Dynamics of dry friction damping in gas turbines: literature survey

Abstract

Vibrating structures, such as gas turbine engines, are assemblages of multiple components interfaced with each other through physical contact. The friction produced at these contact interfaces, by sliding or slipping at the interface, is a source of energy dissipation, resulting in damping effects. This passive approach to damping is exploited by engineers to provide optimal energy dissipation to attenuate vibration in a variety of systems (such as bladed disc assemblies and airfoils), to prevent wear and premature failure. However, the highly nonlinear nature of friction renders accurate predictive modeling of such a problem challenging. This article gives an overview of the current state of the art, and the methods used for mathematical and predictive computational modeling of vibrating systems under the influence of dry friction.

Keywords

Dry friction damping friction models gas turbines harmonic balance method nonlinear problems

1. Introduction

Friction occurs at the physical interface between two bodies in contact while there is a relative movement. It can be seen in all mechanical systems, e.g. shafts, bearings, wheels, brakes, hydraulic and pneumatic cylinders, valves, etc. The cause of friction includes elastic and plastic deformations (or the ploughing effect), molecular adhesion, and surface roughness (Akay, 2002; Bhushan, 2002). Friction was studied extensively in classical mechanical engineering, and, with its wide range of applications, the development in this field has been continuous, with significant contributions such as in Akay (2002), Bowden and Tabor (1954) and Johnson (1987).

Friction opposes the motion of one surface across another surface and is dependent on the nature of the surfaces in contact as well as on the amount of normal force acting on the two surfaces. At contact interfaces, the mechanical energy is converted into thermal energy which is absorbed through friction that provides damping in vibrating structures.

Many vibrating structures, such as gas turbine engines (Figure 1) are an assemblage of multiple components connected with each other under different conditions (welded, bolted, simply hitched, etc). The friction produced at these contact interfaces is a source of energy dissipation, and it produces damping effects. This passive approach of damping is exploited by engineers, and external dampers are also used to provide optimal energy dissipation caused by sliding or slipping in the contact area (see, for example Zucca et al., 2012).

A number of studies have provided use of friction damping as a means of reducing vibration in a variety of systems, such as bladed disc assemblies (Sinha and Griffin, 1984; Griffin and Sinha, 1985) and airfoils (Griffin, 1980; Wang, 1997) so as to limit the occurrence of wear and premature failure (Poudou, 2007).

Figure 1 shows a typical bladed disc assembly along with a single blade/disc sector. The blade and disc interface utilizes the dissipative quality of (dry) friction caused by slip in the contact area due to relative motion triggered by vibration (Meguid et al., 2000). The under-platform damper is used in the turbine assembly to reduce blade vibration amplitude and fatigue damage.

Frictional systems are highly nonlinear and the performance depends on several parameters. In general, the modeling of nonlinear systems of friction depends on a few key parameters, namely: material properties, contact conditions, temperature, load frequency, etc. Researchers have, over the years, developed several models by selecting these parameters in order to optimize the accuracy of their model for their specific problems as well as developing a general framework for prediction of dynamic behavior of frictionally damping systems.

Friction-related problems can be said to belong to three fields: contact mechanics, tribology and nonlinear dynamics. Therefore, modeling of a dry friction damping problem can be divided into three steps:

Modeling of the contact interface or friction modeling

Modeling of the frictionally damped vibrating systems

Solving the resulting nonlinear equations

The following sections give a detailed overview of these steps.

2. Modeling dry friction damping problem

2.1. Friction modeling

Accurate modeling of frictional contact interfaces forms the basis for developing reliable methods for analysis of nonlinear mechanical systems. Akay (2002) and Tworzydlo et al. (1998) give a very thorough and detailed analysis of the complex phenomena of friction at contact interfaces.

The most commonly used friction model is the Coulomb friction model, which is based on the classic laws of friction discovered by Leonardo da Vinci and developed by Charles-Augustin de Coulomb, after whom the model is named (Amontons, 1699; Dowson, 1979). Coulomb examined the effect of four main factors on friction: the characteristics of the materials in contact, the area of the contact surface, the normal load; and the time of repose. Although the Coulomb friction model does not always properly represent the friction behavior in a contact, it is often used to describe the friction in mechanical contacts because of its mathematical simplicity (Andersson et al., 2007).

In relation to gas turbine engines, the early work on friction damping used simple Coulomb models. The friction dampers in gas turbines tend to be of two kinds – blade to ground (friction between the vibrating structure and a relatively rigid structure, e.g. blade and coverplate) or blade-to-blade (friction between two vibrating structures) (Griffin, 1980). The sliding occurring at the contact interface causes energy dissipation which in turn reduces vibratory stresses. These friction models can be one-dimensional (Den Hartog, 1931; Mindlin, 1949; Cigeroglu et al., 2006), planar (Sextro, 1997; Sanliturk and Ewins, 1996), or three-dimensional (Chen et al., 2000; Popp et al., 2003), where the relative motion is considered one-, two- or three-dimensional, respectively.

Several methods based on Coulomb modeling were developed to analyze frictional structures such as bladed disc assemblies by Petrov (2005), Charleux et al. (2006), Cigaroglu et al. (2009), Firrone and Zucca (2011), Csaba (1998); tip shrouds by Yang and Menq (1998), Petrov and Ewins (2003), Menq et al. (1991), Wagner and Griffin (1990); bladed disc with wedge dampers by Sanliturk and Ewins (1996), Petrov (2007), Petrov and Ewins (2006), Panning et al. (2003), Hohl et al. (2008); mechanical joints by Petrov and Ewins (2006), Ferri (1995), Wentzel (2006), Putignano et al. (2011); and torsional structures such as rotor shafts and clutches by Duan and Singh (2005).

Due to the highly nonlinear nature of dry friction, dynamic analysis of such structures is difficult (Cigeroglu, 2007; Cigeroglu et al., 2009) and thus these methods were developed for specific cases. Cigeroglu et al. (2009) tried to present a general method to analyze structures comprising frictional contacts. Their method considered the two general contacting structures to be constrained or unconstrained to analyze the behavior of such systems.

There are other improved friction models as well, which give a better representation of the physical model, such as the Stribeck friction model, which is an extension of Coulomb model by replacing the constant Coulomb friction force with a velocity-dependent function (Stribeck curve) (Söderberg, 2009).

However, Coulomb and Stribeck friction models only give acceptable representation of the physical model when there is large displacement (so-called “macro-slips”) at the interface. Also, these models fail to determine friction force at zero sliding speed (Andersson et al., 2007).The response at the start and reverse of motion is not captured by these models. Although a sign function can be used to capture the change in direction of velocity, this requires repeated condition checks and an intricate simulation procedure. This problem and the said workaround has been employed in many recent works such as the models proposed by Petrov and Ewins (2003, 2004, 2006), Sanliturk and Ewins (1996), Cigeroglu et al., 2009), etc.

Replacing the sign function with a more suitable continuous function, e.g. the hyperbolic tan function can be used to improve the accuracy of the model. Another idea is given by Dankowicz in which the normal dynamic is coupled. Although these modifications improve the representation of the friction models to the actual behavior of systems, they still do not accurately represent small displacements (or microslips) (Söderberg, 2009).

The so-called macro-slip approach is widely used due to its simple mathematical implementation, and works well if the normal load is small. Microslip, or partial slip, of the friction interface is important in high-precision and high-control applications like bladed discs, shafts, etc, where the friction contact pressure is large, so models that can process friction at microslips level are needed.

Microslip models assume that the coefficient of friction is a function of displacement of the contact surfaces. The history of microslips model can be traced back to Courtney-Pratt and Eisner (1957) and before them, Mindlin (1949), where he developed an analytical model of microslip at the contact interface of two spheres.

Many researchers, such as Menq et al. (1986), have verified the effect and significance of microslip on frictionally damped vibrating structures. However, formulating microslip effects causes mathematical complexity and it is seen that this effect is usually employed in simple (mostly one-dimensional) contact problems. Cigeroglu et al. (2006) proposed a one-dimensional model and compared the results with a macro-slip model. Sanliturk et al. (2001) verified their one-dimensional model using experimental data. Sanliturk and Ewins (1996) also combined the macro and microslip effects for one- and two-dimensional models in turbomachinery joints.

Cigeroglu et al. (2006) developed a one-dimensional dynamic microslip friction model. This microslip friction model was further developed for two-dimensional displacement by Cigeroglu (2007), in which he proposed to use a new model with harmonic balance method in frictionally constrained structures with varying normal load. To demonstrate the application, the model is applied on a blade in contact with the ground. Carrying on their work, Cigeroglu et al. (2007) implemented this model on wedge-shaped underplatform dampers in a bladed disc assembly. They allowed the wedge damper to undergo three‐dimensional translation and rotation along with the elastic deformation while the damper was constrained only by friction contacts.

Burdekin et al. (1978) proposed a microslip model which considers the asperities as prismatic rods of equal stiffness. Hagman (1993) proposed a similar model that considered contacting asperities to be spherical objects of constant radius. Olofsson (1995) and Olofsson and Hagman (1997) consider the asperities to be elliptical bodies. Asai et al. (2009) veriﬁed Hagman and Olofsson’s model experimentally and have shown that the Hagman and Olofsson elasto-plastic theory can be used for turbine blade problems. However, the problem is highly nonlinear and thus not simple. More recently Rao et al. (2010) adopted the Oloffson-Hagman microslip damping model to determine the friction coefficient that is a function of various parameters such as the tangential and normal forces at the contact interface, tangential stiffness of the contact and slip amplitude at a resonance so as to capture all the dependences of friction force.

There are a few other models for microslips, such as the ones proposed by Dahl (1977) and Dankowicz (1999). Both Dankowicz and Dahl’s models are based on the observation that friction force is a function of displacement and they modeled the microslip as a first-order differential equation. The contact is represented by springs and does not take sliding or separation into account. This assumption does not always depict the true physical phenomenon, for example, for tangential loads, where sliding must exist at the friction interface, leading to energy dissipation and damping. Laxalde et al. (2008) used Dahl’s model to investigate the use of friction ring dampers in bladed discs. Canudas et al. (1995) proposed a model that combines all three – Stribeck, Dahl and Dankowicz models. Canudas’s model is able to capture a lot of experimentally observed friction characteristics such as microslips, hysteretic of friction force and relative velocity, stick-slip motion, etc. and hence has been used successfully by a number of researchers. It was also further refined and extended by Lampaert et al. (2002) and Swevers et al. (2000). Bauchau and Ju (2006) have proposed an improved extension of the model for joint friction.

Although the above models are successfully employed by researchers to solve friction problems in gas turbines, because of the highly nonlinear nature of friction phenomena, application of these models are problem-dependent and developing a general framework to solve different contact problems has not been attempted so far.

2.2. Modelling of frictionally damped systems

The classical model to analyze the nonlinear behavior in systems was first proposed by Den Hartog (1931). In his work Den Hartog obtained the exact steady-state solution for a single-degree-of-freedom system containing combined viscous and Coulomb damping. Later in 1950, Den Hartog simplified the friction problem with harmonic excitation to get a damped periodic response. Many researchers later extended his work by adding more complexities to his model, such as analyzing the system under harmonic base excitation by Levitan (1960) and Hundal (1979), external excitation on an impact and friction oscillator (Hinrichs et al., 1998), and many others (Haessig and Friedland, 1990; Liang and Feeny, 1998a,b). Yeh (1966) extended Den Hartog’s model to two-degrees-of-freedom systems. Jacobson and Ayre (1958) compared viscous and dry friction damping and established that energy dissipation is larger due to friction damping than to viscous damping. Ibrahim (1994) held experimental and numerical analysis to establish this to be true for small oscillations.

Den Hartog’s friction model was based on pure slip behavior and used only the static coefficient of friction. Shaw (1986) extended Den Hartog’s method by including a static coefficient of friction that was different from the dynamic coefficient of friction and considered the stick-slip behavior of friction. This was further extended and implemented on a frictional system under periodic oscillation by Feeny (1992). Hong and Liu (2000, 2001) said that although Den Hartog’s model is correct, it is incomplete if the behavior of the constitutive friction force during sliding and sticking conditions and the transition between the two is not considered. However, Den Hartog’s analytical model was still not practical for more complex systems with multiple nonlinearities such as multi-degrees-of-freedom systems because his model is piecewise linear and cannot handle multiple nonlinearities at the same time in the governing equations.

A benchmark paper which is repeatedly cited in most studies for friction damping is the review paper by Griffin (1980). Other rather recent survey papers on friction phenomena in dynamic structures are given by Awrejcewicz and Olejnik (2005) and Kerschen et al. (2006). These survey papers give a good overview of the development of work on friction damping in gas turbines, giving a short description of various methods used for modeling and solution of such complex nonlinear systems. One such method by Nayfeh and Mook (1995), which is also one of the most popular methods developed for approximating the frequency response of nonlinear systems, is the harmonic balance (HB) method. This is a frequency-domain method which is time efficient compared to time-domain methods. This type of method is useful for the analysis of systems with steady-state solutions (Sinha and Griffin, 1984; Griffin and Sinha, 1985; Albertson and Gilbert, 2001; Petrov and Ewins, 2003; Petrov, 2005; Legrand et al., 2006). The harmonic balance method assumes that the input harmonic excitation in a system consists of a number of steady-state sinusoids and therefore its response is also harmonic and a sum of steady-state sinusoids with the same frequency as that of excitation, while the amplitude and phase remain unknown and the friction force is calculated in terms of these unknowns. The nonlinear force and all the spatial variables are then expressed in terms of Fourier series, obtaining a set of nonlinear algebraic equations by grouping the harmonic coefficients. Finally, the algebraic equations are solved for the coefficients of the Fourier series.

HB methods have been used to solve nonlinear dynamical problems for many years. Their application to various practical problems have been discussed in a number of papers (Sinha and Griffin, 1984; Griffin and Sinha, 1985; Albertson and Gilbert, 2001; Petrov and Ewins, 2003; Petrov, 2005). Ferri and Dowell have done a significant amount of work in friction damping analysis using the HB method in joint papers (Ferri and Dowell, 1985, 1988) and separate papers (Dowell, 1983; Ferri, 1986). Whiteman and Ferri (1996) used the HB method on a beam-like structure and compared the results obtained with exact time-domain methods. They established that the error between the harmonic balance and time integration results depends on the system and excitation parameters and at low excitation frequencies, the error between harmonic balance and time integration results is quite significant.

In the context of turbine engines, Griffin (1980), Sinha and Griffin (1984), Petrov and Ewins (2006) and Cigeroglu et al. (2009) have applied the HB method and its extensions to the study of the forced response of gas turbine blades and discs. Griffin (1980) and Sinha and Griffin (1984) have comprehensively analyzed the dynamics of flutters and friction dampers in mistuned bladed disc assemblies.

The underlying principle of the HB method is to take only the fundamental harmonics into account. However, more harmonics must be considered for cases with strong nonlinearities and the resonant frequency of the system is an integer multiple of the fundamental excitation frequency (Cigeroglu et al., 2009). The HB method formulated with multiple harmonics is called the multi-harmonic balance method (MHBM). Petrov and Ewins (2003) used MHBM to develop a new friction model for bladed discs. Later (2006) these authors and Wang and Chen (1993) included sticking and slipping transition times in the friction damping formulation using a multi-harmonic approach. This approach is not suitable for complex multi-harmonic systems and the work found in the literature involves only simple, often one-dimensional problems.

A drawback of the HB method is the complexity in formulation of the nonlinear algebraic equations in terms of Fourier series, as this cannot be applied in every system. Also, the problem must be reformulated whenever more harmonic terms are added because the final algebraic equations are formulated using the harmonic coefficients. Moreover, because it is a steady-state method, it does not consider the initial transients and the complicated algebraic formulation is incapable of capturing nonlinearities involved in friction damping that cannot be directly evaluated in the frequency domain.

To overcome these deficiencies, an extension of the harmonic balance method was introduced by Lau and Cheung (1981) as the incremental harmonic balance (IHB) method. Lau and Cheung originally presented this method for quasi-periodic vibrations.

The IHB method is equivalent to the HB method plus the Newton-Raphson method. Ferri (1986) has shown their equivalence. Ferri applied the IHB method on multi-degree-of-freedom systems under periodic excitation and verified the accuracy of the method by comparing the results with the time-domain method. Later Chen et al. (2000) combined the IHB method with the finite element method to nonlinear vibration analysis problems. Pierre and Dowell (1985), and Pierre et al. (1985) extended the IHB approach to analyze the dynamic behavior of systems with perfectly plastic friction dampers. Ferri and Dowell (1988) used a multi-harmonic method and established their superior accuracy over one-harmonic methods and conducted experiments to verify the method. Wei and Pierre (1989) applied a similar approach to study localised forced vibrations in weakly coupled mistuned assemblies. Lau and Zhang (1992) extended the IHB method to analyze systems with piecewise-linear stiffness characteristics. Generally the friction damper elements in these nonlinear systems are described as rigid friction dampers (Pierre and Dowell, 1985; Pierre et al., 1985). Guillen et al. (1999) presented a technique to predict the vibration behavior with flexible dampers.

Dowell (1986) has used the IHB method to investigate the boundary supports of clamped beams and plates. Cheung et al. (1991) derived a formulation for third-order general nonlinear systems. Progressive research has established the usability of the IHB method especially because of its sophistication in problem formulation and computational implementation. In the IHB method, the increment is updated at each iteration step, and hence the number of harmonics can be changed at these points without hindering the continuation (Dowell, 1986). However, good initial values are required for the efficiency of this method. The inaccuracy of the initial solution may lead to non-convergence of the iteration in the method. Choosing an appropriate guess value is not always easy, but there are some techniques available to help decide on a fitting initial value. One such method is the homotopy method proposed by Lau and Cheung (1981). An extension to Lau’s homotopy analysis method is proposed by Chen and Liu (2009).

A problem with the harmonic balance type methods is their complexity (sometimes inability) to capture the nonlinearities in the response and the friction force. An alternative method, which is regarded as a breakthrough in the area of solving nonlinear problems, was introduced by Cameron and Griffin (1989). This method is called the alternating frequency-time domain (AFT) method. As the name suggests, the method switches between frequency and time domains. Research suggested that nonlinearities are more accurately solved in the time domain whereas the vibration problem as a whole is solved efficiently in the frequency domain. In AFT methods, the nonlinearities due to the friction force are calculated in the time domain and transformed back in the frequency domain to be solved with the rest of the equation using simple Newton-like numerical schemes (Narayanan and Sekar, 1998). Sinou et al. (2003) have successfully applied the AFT method on a multi-degree-of-freedom system with friction nonlinearity.

Both IHB and AFT methods are iterative procedures. Research suggests that these methods are good for solving nonlinear problems, especially polynomial nonlinearities. However formulation of these nonlinearities in the IHB method can be very complicated and the AFT method can have slow convergence, thus not having much advantage over simple time-domain methods.

It is seen that the AFT method is accepted by the current researchers as the most suitable generalized method to solve nonlinear dry friction problems in gas turbines. More recent research shows the ease of AFT schemes in handling nonlinearities due to friction force as well as the harmonic features of the excitation. Liu et al. (2010a) showed the application of the AFT method in vibration analysis of shrouded blade systems, Laxalde et al. (2008) showed bladed discs with friction ring dampers, Leib et al. (2008) showed the efficiency of the AFT schemes over time-domain techniques in their investigation of two overlaying beams under harmonic vibrations over a very large range of frequency. Liu et al. (2010b) introduced a fast anti-alias Fourier transform as an extension to the AFT method to improve the accuracy of the calculation. Guillen and Pierre (1999) modified the AFT method and proposed a hybrid frequency-time (HFT) domain method.

As an improvement to the AFT method, in the HFT method the resulting nonlinear algebraic equations are solved using a modified Broyden’s method. The use of this modified method proved to be useful to efficiently solve large scale frictionally damped systems. Poudou and Pierre (2003), and Poudou (2007), further extended the HFT method using order reduction techniques to get a compact set of nonlinear equations and then using a hybrid powell algorithm (Poudou and Pierre, 2003; Powell, 1970) to solve these resulting equations. Poudou demonstrated that this extended HFT method can efficiently solve large-scale, industrial structural systems, by applying it to tuned and mistuned bladed disc assemblies. These works established that complex large scale friction damped systems can be efficiently studied with this method.

2.3. Methods for solving nonlinear systems

Modeling the dynamics of any mechanical problem leads to a set of differential equations. The first step is the formulation of the equations representing the physical system. Then the system response is formulated into a set of differential equations using some assumptions for mathematical simplicity, and the formulation is brought down to a set of differential equations using various methods (discussed in the previous section). The last step to achieve the final solution is to solve these resulting set of differential equations, either analytically or when an analytical solution is not possible – numerically.

In relation to friction damping problems, although it is not always easy to bring down the governing equations to a form where an analytical solution is possible, it is achieved in simpler systems such as one-dimensional models. For example, Ding and Chen (2008) presented an analytical method for determining the steady-state response of a single-degree-of-freedom nonlinear system. Cigeroglu et al. (2006) developed a one-dimensional dynamic microslip friction model and solved the resulting partial differential equations analytically to get the steady-state solution of the shear layer.

For multi-dimensional models, solving a nonlinear system involves finding all possible solutions of the polynomial equations of the system (see (Huntley and Johnson, 1983; Rheinboldt, 1998)). For this reason, there is no organized theory or a general method to solve a set of nonlinear equations analytically and often to obtain an analytical solution to a nonlinear equation is impossible and the use of computers becomes necessary. A number of numerical techniques are available in the literature for computation of an approximate solution for such equations. Solutions of polynomial equations are also termed as its roots or zeros (where the polynomial equation is equal to zero).

Numerous root-finding algorithms have been developed, e.g. bisection method, iteration method, Newton’s method, Secant method, Broyden's method, etc (Sastry, 2005). These methods are usually iterative. As opposed to direct methods which deliver an exact solution by solving a problem by a finite series of operations, iterative methods converge to the solution over time. These algorithms start with a guess for the solution, which may be arbitrary (bisection method), semi-arbitrary, or must be well chosen (Newton’s method, Broyden’s method), and run through several iterations until one or more convergence criteria or other limit are met, such as the number of iterations or minimum error. How quickly the method converges to the solution is an important consideration that helps to choose which method to use for a given problem. Past experience of a similar problem, the solution of a simplified approximate model, and the efficiency of the initial guess are the other things that can help in ensuring that a solution, with an acceptable accuracy, and within an acceptable time frame, is reached (Yang, 2008).

Newton’s method or Newton-based methods are quite practical and popular for solving nonlinear systems (Kelley, 1987). Newton’s method for nonlinear equations is based on a linear approximation of the nonlinear system, i.e. the nonlinear problem is replaced by a set of linear equations, and solutions of these equations converge to the solution of the nonlinear problem. Many researchers, such as Yang and Menq (1998) and Petrov and Ewins (2003) used a Newton-type solution procedure to solve the nonlinear differential problem of their proposed friction model to trace multiple solutions in frequency-domain methods discussed in previous sections.

3. Conclusion

Friction damping problems in mechanical systems are complex. In large scale nonlinear systems such as gas turbine engines, predicting the vibration response at frictional contacts becomes even more difficult and complicated because of the associated nonlinearities, the complex shape and geometries, uncertain boundary conditions or loads, fluctuating loads, etc. There is a strong coupling between the dynamics and the friction behavior of such systems (see for instance Awrejcewicz and Olejnik, 2005). Friction models with coefficients described by constant or velocity dependent functions have been developed to model the nonlinear friction phenomenon. The study of the friction behavior established that the implementation of these models is highly problem-specific and depends on the type of application at hand.

The complexity of a multi-dimensional vibrating system with friction modeled using the constant Coulomb friction coefficient leads to uncertainties in prediction of the behavior at contacts as the Coulomb model is not always an accurate depiction of the actual physical system. In response, improved models with velocity-dependent friction coefficients were developed and continue to be used as better approximations of friction behavior.

Friction models of slipping, modeling of frictional damping via the HB method and variations thereof are based on many assumptions, for ease of calculation, which introduce artificial errors and at times may not represent the system well, such as where asperities dominate the friction response. This is widely recognized and recent attempts have been made to develop methods which use fewer assumptions. Techniques in numerical methods such as root finding algorithms are mature and have well-known limitations, for example the requirement for accurate initial values in Broyden’s method (Yang, 2008).

Even though there are several methods available in the literature to analyze the class of problems described above, improvement and further research are required in nonlinear modeling. Models available at present are developed for specific cases which are usable if the same model is required for every analysis. This does not give designers of new components much scope to try alternative methods or solutions.

Summing up, there is still a clear need to better capture the dynamics of friction mechanisms and provide methods for scientists and engineers to use to predict behavior with greater accuracy.

Footnotes

Funding

This work was supported by Rolls-Royce PLC and the Technology Strategy Board under the programme Strategic Investment in Low Carbon Technology.

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