# Friction-induced vibration of a slider on an elastic disc spinning at variable speeds

## Abstract

The friction-induced vibration of a mass–slider with in-plane and transverse springs and dampers in sliding contact with a spinning elastic disc in three different situations of spinning speed, i.e. constant deceleration, constant acceleration and constant speed, is studied. The stick–slip motion in the circumferential direction and separation–re-contact behaviour in the transverse direction are considered, which make the system responses non-smooth. It is observed that the decelerating rotation of the disc can make the in-plane stick–slip motion of the slider more complicated in comparison with constant disc rotation and thereby exerting significant influence on the transverse vibration of the disc, while the accelerating rotation of the disc contributes to the occurrence of separation during the vibration and thus influencing the vibration behaviour of the system. Numerical simulation results show that distinct dynamic behaviours can be observed in the three situations of spinning speed of disc and three kinds of particular characteristics of differences are revealed. The significant effects of decelerating and accelerating disc rotation on the friction-induced dynamics of the system underlie the necessity to consider the time-variant spinning speed of disc in the research of friction-induced vibration and noise.

## Keywords

Friction-induced vibration Mass–slider Spinning elastic disc Stick–slip Separation–re-contact Deceleration Acceleration Nonlinear## 1 Introduction

Discs rotating relative to stationery parts can be found in a wide variety of industrial applications, such as car disc brakes, computer discs, clutches, angular sensors and disc actuators. In the normal operations of these systems, dry friction plays a crucial role. Besides the useful functions they perform, the friction force at the contacting interface may induce unstable vibration of the mechanical components, which greatly influences the performance of these machines or leads to annoying noise. For example, friction-induced noise in cars such as brake squeal is still a major issue in automobile industry today, which may be perceived by customers as quality problems and thereby increase the warrant costs [1].

Dry-friction-induced vibration has been studied extensively, and there are several significant mechanisms proposed to explain the occurrence of friction-induced self-excited vibration: the negative friction slope [2], the stick–slip motion [3], the sprag–slip motion [4] and the mode-coupling instability [5]. The stick–slip motion is characterised by alternating stick and slip regimes. It happens when the static coefficient of friction is greater than the kinetic coefficient of friction or the coefficient of friction decreases with relative velocity [3, 6]. Much of the literature is dedicated to friction-induced stick–slip phenomenon [7, 8, 9]. Popp and Stelter [3] investigated the discrete and continuous models exhibiting stick–slip motion and rich bifurcation and chaotic behaviours were revealed. Two kinds of friction laws: the Coulomb friction with stiction and the friction model with Stribeck effect, were applied. Kinkaid et al. [10] studied the stick–slip dynamics of a four-DOF (degree-of-freedom) system with friction force in two orthogonal directions on the contact plane and found the change in direction of the friction force can excite unstable vibration even with the Coulomb friction law, thereby introducing a new mechanism for brake squeal. In [11], a systematic procedure to find both stable and unstable periodic stick–slip vibrations of autonomous dynamic systems with dry friction was derived, in which the discontinuous friction forces were approximated by a smooth function. Hetzler [12] studied the effect of damping due to Coulomb friction on a simple oscillator exhibiting self-excitation due to negative damping. Tonazzi et al. [13] performed an experimental and numerical analysis of frictional contact scenarios from macro stick–slip to continuous sliding. Feeny et al. [14] presented a historical review of dry friction and stick–slip phenomena in structural and mechanical systems. The sprag–slip concept was firstly proposed by Spurr [4], in which the variations of normal and tangential forces due to the deformations of contacting structures were considered to cause the vibration instability. Hoffmann and Gaul [15] examined the dynamics of sprag–slip instability and found that there were parameter combinations for which the system did not possess a static solution corresponding to a steady sliding state, which could be a sufficient condition for occurrence of sprag–slip oscillation. Sinou et al. [16] studied the instability in a nonlinear sprag–slip model with constant coefficient of friction by a central manifold theory. The mode-coupling instability occurred as some modes of the system became unstable when coupling with other modes as a result of friction-induced cross-coupling force. Hoffmann et al. [17] used a two-DOF model to clarify the physical mechanisms underlying the mode-coupling instability of self-excited friction-induced vibration. The effect of viscous damping on the mode-coupling instability in friction-induced vibration was investigated in [18]. Kang et al. [19] studied the dynamic instability of a thin circular plate with friction interface and established the formulation of modal instability due to the mode coupling of the transverse doublet modes. In the case of high-frequency excitation, some non-trivial effects such as shifting of the equilibrium point and dry friction behaving as linear viscous damping can occur to the system dynamics [20, 21].

There were also other friction-related factors responsible for exciting unstable vibration and noise. Chan et al. [22] revealed the destabilising effect of the friction force as a follower force. By using the LuGre friction model, Feng et al. [23] studied the chaotic motions on an autonomous single-degree-of-freedom oscillator because the friction model contains one internal variable. Besides, Butlin and Woodhouse [24] analysed the sensitivity of friction-induced vibration to parameter changes in idealised systems. Wang and Woodhouse [25] developed a novel tribometer to measure the linearised frequency response function for sliding friction, which could provide the input data needed for squeal prediction. Nordmark et al. [26] explored the possibility to formulate a consistent and unambiguous forward simulation model of planar rigid-body mechanical systems with isolated points of intermittent or sustained contact with rigid constraining surfaces in the presence of dry friction. Saha et al. [27] investigated two different friction models by examining the dynamic responses of a single-degree-of-freedom system exhibiting friction-induced vibration. Marques et al. [28] presented a comprehensive review of literature on friction force models and demonstrated the influence of the various friction models on the dynamic response of the multibody mechanical systems with friction.

The dynamic instabilities of an elastic disc under a mass–spring–damper loading system rotating relative to the disc were studied in [29, 30]. In [22], the parametric resonances of an annular plate excited by a rotating transverse load system with frictional follower force were examined and the results showed the friction force could be a destabilising factor. Ouyang and Mottershead [31] investigated the vibration of a disc excited by two co-rotating sliders on either side of a disc and the moving normal forces and friction couple produced by the sliders were seen to bring about dynamic instability. A slider–mass system driven around the surface of a flexible disc was studied in [32], where the in-plane vibration of the slider was considered and coupled with the transverse vibration of the disc through the normal contact force. Subsequently, Li et al. [33] used a similar model but incorporated the separation and reattachment phenomena considering the possibility of loss of contact due to growing transverse disc vibration and the results highlighted the important role of separation on friction-induced vibration. Hochlenert et al. [34] studied the self-excited vibrations in a moving beam and plate generated by frictional forces and an accurate formulation of the kinematics of the frictional contact in two or three dimensions was established. Kang et al. [35] conducted a comprehensive stability analysis of disc brake vibration with gyroscopic, negative friction slope and mode-coupling mechanisms included by modelling the disc and pads as rotating annular and stationary annular sector plates, respectively. Sui and Ding [36] investigated the instability of a pad-on-disc in moving interactions and a stochastic analysis was carried out. The dynamics of an asymmetric spinning disc under stationary friction loads was investigated in [37], and the analysis showed that the stability boundaries of the system were altered by the loss of axisymmetry of the disc.

The models used to study the friction-induced vibration (FIV) problems in the existing literature usually employ a constant sliding velocity, e.g. constant belt velocity in the slider-on-belt model or constant spinning speed of the disc. There has been little research that has considered the decelerating or accelerating sliding, which should not be neglected as an important influential factor in friction-induced-vibration. In [38], a mathematical model was presented to prove that stick–slip oscillation could be induced by deceleration. Pilipchuk et al. [39] examined the friction-induced dynamics of a two-DOF (degree-of-freedom) ‘belt–spring–block’ model and showed that due to the decelerating belt, the system response experiences transitions which could be regarded as simple indicators of onset of squeal. Recently, Dombovari et al. put forward a method to investigate the stability property of the quasi-stationary solution for a smooth dynamic system with slowly time-varying parameters [40]. However, the work on the friction-induced dynamics under decelerating/accelerating sliding motion is still quite limited. To investigate the influences of decelerating/accelerating sliding on the dynamic behaviour of frictional systems and study the problems such as brake noise in a more realistic model because the braking process is practically a decelerating process for the brake disc, the friction-induced vibration of a mass–slider on a spinning elastic disc at variable speeds is examined in this paper.

The rest of the paper is arranged as follows. In Sect. 2, the system configuration of the slider-on-disc model is introduced and the equations of motion for the system in three different states: stick, slip and separation, are derived. The conditions for the transitions among these states are determined. Subsequently, the numerical simulation and analysis are conducted to investigate the distinct dynamic behaviours of the system in the three different situations of spinning speed of disc in Sect. 3 and to help reveal the effects of deceleration and acceleration on the friction-induced dynamics of the system, the system responses under the decelerating and accelerating sliding motion are compared with the results under constant sliding speed. The significant differences that the deceleration and acceleration make to the vibration behaviour of the frictional system from that in the constant disc speed underlie the necessity to consider the time-variant spinning speed in the research of friction-induced vibration and noise. Finally, in Sect. 4 the conclusions on the effects of the decelerating and accelerating sliding motion on the dynamics of the frictional system are drawn.

## 2 Model description and theoretical analysis

*c*is the acceleration of the disc speed and always positive. In Eq. (3), \({\varOmega }_{\mathrm{c}}\) is a constant value which is independent of time.

### 2.1 Circumferential stick–slip vibration of the slider

*m*is the slider’s mass, and

*N*represents the normal force between the disc and the slider. \(\mu \) is the kinetic friction coefficient, and here, it is taken as a function of the relative velocity [41] as follows,

### 2.2 Transverse vibration of the disc

*t*. When the slider is in contact with the disc, the normal displacement

*z*(

*t*) of the slider equals to the local transverse displacement of the disc at \(({r_0, {\varphi }(t)})\) in the space-fixed coordinate system [42], i.e.

*h*is the thickness of the disc. During the slip phase, the friction force reads,

*k*and

*l*denote the number of nodal circles and nodal diameters, respectively, \(C_{kl} (t)\), \(D_{kl} (t)\) are modal coordinates and \(R_{kl} (r)\) is a combination of Bessel functions satisfying the inner and outer boundary conditions of the nonrotating disc and orthogonality conditions. And the equations of motion with respect to the modal coordinates can be obtained from Lagrange’s equations,

*T*and

*U*represent the kinetic energy and strain energy of the disc, respectively, and \(P_{kl}\) and \(Q_{kl}\) represent the generalised forces obtained from the virtual work of the normal force and bending moment acting on the disc.

*A*is the area of the disc surface, \(\rho \) is the density of material, \(D = \frac{Eh^{3}}{12 ({1-\nu ^{2}})}\) is the bending rigidity, and

*E*and \(\nu \) are the Young’s modulus and the Poisson’s ratio of the disc material, respectively.

### 2.3 Coupled in-plane and out-of-plane vibration

*k*nodal circles and

*l*nodal diameters of the corresponding nonrotating plate, and

*f*is given by Eqs (21)–(23) for the situations of decelerating disc, accelerating disc and constant disc speed, respectively, and \({\varphi }, {\dot{{\varphi }}}, \ddot{{\varphi }}\) during the stick phase for the three different situations are given in Eqs. (7)–(12). The condition for remaining in the stick state, which is given in Eq. (14), is thus obtained as

### 2.4 Separation and re-contact

*p*; thus, the distributed load on the disc due to the impact is \(-p\delta ({r-r_0}) \delta ({\theta -{\varphi }(t)}) \delta ({t-t_r})\), which causes the equations of motion of the disc to become,

The values of the constant system parameters

| | \(r_0\) | \(\rho \) | | | \(\nu \) |
---|---|---|---|---|---|---|

0.044 m | 0.12 m | 0.1 m | \(7200\,{\hbox {kg/m}}^{3}\) | 150 GPa | 0.002 m | 0.211 |

| \(k_z \) | \(r_0^2 k_{\varphi }\) | \(N_0 \) | \(\mu _0 \) | \(\mu _1 \) | \(\mu _s \) |
---|---|---|---|---|---|---|

0.1 kg | \(3\times 10^{4}\,\hbox {N/m}\) | \(2 \times 10^{3}\,\hbox {N} \, \hbox {m/rad}\) | 500 N | 0.6 | 0.35 | 0.8 |

## 3 Numerical simulation and analysis

The basic system parameters whose values are constant in the numerical examples are listed in Table 1. It should be noted that numbers *k* and *l* in the expression of the transverse displacement of the disc can be chosen to include as many modes as needed to represent the dynamics of the system with acceptable accuracy. To avoid excessive computations, the modal series in Eq. (24) are truncated at suitable values of indices *k* and *l*. The first seven natural frequencies of the disc are 1492, 1517, 1517, 1824, 1824, 2774 and 2774 rad/s, which are 237, 241, 241, 290, 290, 442 and 442 in Hz. It is found that the first seven disc modes (one single mode with zero nodal circle and zero nodal diameter and three pairs of doublet modes with zero nodal circle and one, two or three nodal diameters) are good enough in terms of the convergence of the results.

### 3.1 Stable sliding equilibrium under the constant speed and the effects of time-variant speed

In this subsection, the dynamic responses in the three different situations of disc speed are obtained and compared to reveal the effects of time-variant disc speed on the friction-induced dynamics of the system.

*fsolve*in MATLAB. Then, the Lyapunov stability at this equilibrium point is investigated. That is, if the solutions of Eqs. (33), (34) and (36) with a small initial perturbation from the equilibrium point converge to the equilibrium point with time approaching infinity, the sliding equilibrium under study is considered to be asymptotically stable; while if the solutions move away from the equilibrium point with time increasing, the sliding equilibrium under study is unstable. Based on the system parameters listed in Table 1, the regions of stability with respect to four parameters \(c_z\), \(c_{\varphi }\), \(\alpha \) and \({\varOmega }_{\mathrm{c}}\) which are found to have significant effects on the stability are obtained. Figure 3 illustrates some combinations of \(c_z\) and \(r_0^2 c_{\varphi }\) which correspond to stable sliding equilibriums with different values of \(\alpha \) under two different constant spinning speeds \({\varOmega }_{\mathrm{c}} =1\) and \(10\, \hbox {rad}/\hbox {s}\).

### 3.2 Non-stationary dynamic behaviour under the time-variant disc speed

### 3.3 Separation and impact during vibration

## 4 Conclusions

- 1.
In the situation of constant speed, a sliding equilibrium of the system can be found. The parameter combinations corresponding to the stable or unstable equilibrium points in the sense of Lyapunov stability are identified.

- 2.
For the system with the parameter combinations corresponding to the stable sliding equilibrium in the situation of constant speed, the vibration starting from an initial condition near the equilibrium point decays with time and ceases eventually, while in the situation of time-varying disc speed, stability may change with time due to the variation of disc speed with time, resulting in an interesting phenomenon that the system vibration decays with time in the early stage but grows in the later stage. This kind of time-varying characteristic of friction-induced vibration results from the negative-slope friction force–relative velocity relationship in the situation of decelerating disc and the speed-dependent instability caused by the moving load in the situation of accelerating disc.

- 3.
The time-variant disc speed increases the non-stationary characteristics of the system dynamics as opposed to the constant disc speed, especially the in-plane motion of the slider, which means there are more shifts of frequency spectra of the dynamic responses throughout the process in the situation of time-variant disc speed than that in the situation of constant speed.

- 4.
In the situation of decelerating disc, separation is more inclined to happen in the case of high initial disc speed and long decelerating process. When impact is considered, the transverse vibration of the disc becomes lower than without.

## Notes

### Acknowledgements

The first author is sponsored by a University of Liverpool and China Scholarship Council joint scholarship.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Trapp, M., Karpenko, Y., Qatu, M., Hodgdon, K.: An evaluation of friction- and impact- induced acoustic behaviour of selected automotive materials, part I: friction-induced acoustics. Int. J. Veh. Noise Vib.
**3**(4), 355–369 (2007)Google Scholar - 2.Mills, H.R.: Brake squeak. Technical report 9000 B, Institution of Automobile Engineers (1938)Google Scholar
- 3.Popp, K., Stelter, P.: Stick–slip vibrations and chaos. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**332**(1624), 89–105 (1990)zbMATHGoogle Scholar - 4.Spurr, R.T.: A theory of brake squeal. ARCHIVE Proc. IMechE Automob. Div.
**1947–1970**(1961), 33–52 (1961)Google Scholar - 5.North, N.R.: Disc brake squeal. Proc. IMechE
**C38**(76), 169–176 (1976)Google Scholar - 6.Elmaian, A., Gautier, F., Pezerat, C., Duffal, J.M.: How can automotive friction-induced noises be related to physical mechanisms? Appl. Acoust.
**76**, 391–401 (2014)Google Scholar - 7.Leine, R.I., van Campen, D.H., de Kraker, A., van den Steen, L.: Stick–slip vibrations induced by alternate friction models. Nonlinear Dyn.
**16**(1), 41–54 (1998)zbMATHGoogle Scholar - 8.Luo, A.C.J., Gegg, B.C.: Stick and non-stick periodic motions in periodically forced oscillators with dry friction. J. Sound Vib.
**291**(1–2), 132–168 (2006)MathSciNetzbMATHGoogle Scholar - 9.Oestreich, M., Hinrichs, N., Popp, K.: Bifurcation and stability analysis for a non-smooth friction oscillator. Arch. Appl. Mech.
**66**(5), 301–314 (1996)zbMATHGoogle Scholar - 10.Kinkaid, N.M., O’Reilly, O.M., Papadopoulos, P.: On the transient dynamics of a multi-degree-of-freedom friction oscillator: a new mechanism for disc brake noise. J. Sound Vib.
**287**(4–5), 901–917 (2005)Google Scholar - 11.Van de Vrande, B.L., Van Campen, D.H., de Kraker, A.: An approximate analysis of dry-friction-induced stick–slip vibrations by a smoothing procedure. Nonlinear Dyn.
**19**(2), 159–171 (1999)zbMATHGoogle Scholar - 12.Hetzler, H.: On the effect of nonsmooth Coulomb friction on Hopf bifurcations in a 1-DoF oscillator with self-excitation due to negative damping. Nonlinear Dyn.
**69**(1–2), 601–614 (2012)MathSciNetGoogle Scholar - 13.Tonazzi, D., Massi, F., Baillet, L., Culla, A., Di Bartolomeo, M., Berthier, Y.: Experimental and numerical analysis of frictional contact scenarios: from macro stick–slip to continuous sliding. Meccanica
**50**(3), 649–664 (2015)Google Scholar - 14.Feeny, B.F., Guran, A., Hinrichs, N., Popp, K.: A historical review of dry friction and stick–slip phenomena. Appl. Mech. Rev.
**51**(5), 321–341 (1998)Google Scholar - 15.Hoffmann, N., Gaul, L.: A sufficient criterion for the onset of sprag–slip oscillations. Arch. Appl. Mech.
**73**(9–10), 650–660 (2004)zbMATHGoogle Scholar - 16.Sinou, J.J., Thouverez, F., Jezequel, L.: Analysis of friction and instability by the centre manifold theory for a non-linear sprag–slip model. J. Sound Vib.
**265**(3), 527–559 (2003)Google Scholar - 17.Hoffmann, N., Fischer, M., Allgaier, R., Gaul, L.: A minimal model for studying properties of the mode-coupling type instability in friction induced oscillations. Mech. Res. Commun.
**29**(4), 197–205 (2002)zbMATHGoogle Scholar - 18.Hoffmann, N., Gaul, L.: Effects of damping on mode-coupling instability in friction induced oscillations. Z. Angew. Math. Mech.
**83**(8), 524–534 (2003)zbMATHGoogle Scholar - 19.Kang, J., Krousgrill, C.M., Sadeghi, F.: Dynamic instability of a thin circular plate with friction interface and its application to disc brake squeal. J. Sound Vib.
**316**(1–5), 164–179 (2008)Google Scholar - 20.Fidlin, A.: Nonlinear Oscillations in Mechanical Engineering, pp. 213–263. Springer, Berlin (2005)Google Scholar
- 21.Thomsen, J.J.: Some general effects of strong high-frequency excitation: stiffening, biasing and smoothening. J. Sound Vib.
**253**(4), 807–831 (2002)Google Scholar - 22.Chan, S.N., Mottershead, J.E., Cartmell, M.P.: Parametric resonances at subcritical speeds in discs with rotating frictional loads. IMechE J. Mech. Eng. Sci.
**208**(6), 417–425 (1994)Google Scholar - 23.Li, Y., Feng, Z.C.: Bifurcation and chaos in friction-induced vibration. Commun. Nonlinear Sci. Numer. Simulat.
**9**(6), 633–647 (2004)zbMATHGoogle Scholar - 24.Butlin, T., Woodhouse, J.: Sensitivity of friction-induced vibration in idealised systems. J. Sound Vib.
**319**(1–2), 182–198 (2009)Google Scholar - 25.Wang, S.K., Woodhouse, J.: The frequency response of dynamic friction: a new view of sliding interfaces. J. Mech. Phys. Solids
**59**(5), 1020–1036 (2011)Google Scholar - 26.Nordmark, A., Dankowicz, H., Champneys, A.: Friction-induced reverse chatter in rigid-body mechanisms with impacts. IMA J. Appl. Math.
**76**(1), 85–119 (2010)MathSciNetzbMATHGoogle Scholar - 27.Saha, A., Wiercigroch, M., Jankowski, K., Wahi, P., Stefański, A.: Investigation of two different friction models from the perspective of friction-induced vibrations. Tribol. Int.
**90**, 185–197 (2015)Google Scholar - 28.Marques, F., Flores, P., Claro, J.P., Lankarani, H.M.: A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems. Nonlinear Dyn.
**86**(3), 1407–1443 (2016)MathSciNetGoogle Scholar - 29.Iwan, W.D., Stahl, K.J.: The response of an elastic disk with a moving mass system. ASME J. Appl. Mech.
**40**(2), 445–451 (1973)Google Scholar - 30.Iwan, W.D., Moeller, T.L.: The stability of a spinning elastic disk with a transverse load system. ASME J. Appl. Mech.
**43**(3), 485–490 (1976)Google Scholar - 31.Ouyang, H., Mottershead, J.E.: Dynamic instability of an elastic disk under the action of a rotating friction couple. ASME J. Appl. Mech.
**71**(6), 753–758 (2005)zbMATHGoogle Scholar - 32.Ouyang, H., Mottershead, J.E., Cartmell, M.P., Brookfield, D.J.: Friction-induced vibration of an elastic slider on a vibrating disc. Int. J. Mech. Sci.
**41**(3), 325–336 (1999)zbMATHGoogle Scholar - 33.Li, Z., Ouyang, H., Guan, Z.: Friction-induced vibration of an elastic disc and a moving slider with separation and reattachment. Nonlinear Dyn.
**87**(2), 1045–1067 (2017)Google Scholar - 34.Hochlenert, D., Spelsberg-Korspeter, G., Hagedorn, P.: Friction induced vibrations in moving continua and their application to brake squeal. ASME J. Appl. Mech.
**74**(3), 542–549 (2007)zbMATHGoogle Scholar - 35.Kang, J., Krousgrill, C.M., Sadeghi, F.: Comprehensive stability analysis of disc brake vibrations including gyroscopic, negative friction slope and mode-coupling mechanisms. J. Sound Vib.
**324**(1–2), 387–407 (2009)Google Scholar - 36.Sui, X., Ding, Q.: Instability and stochastic analyses of a pad-on-disc frictional system in moving interactions. Nonlinear Dyn.
**93**(3), 1619–1634 (2018)Google Scholar - 37.Kang, J.: Moving mode shape function approach for spinning disk and asymmetric disc brake squeal. J. Sound Vib.
**424**, 48–63 (2018)Google Scholar - 38.Van De Velde, F., De Baets, P.: Mathematical approach of the influencing factors on stick–slip induced by decelerative motion. Wear
**201**(1–2), 80–93 (1996)Google Scholar - 39.Pilipchuk, V., Olejnik, P., Awrejcewicz, J.: Transient friction-induced vibrations in a 2-DOF model of brakes. J. Sound Vib.
**344**, 297–312 (2015)Google Scholar - 40.Dombovari, Z., Munoa, J., Kuske, R., Stepan, G.: Milling stability for slowly varying parameters. Procedia CIRP.
**77**, 110–113 (2018)Google Scholar - 41.Bengisu, M.T., Akay, A.: Stability of friction-induced vibrations in multi-degree-of-freedom systems. J. Sound Vib.
**171**(4), 557–570 (1994)zbMATHGoogle Scholar - 42.Ouyang, H.: Moving-load dynamic problems: a tutorial (with a brief overview). Mech. Syst. Signal Process.
**25**(6), 2039–2060 (2011)Google Scholar - 43.Chung, J., Oh, J.E., Yoo, H.H.: Non-linear vibration of a flexible spinning disc with angular acceleration. J. Sound Vib.
**231**(2), 375–391 (2000)Google Scholar - 44.Stancioiu, D., Ouyang, H., Mottershead, J.N.: Vibration of a beam excited by a moving oscillator considering separation and reattachment. J. Sound Vib.
**310**(4–5), 1128–1140 (2008)Google Scholar - 45.Pollard, H., Tenenbaum, M.: In: Tenenbaum, M., Pollard, H. (eds.) Ordinary Differential Equations. Harper & Row, New York (1964)Google Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.