## Abstract

A ring damper can be affixed to a rotating base structure such as a gear, an automotive brake rotor or a gas turbine's labyrinth air seal. Depending on the frequency range, wavenumber and level of preload, vibration of the base structure can be effectively and passively attenuated by friction that develops along the interface between it and the damper. The assembly is modelled as two rods that couple in longitudinal vibration through spatially distributed hysteretic friction, with each rod having periodic boundary conditions in a manner analogous to an unwrapped ring and disc. As is representative of rotating machinery applications, the system is driven by a travelling wave disturbance, and for that form of excitation, the base structure's and the damper's responses are determined without the need for computationally intensive simulation. The damper's performance can be optimized with respect to normal preload, and its effectiveness is insensitive to variations in preload or the excitation's magnitude when its natural frequency is substantially lower than the base structure's in the absence of contact.

## 1. Introduction

Friction damping refers to the conversion of kinetic energy associated with the motion of vibrating surfaces to thermal energy through friction between them (Akay 2002). Applications of friction damping range from bean-bag dampers with granular materials to devices such as the Lanchester damper for torsional vibration. Distributed contact friction dampers are one means to reduce the vibration of continuous mechanical systems, including structural beams, power transmission shafts and plates. A ring damper is one embodiment of a distributed contact damper, and it can be used to passively control bending and in-plane vibration. Ring dampers can be affixed to the periphery of automotive disc (Wickert & Akay 1999) and drum (Wickert & Akay 2000) brakes to reduce squeal noise. In gear train and power transmission applications, ring dampers comprising a snap ring having a circular cross section are used to prevent fatigue failure. Split ring dampers are also used to prevent excessive vibration of annular air seals in gas turbine engines (Niemotka & Ziegert 1993).

In each of those applications, vibration of a rotating structure is excited by forces that are applied in a stationary reference frame. From the structure's viewpoint, the excitation takes the form of a moving load that is periodic in both time and space (Wildheim 1981), and that generates travelling wave response in an axisymmetric structure. In the simplest case, the travelling wave excitation and the ensuing response can be represented by the first term of Fourier expansions.

Ferri (1995) explored mathematical models of friction and solution techniques for friction–vibration interaction problems. Friction can be represented by an *sgn* function (Dowell 1983; Dowell & Schwartz 1983; López *et al*. 2004), a hysteresis loop (Menq *et al*. 1986*a*,*b*) or through displacement-dependent surface stiffness (Whiteman & Ferri 1996, 1997). Compared to the *sgn* model, which is discontinuous with respect to relative velocity between the contacting surfaces, the hysteretic model incorporates interfacial stiffness. Microslip is an extension that becomes important when the normal load is high or when local motions are small (Menq *et al*. 1986*a*). The solid friction model (Dahl 1976; Bliman 1992), originally developed in the context of rolling element bearings, is a smoothened form of the classical piecewise-linear hysteresis loop. The alternative ‘LuGre’ (Do *et al*. 2005) model exhibits very slow dynamics during the sticking phase of interfacial contact and very fast dynamics during the slipping stage. A topic of particular importance in the design of damper systems is the level of normal force that maximizes dissipation. With application to turbine blades, Griffin (1980) derived approximate expressions for the optimal load and the stress reduction achieved by the damper, while neglecting the damper's mass relative to the mass of the blade. Either small normal forces (where little friction develops), or very large ones (where the interface exhibits little slip), resulted in suboptimal damping. Cameron *et al*. (1990) outlined a procedure that integrated analytical models, finite-element models and experimental data to guide the optimization of dampers.

Unlike conventional turbine blade dampers, a ring damper has a spatially distributed friction interface, and such dampers are used on the wheels of some railway vehicles to reduce the squeal noise of passing trains. López *et al*. (2004) studied the behaviour of ring dampers by approximating them as either one- or two-degree-of-freedom systems, although the level of vibration reduction in the base structure was not explicitly addressed. In application to aircraft engine gear trains, the ring's weight is a key variable that can be adjusted in design to achieve significant dissipation; an empirical relation between the weights of the ring and gear was offered by Drago & Brown (1981), but without derivation. In an investigation of split ring dampers on labyrinth seals in a gas turbine's compressor, a static model for the damper's performance was developed and applied in optimization (Niemotka & Ziegert 1993).

By way of motivation for what follows, figure 1 shows an experimental disc having a ring damper that is preloaded against a groove on the disc's outer periphery. With proper choice of radial pressure between the ring and the disc, the ring effectively attenuates the disc's bending and in-plane vibration. The in-plane frequency response of the disc is shown in figure 2*a*, and it was measured in the radial direction using an instrumented force hammer, an accelerometer and standard modal testing methods. A similar test was performed on the disc without the damper (figure 2*b*). In each case, the disc or the disc–ring system was suspended by wires, and the accelerometer was located on the disc's inner periphery. With the ring damper in place, the response amplitude was reduced by an average of 89% in the first three modes at 5.3, 12.6 and 13.1 kHz, respectively.

When a ring damper is attached to a disc or another base structure, a spatially distributed friction interface forms, and vibration energy is dissipated through the relative motion between the two systems. The preload and the relative mass and stiffness of the damper and the base structure can in principle be tuned to maximize dissipation and attenuate the base structure's vibration. If the preload is small, little energy is dissipated owing to the relatively small magnitude of the friction force. Conversely, if the preload is too large, insufficient slippage occurs at the interface, and again little energy is dissipated. An objective here is to study the characteristics of friction damping in the context of two simplified models, termed mutually coupled and mono-coupled, for longitudinal vibration of an ‘unwrapped’ ring damper and disc system that responds to travelling wave excitation. Each subsystem is viewed as a rod having periodic boundary conditions. In the mutually coupled model, the base structure is driven by a travelling wave force, and energy is dissipated as the responses of the base and damper rods couple dynamically through a spatially distributed hysteretic interface. This model is representative of situations in which the subsystems have similar masses, and dynamic coupling between the two is significant. The reduction of the base structure's amplitude is described in §2. In the companion mono-coupled model of §3, the base structure's motion is instead specified to be a travelling wave of known amplitude that, in turn, drives the damper's vibration, but not vice versa. This model is shown to be representative of situations in which the base structure is massive when compared with the damping element. The quantity of energy that is dissipated along the friction interface per cycle of vibration is taken as a metric of the damper's performance. These two viewpoints are compared in §4, and the usefulness and limitations of the simpler approach are set forth.

## 2. Mutually coupled base and damper subsystems

### (a) Vibration model

The discrete model of two rods in frictional contact, representing in an analogous viewpoint an unwrapped base disc and attached ring damper, is shown in figure 3. The hysteretic interface between the rods is modelled by the serial combination of tangential stiffness and dry friction. The parameters that vary with the number of elements *N* used to segment each rod are(2.1)where *M*_{B} and *K*_{B} are the base rod's mass and axial stiffness, respectively; *M*_{D} and *K*_{D} are the damper rod's mass and axial stiffness, respectively; *K*_{F} is the tangential stiffness of the entire interface; and *p*_{0} is the normal preload per unit of length. The base rod is excited by a travelling wave with known force *f*_{0} per unit of length and frequency *ω*. The force on the *i*th element of the base at *m*_{b} is(2.2)for *i*=1, 2, …, *N*, where *n* is the spatial wavenumber of the excitation. Since the base and damper rods are each axisymmetric owing to periodic conditions at *i*=1 and *N*, and since the excitation assumes the special form of a travelling wave, masses *m*_{b} and *m*_{d} respond with displacements(2.3)and(2.4)Here *α* and *β* denote the phase difference between the damper's and the base's responses relative to the excitation, respectively.

Equations (2.3) and (2.4) are exact during pure sticking motion along the interface, and the expressions are first-term approximations in the sense of harmonic balance when the interface responds with a combination of sticking and slipping. Over one cycle of steady-state response, each element of the base, or damper, subsystem exhibits the same motion, the only difference between stations being a phase shift from one location to another. Therefore, the two (unknown) parameters *Y*_{0} and *β* fully describe the base's response, and the parameters *X*_{0} and *α* capture the damper's motion. The relative displacement along the interface becomes(2.5)and the equations of motion become(2.6)and(2.7)Here, *f*_{1} denotes the friction force at station *i*=1, which is a function of *r*_{i} and the response's time history. During steady-state vibration, the interface either (i) sticks and the response is dominated by *k*_{t} or (ii) responds in a combination of sticking and slipping. In §2*b*, the solutions for those two regimes are developed.

### (b) Sticking and sticking/slipping phases

When the interface sticks, the force applied to the base's first element is(2.8)The model is then non-dimensionalized in terms of the quantities(2.9)where *L* is the length of each rod (analogously, the circumference of the ring damper). With *μ* being the coefficient of friction, the displacement at which slip commences is(2.10)Parameter denotes the first flexible body natural frequency of the damper rod in the absence of contact with the base rod. With periodic boundary conditions, integer *n* values of the frequency ratio *η* correspond to the *n*th natural frequency in the absence of any contact. The natural frequency ratio between the base and the damper becomes , and corresponds to the base's *n*th natural frequency in the absence of contact.

The interface's tangential stiffness can be estimated by the shear expression , where *G* is the shear modulus; and *w* and *h* are the width and the height of the damper's cross section, respectively. With the expression for the damper's stiffness, where *E* denotes elastic modulus, the stiffness ratio becomes(2.11)with *ν* as the Poisson's ratio of the damper's material, taken subsequently as *ν*=0.3.

By substituting equations (2.2)–(2.4) and (2.8) into equations (2.6) and (2.7), and by balancing the harmonic coefficients, the dimensionless response amplitudes *X* and *Y*, and the phases *α* and *β*, for the sticking phase are the solutions of(2.12)with(2.13)(2.14)(2.15)and(2.16)As the number of discretization elements for the rods grows relative to the wavenumber, *N*/*n*→∞, the matrix elements admit the approximation(2.17)and the predicted response amplitude and the phase become independent of *N*. Parameters *ζ*_{b} and *ζ*_{d} are the modal damping ratios1 for the base and the damper subsystems, respectively. The approximations in equation (2.17) for large *N*/*n* effectively eliminate the need for convergence analysis with respect to *N*.

Alternatively, when the interface at least partly slips, the harmonic balance method (Caughey 1960; Koh *et al*. 2005) is useful to develop an approximate solution for steady-state amplitude and phase. Direct numerical simulation of the transient response for a hysteretic continuous system can be computationally intensive, and even prohibitively so, owing to the large number of degrees of freedom, possible states, and locations of sticking and slipping that must be tracked during each time step (Berger & Krousgrill 2002). A detailed velocity tracking scheme was developed to simulate the transient response of a beam having bolted joints (Song *et al*. 2004), but only a single hysteresis element was included in the simulation that illustrated the method. In the present case, by virtue of stipulating axisymmetry and travelling wave excitation, an analytical solution is obtainable for arbitrary *N*.

Figure 4 shows the bilinear relation between *f*_{i} and *r*_{i} along the interface. In a one-term harmonic approximation, the force applied to the first element of the base is(2.18)where and *ψ* is the phase associated with *r*_{1} in equation (2.5). The Fourier coefficients are(2.19)In terms of the dimensionless parameters and , the coefficients are given by(2.20)where . By substituting equations (2.2)–(2.4), (2.18) and (2.20) into equations (2.6) and (2.7), and by balancing the harmonics, the equation governing relative amplitude *R* becomes(2.21)with(2.22)

### (c) Preload and amplitude reduction

When the damper is lightly preloaded against the base, dissipation is low owing to the correspondingly small value of the friction force. Likewise, when the damper is highly preloaded, little slippage takes place along the stiffness-dominated interface and again little dissipation develops. With a view towards optimizing the contact pressure, figure 5*a* shows the base's frequency response for different levels of dimensionless preload *P*. The relative motion along the interface is compared in figure 5*b* to the slip displacement *S*. In figure 5*a*, when the preload is as low as 0.1, the response has one resonant peak near *η*=4, and that frequency is nearly the same as that for the base alone (*P*=0).2 In figure 5*b* and also for *P*=0.1, *R*/*S*≫1 over most of the frequency range depicted, consistent with the interface assuming a ‘nearly pure’ slipping state. These characteristics in *Y* and *R* suggest that under low preload, the dynamics of the base and the damper are lightly coupled, and the base vibrates in a manner nearly undisturbed by the damper.

As preload increases, the sticking phase dominates a greater portion of the frequency range. For instance, the frequency corresponding to the first resonance of *Y* decreases through *P*=1, and then increases slightly with preload (e.g. at *P*=10 or 50). With growing preload, a second resonant peak in *Y* becomes discernible at *P*=0.5 in figure 5*a*, and its magnitude increases as *P* grows further. At the largest preloads, the interface sticks over nearly the entire extent of the frequency range as shown. In the neighbourhood of the second peak's resonance, the base and the damper vibrate out-of-phase from one another. With large preload, the damper and base are strongly coupled by tangential stiffness, and the combined system responds in a nearly linear manner. When the excitation frequency coincides with the damper's natural frequency in the absence of contact (), *R*/*S*≪1 for all preloads, and slipping does not occur. Thus, to maximize performance of such a damper, the condition *η*=*n* should be avoided.

As shown in figure 5*a*, the preload can be tuned to reduce the base's maximal response *Y*^{max}, and that value is compared with the amplitude(2.23)that develops in the damper's absence. Figure 6 shows the manner in which *Y*^{max} changes with preload for various *M*. Under large preload, the base is nearly pinned at each contact point with the damper. For such a nearly linear combined system, the value of *Y*^{max} is set primarily by mass and stiffness, and *Y*^{max} is relatively insensitive to preload, as for *P*>40 and *M*=100 in figure 6. When *M* is smaller, and the damper rod is heavier and stiffer, the damper is more effective at controlling the base's maximal response, and over a wide variation in *P*.

### (d) Excitation amplitude

Figure 7*a* shows the base's frequency response for different levels of excitation. The relative displacement along the interface is compared with the slip displacement for the same conditions in figure 7*b*. Transitions from nearly pure sticking (e.g. at *F*=0.01), to a combination of sticking and slipping (*F*=1), and eventually to nearly pure slipping (*F*=10) develop as the excitation's amplitude is increased. When *F*=0.01, for instance, *Y*^{max} is only slightly smaller than , and two resonances are present over the frequency range shown. The base and the damper couple strongly at that low excitation level, and the system behaves in a nearly linear fashion. At the larger excitation level of *F*=10, on the other hand, the interface responds with nearly pure slipping in the neighbourhood of resonance, and any reduction in the base's amplitude is minimal. The damper is effective in controlling the base's vibration over a range of excitation amplitude when the interface experiences a combination of sticking and slipping motions. Pure sticking occurs at the conditions *η*=*n* in each case, implying that the damper's natural frequency in the absence of contact is preferentially designed to be away from the excitation's frequency. Figure 8 summarizes the manner in which *Y*^{max} changes with the excitation's amplitude. When *F* is either very small or very large, the base's response is nearly linear, and little friction damping develops. The relatively flat portion of each curve in figure 8 corresponds to a combination of sticking and slipping motions, and a heavier and stiffer damper is evidently most effective in controlling the base's response over a wide range of excitation amplitude.

### (e) Natural frequency and mass ratios

In an alternative viewpoint, given the same levels of preload and excitation, the damper's effectiveness varies with the natural frequency ratio *γ* between the base and damper subsystems.3 Figure 9*a* shows the manner in which the normalized *Y*^{max} changes with preload and frequency ratio; conversely, figure 9*b* represents the results for constant preload. A damper with a natural frequency which is much lower than that of the base (namely, a system with larger *γ*) is most effective in controlling the base's response over a range of preload given the same excitation, or over a wider range of excitation amplitude given the same preload. When , the damper is least effective in controlling the base's response, and that condition is preferentially avoided when designing the damper.

## 3. Mono-coupled base and damper subsystems

### (a) Damper's vibration model and response

When the excitation frequency is away from *nγ*, the base's response is influenced little by the damper's motion so long as the damper is light (*M*≫1). With a low-mass damper attached, the base's amplitude is insensitive to preload at frequencies away from *nγ*. In that event, mutual coupling between the base and the damper can be neglected, and the mono-coupled system model, in which the base's motion drives the damper's but not vice versa, provides a simpler approximation for the problem at hand. The discrete representation of the damper rod under base excitation is shown in figure 10. The base is specified to vibrate in the steady state at constant amplitude in response to its travelling wave excitation, and that motion serves as base excitation—through the friction interface—to the damper subsystem. At each contact point with the damper, the base's displacement is specified to be(3.1)and the damper responds with the displacements(3.2)where *ϕ* denotes the phase difference between motions of the damper and the base.

The damper's equation of motion is given by equation (2.7). When the interface is in the sticking phase, the damper's amplitude and phase are determined from(3.3)with(3.4)and(3.5)where approximations similar to those in equation (2.17) are made in the limit *N*/*n*→∞. When the interface is instead in the sticking/slipping regime, the friction force applied to a damper mass *m*_{d} at station *i*=1 is approximated by equation (2.18). By balancing the harmonic coefficients in equation (2.7),(3.6)from which the relative amplitude *R* and its phase *ψ* can be determined.

As the interface slips, energy is dissipated based on the area enclosed by trajectories in the hysteresis loop. The energy dissipated per cycle of vibration across the entire interface is given by(3.7)and by(3.8)in terms of the non-dimensional preload , which is taken as a measure of the damper's performance.

### (b) Preload and energy dissipation

For each excitation frequency, an optimal preload exists at which dissipation is maximized. In figure 11, at the higher values of *η*, the maximum energy dissipation that can be achieved is greater than the dissipation at lower-frequency ratios. Likewise, as *η* increases, the damper locks up at ever higher preloads. By designing the damper's natural frequency to be much lower than the excitation frequency, the damper's sensitivity to preload can be reduced. At each excitation frequency, the maximum dissipation that can be achieved, , is shown in figure 12*a*, together with the corresponding optimal preload value, , in figure 12*b*. Both quantities increase for *η*>1. At , little energy is dissipated for any value of preload. When it is excited at the *n*th natural frequency, the damper follows the base with nearly identical amplitude. If the frequency range of the base's vibration is known, can be chosen appropriately to provide dissipation over the entire range. By designing the damper's natural frequency to be well below the base's excitation frequency, energy dissipation can be maximized. This conclusion is consistent with the finding of López *et al*. (2004) in their analysis of a one-degree-of-freedom ring damper model.

### (c) Excitation amplitude

Figure 13*a* shows the influence of *Y* on dissipation at different excitation frequencies. The corresponding changes in *θ*^{*} shown in figure 13*b* identify the transitions from pure sticking (in which case *θ*^{*}=*π*), to a combination of sticking and slipping (0<*θ*^{*}<*π*), and ultimately to nearly pure slipping () as *Y* increases. When the base vibrates with small amplitude at low frequencies, the relative amplitude between it and the damper is also small, and the interface sticks. A threshold value of *Y* exists at each frequency below which no energy is dissipated. When the combination of sticking and slipping motions commences, *E*_{d} increases nonlinearly with *Y*. At a critical amplitude, *θ*^{*} rapidly approaches zero, the condition of nearly pure slipping motion along the interface. Beyond that point, the dependence of *E*_{d} on *Y* is almost linear; however, *θ*^{*} does not precisely vanish owing to the interface's finite tangential stiffness.

## 4. Comparisons

The mono-coupled model provides a good approximation to the behaviour of the base and damper subsystems when the mass of the latter is relatively small. At a representative excitation frequency, figure 14*a* compares the dissipation per cycle as predicted by the mutually coupled and the mono-coupled vibration models.4 As lighter damper designs are considered, the value of *E*_{d} as predicted by the mutually coupled model approaches the limit as estimated by the simpler mono-coupled model. With *M*=50, for instance, the two models are in very close agreement, and the mono-coupled model reproduces the energy dissipation characteristics of the mutually coupled base and damper subsystems.

However, since the mono-coupled model does not consider the base's physical properties, no direct comparison can be made with respect to excitation frequency and the base's natural frequency in the absence of contact (*Ω*_{B}). With *M* being held constant and *γ* varied, the predictions of *E*_{d} over a range of preload are shown in figure 14*b*. When the excitation frequency coincides with *Ω*_{B} (namely, when *η*=*nγ*), the levels of dissipation predicted by the two models differ significantly. However, as *nγ* shifts away from the excitation's frequency, either above or below it, *E*_{d} as predicted by the mutually coupled model approaches the mono-coupled model's estimate. In fact, the further separated *nγ* is from the excitation's frequency, the closer are the two models' predictions.

## 5. Summary

The longitudinal vibration of a base rod and an attached damper rod, each having periodic boundary conditions, is examined as the two couple through a spatially distributed hysteretic friction interface. The system is an analogue, in the first approximation, for the use of a ring damper in attenuating vibration of a disc-like structure that is subjected to travelling wave excitation by virtue of its rotation. The dissipation characteristics of the base–damper system are studied in the context of two simplified models: the mutually coupled model which considers dynamic interaction between the base and the damper, and the mono-coupled model which stipulates that, aside from dissipation, the base's motion is undisturbed by the damper. The rotational periodicity of the damper and the base and the form of excitation facilitate the analytical solution of nonlinear continuous system models having spatially distributed friction. Even for arbitrary degree of freedom in the rod models, the analysis reduces to a set of either four (in the mutually coupled model) or two (in the mono-coupled model) algebraic equations. Usage of the mutually coupled model is appropriate for situations where the base and the damper have similar masses. In order for the damper to be insensitive to variations in either the preload or the excitation's magnitude, its natural frequency in the absence of contact is preferentially much lower than that of the base. When the damper is significantly lighter than the base, and the excitation frequency is well separated from the base's natural frequency in the absence of contact, the simpler mono-coupled model provides a good approximation to the coupled base–damper system's response.

## Acknowledgments

The research described in this manuscript is based upon work conducted while one of the authors (A.A.) served at the National Science Foundation. The authors wish to thank Prof. Jerry Griffin and Prof. Stephen Garoff for their valuable discussions and suggestions on the subject of friction damping.

## Footnotes

One contribution of 8 to a Theme Issue ‘Experimental nonlinear dynamics I. Solids’.

↵The proportional damping levels for the base (b) and damper (d) are taken as

*c*_{b(d)}=(2*ζ*_{b(d)}/*Ω*_{B(D)})*k*_{b(d)}, where*Ω*_{B}is the base's first flexible body natural frequency in the absence of contact.↵With

*γ*=4 and*n*=1,*η*=4 corresponds to the base's natural frequency in the absence of contact with the damper.↵For simplicity, the stiffness ratio

*K*is held constant, and*M*is illustratively varied to generate different*γ*.↵Different values of

*M*are taken in evaluating*E*_{d}through the mutually coupled model while maintaining the same natural frequency ratio. The base's amplitude is specified to be*Y*=0.01 in the mono-coupled model. In the mutually coupled model, the excitation's amplitude, , is adjusted for each*M*to realize*Y*_{ND}=0.01.- © 2007 The Royal Society