coupled with highly selective filters is used to identify both fundamental frequency and harmonic amplitudes in the temporal signal, at every moment. This tool enables to extract the harmonic amplitudes of the output signal as well as the input signal. The input of the Harmonic Balance Method can then be either an ideal mono-harmonic signal or a multi-harmonic experimental signal. Finally, the present work focuses on the comparison of experimental and simulated results. From experimental output harmonics and numerical simulations, it is shown that it is possible to distinguish the nonlinearities of the clamped–clamped beam and the effect of the non-ideal input signal.     

Nonlinear vibration and stability of a rotary truncated conical shell with intercoupling of high and low order modals

Truncated conical shell is an important structure that has been widely applied in many engineering fields. The present paper studies the internal dynamic properties of a truncated rotary conical shell and considers the intercoupling of the high and low order modals by utilizing the Harmonic Balance Method. To disclose the detailed intercoupling characteristics of the high order and low order modals of the system, a truncated shallow shell is studied and the internal response properties of the system are investigated by using the Multiple Scale Method. The nonlinear dynamic stabilities of the system are also analyzed using the Incrementation Harmonic Balance Method. Abundant dynamic characteristics are found in the research. The research results show that the high order modals of rotating conical shells have a significant effect on the curves of vibration amplitude and frequency of the shells.     

Nonlinear Vibrations with Friction: A Multiharmonic Approach

Turbine blades are subjected to vibrations caused by static and dynamic loads. To avoid damage, the arising dynamic stresses can be reduced by generating friction contacts between the blades leading to additional damping. The resulting nonlinear equations of motion of the blade dynamics can be solved by the Multiharmonic Balance Method (MHBM), which is described in this paper. With the MHBM, it is possible to consider a multiharmonic excitation as well as a response consisting of several harmonics. In this paper, the MHBM is applied to both a simple 1 DOF nonlinear oscillator and a cyclic model of a blade integrated disk (blisk). (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibrations of linear structures with spatial local nonlinearities

Classical Modal Analysis can be applied to linear systems if the corresponding damping matrix is proportional to the mass or/and stiffness matrices. Otherwise, e.g., in case of structures with single external damping devices, an alternative or approximate solution procedure for determining the dynamic response has to be chosen, compare [1]–[3]. Vibration problems of linear structures with spatially localized nonlinearities are related to those non-classically damped systems. Such systems are characterized by the fact that their nonlinear behavior is largely restricted to a limited number of single points in the structure. The objective of this paper is to present an approximate semi-analytical procedure for analyzing the steady-state harmonic response of those locally nonlinear structures, where special emphasis is laid on beams with single nonlinear devices. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Vibrational Behavior of Coupled Bladed Disks with Variable Rotational Speed

Low pressure steam turbine blades are subjected to high static and dynamic loads during operation. These loads strongly depend on the turbine's rotational speed, leading to entirely new load conditions. To avoid high dynamic stresses due to the forced vibrations, a coupling of the blades, such as shrouds or snubber coupling, is applied to reinforce the structure. In this work the influence of the rotational speed on the vibration behavior of shrouded blades is investigated. Two fundamental phenomena are considered: the stress stiffening and the spin softening effect. Both effects are caused by centrifugal forces and affect the structural mechanical properties, i.e. the stiffness matrix K , of the rotating system. Since the rotational speed Ω appears quadratically, it is possible to derive the stiffness matrix as a second order matrix polynomial in Ω² [3]. In the case of shrouded blades, contact forces between neighboring blades must be taken into account. The contact status and the pressure distribution in particular is strongly influenced by the rotational speed, respectively, centrifugal forces, caused by the untwisting and radial deformation of the blades. For the calculation, a three dimensional structural mechanical model including a spatial contact model is considered. The solution of the nonlinear equations of motion is based on the well known Multiharmonic Balance Method [2]. Here, the nonlinear forces are computed in the time domain and transferred in the frequency domain by the use of the Fast Fourier Transformation (FFT), also known as the Alternating Frequency Time method (AFT) [1]. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of jointed structures using zero thickness interface elements

An important contribution to global damping of mechanical devices is structural damping due tomicroslip effects with friction in joint interfaces. In order to investigate the mechanical behaviour in these contact interfaces numerically, a contact element in the context of Finite Element Method (FEM) is presented. The suggested element is an isoparametric zero thickness element which is well suited for the present problem because the contact area is known and only small relative displacements occur. Arbitrary linear or nonlinear constitutive contact models for normal and tangential contact behaviour can be implemented. Using a proper parametrisation of the contact area, it is possible to apply the element in contact interfaces lying arbitrarily in space and in interfaces discretized with distorted elements. This method is described before a numerical example is compared with experimental results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic analysis and passive control of viscoelastically damped nonlinear dynamic contacts

This paper presents an analytical and finite element study on nonlinear contact dynamics and controls. Nonlinear dynamic contacts between eccentrically supported masses and simply supported beams are studied. Passive control of the dynamic contacts using viscoelastic dampers is also proposed and evaluated. A nonlinear contact finite element is modeled by a set of nonlinear stiffness and damping polynomial functions; and a nonlinear viscoelastic finite element is modeled by a Standard Linear Model with frequency-dependent nonlinear stiffness and damping functions. Analyses show that the dynamic contact force increases as the initial gap increases. Application of viscoelastic dampers can effectively reduce contact loads and prevent dynamic contacts. A simple design equation is also proposed.     

Experimental and theoretical investigation of creep groan of brakes through minimal models

Creep groan of brake systems is a low frequency vibration phenomenon occurring at low speeds which can make passengers feel uncomfortable. This phenomenon is caused by the stick-slip-effect resulting in limit cycle oscillations with frequencies lower than 200 Hz. For the experimental investigation of this problem, an idealized brake test rig is designed concentrating on the investigation of the frictional contact by realizing low damping and small disturbances in the system. Different sensors are utilized in the test rig. Limit cycles and bifurcation effects can be observed in the experimental results. With respect to modeling, a one degree-of-freedom (DOF) model using Coulomb's friction law and a two DOF model using the bristle friction law are considered. In a comparative study of experimental and simulation results, the parameters of both friction laws can be identified from the dynamic experimental results, such as the static and dynamic friction coefficients, contact stiffness and Stribeck velocity. Experimental and theoretical results show a very good concordance. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An easy trick to a periodic solution of relativistic harmonic oscillator

In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is investigated by Homotopy perturbation method. Selection of a linear operator, which is a part of the main operator, is one of the main steps in HPM. If the aim is to obtain a periodic solution, this choice does not work here. To overcome this lack, a linear operator is imposed, and Fourier series of sines will be used in solving the linear equations arise in the HPM. Comparison of the results, with those of resulted by Differential Transformation and Harmonic Balance Method, shows an excellent agreement.     

On the oscillations of a thin-walled structure containing a fluid, in the presence of a hydrodynamic damper : PMM vol. 43, no. 6, 1979, pp. 1095–1101

A model of a hydrodynamic oscillation damper is proposed. The model is used to obtain the equations describing longitudinal oscillations of a structure which includes a shell partially filled with fluid, and contains a hydrodynamic damper. It is shown that the use of the damper leads to considerable increase in the damping of the oscillations of specified frequencies within the structure.

In modern technology one encounters various types of problems connected with restricting the amplitudes of the axisymmetric vibrations of shells and of the longitudinal oscillations of structures consisting of shells partially filled with fluid. Various devices have been proposed [1] for solving these problems. All these devices have a common feature, namely an elastic shell filled with gas and placed in the fluid. The natural frequency of oscillations of such a shell in a fluid can be tuned to required frequency. The effect of such a device is analogous to the effect of a dynamic vibration damper in mechanical systems [2]. A part of the fluid contained in the shell serves as the active mass of the dynamic damper, and for this reason we shall call such devices the hydrodynamic vibration dampers.     

On some regularities in dynamic response of cyclic periodic structures

The paper deals with cyclic periodic structures modelling bladed disk assemblies of blades with friction elements for vibration damping. These elements placed between adjacent blades reduce the vibration amplitudes by means of dry friction resulting from centrifugal forces acting on the elements and relative displacements of the blades. However, the application of these friction elements results in an additional dynamical coupling which together with mistuning of some system parameters (e.g., blade eigenfrequency or contact parameters) may cause localization of vibration. In the present paper a linear approximation of such a system is investigated. The structure composed of cyclic periodic cells modelled each as a clamped-free beam interacting with each other by means of viscoelastic elements of complex stiffness is applied for dynamic system analysis. In case of free vibrations as well as in case of steady-state dynamic response to a harmonic pressure field, a perfect periodic structure and the structures with periodically mistuned parameters (blade eigenfrequencies and contact parameters) are studied. Some regularities in the dynamic response of the systems with mistuning have been noticed. Despite the fact that only a linear approximation has been used, the results and conclusions can be applied for models which describe the blade interaction in a nonlinear way.     

非线性系统动力分析的模态综合技术

各种模态综合方法已广泛应用于线性结构的动力分析,但是,一般都不适用于非线性系统. 本文基于[20][21]提出的方法,将一种模态综合技术推广到非线性系统的动力分析.该法应用于具有连接件耦合的复杂结构系统,以往把连接件简化为线性弹簧和阻尼器.事实上,这些连接件通常具有非线性弹性和非线性阻尼特性.例如,分段线性弹簧、软特性或硬特性弹簧、库伦阻尼、弹塑性滞后阻尼等.但就各部件而言,仍属线性系统.可以通过计算或试验或兼由两者得到一组各部件的独立的自由界面主模态信息,且只保留低阶主模态.通过连接件的非线性耦合力,集合各部件运动方程而建立成总体的非线性振动方程.这样问题就成为缩减了自由度的非线性求解方程,可以达到节省计算机的存贮和运行时间的目的.对于阶次很高的非线性系统,若能缩减足够的自由度,那么问题就可在普通的计算机上得以解决. 由于一般多自由度非线性振动系统的复杂性,一般而言,这种非线性方程很难找到精确解.因此,对于任意激励下系统的瞬态响应,可以采用数值计算方法求解缩减的非线性方程.     

On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators

Harmonic Balance is a very popular semi-analytic method in nonlinear dynamics. It is easy to apply and is known to produce good results for numerous examples. Adding an error criterion taking into account the neglected terms allows an evaluation of the results. Looking on the therefore determined error for increasing ansatz orders, it can be evaluated whether a solution really exists or is an artifact. For the low-error solutions additionally a stability analysis is performed which allows the classification of the solutions in three types, namely in large error solutions, low error stable solutions and low error unstable solution. Examples considered in this paper are the classical Duffing oscillator and an extended Duffing oscillator with nonlinear damping and excitation. Compared to numerical integration, the proposed procedure offers a faster calculation of existing multiple solutions and their character.     

非线性阻尼非线性刚度隔振系统随机动力学特性研究

针对随机激励环境,同时引入刚度和阻尼非线性来提高隔振系统的隔振性能.刚度和阻尼非线性分别是由水平弹簧和水平阻尼的几何布置获得.通过求解Fokker-Planck-Kolmogorov(FPK)方程等效非线性随机振动方程来研究非线性隔振系统在随机激励下的隔振性能,并使用路径积分和Monte-Carlo数值方法进行验证.在此基础上研究刚度非线性和阻尼非线性对隔振系统在随机激励下力传递率及其概率分布的影响.研究表明随着噪声强度的增加,非线性阻尼抑制振动的能力增强,但是在较小的随机激励下线性阻尼优于非线性阻尼.