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Choices of excitation signals are important in engineering sciences and in physical simulations; a sufficient excitation can be critical in modelling a complicated nonlinear dynamic system. The discontinuous dynamic of a non-linear, friction-induced with two idealized periodical forced oscillators is studied. The dry friction in the system follows the classical Coulomb law, and various friction characteristics of dry friction laws in engineering sciences. To capture the presence of the two driving forces, the system must be studied as a function of their frequency-modulated and its equivalent amplitude modulated waveforms. Our numerical investigation shows a rich dynamical behaviour including periodic, quasi-periodic motions, thus a variable dynamics phenomenon among others; such as modulated waves, modulated stick-slip, periodic oscillation, and periodic stick-slip. It seems that such excitation forces can be used to conveniently identify the existence of nonlinearity, dry friction effects, and strength degradation in the system. The results achieved via the Coulomb’s law are compared with those obtained via two others particular friction laws: the complete model with Stribeck effect and Coulomb viscosity.

In general, there are many different types of dry friction models and it is crucial to appropriately choose one which best suits to the modelled problem, however choices of excitation forces have been widely studied for the modal testing community [

From the mathematical point of view, the appearance of the discontinuous differential equations is usual, where the character of this discontinuity depends on the friction character adopted [

The “mass on belt system” falls within the category of hybrid system or switching system [

The system under investigation is shown in _{0}. Consider two harmonic driving forces exerting on the mass with the same amplitude p_{0}, but different frequencies w_{1} and w_{2}, these forces are defined as

The two excitation frequencies are proportional to each other in such a way that _{s} and the smaller kinetic coefficient µ_{k}) or other frictions laws. Since the mass contacts the moving belt with friction, the mass can move along or rest on the belt. The equation of motion for such a friction-induced oscillator is:

where x denotes the displacement of the mass. F_{N} is the normal force in the contact area, i.e. The weight of the block

With a non smooth transition, the resulting motion also shows a non smooth behaviour. The stick-slip systems belong to the class of non smooth systems, such as systems with stops, impacts, backlash or hysteresis. The nature of dynamic friction forces developed between bodies in contact is extremely complex and affected by a long list of factors: the constitution of the interface, the time scales and the frequency of the contact, the response of the interface to normal forces, inertia and thermal effects [

where a_{1}and a_{3}are friction coefficient, and

The relative velocity of the contact can be defined as:

The significant role of various dry friction laws in engineering sciences can be illustrated in

The equation of motion Equation (2) can be normalized using:

where τ is dimensionless time coordinate, w_{0} is the frequency of free oscillations of the mass.

Equation (2) can be rewritten then in a dimensionless form as:

The choice of one of the friction laws shown in _{3} (dimensionless friction coefficient).

In the case where b_{1} = b_{3} = 0, one can treat Equation (7) analytically to show amplitude motion which occurs in the sliding phase. Assume that the motion starts when the block sticks on the belt,

where

With the initial conditions

It is noted that the upper and lower part of the compound signs in Equation (8) corresponds to the case of _{1} and c_{2} are constants.

An important idea associated with the response of an oscillator to the periodic force is resonance i.e., when the natural frequency of the oscillator is equal to the frequency of the periodic force.

We further our study, in order to know how the response of the system is affected by the two harmonic driving forces. Numerical investigation will be done firstly, by using Coulomb’s law and secondly with numerical investigation by assuming two others particular friction laws as mention above.

In order to understand dynamical processes in our model, Equation (7) has been integrated numerically using fourth-order Runge-Kutta scheme with time step Δt = 0.04. The results of this numerical investigation are shown in Figures 4-7, in which we obtained unless otherwise specified [_{v} = 1,

Through a series of numerical simulations, it seems that such excitation forces can be used to conveniently reproduce the stick-slip behaviour of ice streams in glaciology [

types of modulation: simple wave modulation and stick-slip modulation. Acronyms like “WM” and “SSM” stand for “wave modulated” and “stick-slip modulated”, respectively. Depending on the amplitude u_{0} and the ratio of the two frequencies

a) Wave modulated: we observed two types; wave modulated 1 (WM_{1}) which corresponds to the angular modulated wave and wave modulated 2 (WM_{2}) which refers to amplitude modulated. _{0} = 0.2 for WM_{1} and_{0} = 0.05 for WM_{2}. The Block follows merely the dynamics of the excitation force F_{e}(t). The two driving forces predominate the dynamics of the system and the effect of dry friction is not really observed.

b) Stick-slip modulated: here the dynamics of the system presents modulated motion with stick phase. We observed two types regarding the form of the modulation of excitation force. _{1}) which corresponds to the angular modulated stick-slip with the corresponding time velocity obtained with the set parameter n = 0.1, u_{0} = 1.15 and n = 0.2, u_{0} = 3.05. At the stick phase, the velocity remains constant and equal to the driving velocity V_{v} = 1, but the amplitudes of the block velocity during slipping are modulated. The stick phase is significantly presents here because of the effect of dry friction which predominated the dynamics of the system in comparison with the first case where the excitation forces influence significantly.

In _{2}) which corresponds to the amplitude stick-slip, this phenomenon occur for the set parameter n = 1.15, u_{0} = 0.7.

c) Periodic stick-slip and periodic oscillation: regarding the form of the excitation force, periodic stick slip occurs with integer values range of n. As n becomes an integer multiple, n, of the forcing ratio of the excitation frequency, the system has gone into periodic oscillation. The force degenerates to a single excitation case for n = 1. _{0} = 1.65, n = 1.0.

As an aid in describing and understanding nonlinear systems, someone can introduce maps in this section. In order to develop a meaningful understanding of friction experiments, and to predict dynamic system response and performance, an influence of the friction model must be studied. Although models predict well-defined stick-slip frequencies, intervals between successive stick-slip events have relatively broad distributions. For a better understanding of the map, the x-axis represents the amplitude of the applied u₀ response while, the y-axis represents the excitation frequency n. The first curve indicated the boundary between the Wave Modulated(WM) motion and the Stick-Slip Modulated (SSM), and the second curve indicated the limit between the end of (SSM) motion and the continue intermittent (WM). From the parameter maps, it is observed that specific motion lies on the special region of the parameters.

_{0} plane), which presents regions of the ratio excitation frequency that correspond to different regimes of the motion of the block. Wave modulated 1 (WM_{1}) and

(a) (b)

(c) (d)

Figures 8. Parameters map of (u_{0}; n) using Coulomb law: (b_{1} = 0.0; b_{3} = 0.0; λ= 1/2; V_{v} = 1): (a) η = 1/9; (b) η = 1/5; (c) η = 5/3; (d) η = 7/3.

Wave modulated 2 (WM_{2}) motion occur, respectively, to the left and right at the first and second line (with n < 1) and to the left-right of the first-second line (with n > 1). The system exhibits stick-slip modulated (SSM) motion in the range of parameters between these two curves. The two lines describe the n dependence of the applied u₀ corresponding to the transitions between different states of motion. The parameter η gives the orientation of these two curves. For instance, in

As an interesting details, one can observe that if w_{1} smaller than the natural frequency w₀, the bandwidth became bigger as n increase and for weakest values of η, the stick-slip modulated begin for very large values of u₀, the response to a periodic oscillation is described in all figures in parameter maps for integer ratio n outcast stick-slip band. The bright reflecting type points indicated the stick-slip modulated inside stick-slip band and the dash one represent the periodic stick-slip. For larger values of the excitation amplitude depending on the excitation frequency, the friction oscillator shows periodic motions modulated.

When n is very weak, we have u₀ very large in

(a) (b)

(c) (d)

Figures 9. Parameters map of (u_{0}; n) using Coulomb + viscosity: (b_{1} = 0.2; b_{3} = 0.0; λ= 1/2; V_{v} = 1): (a) η = 1/9; (b) η = 5/17; (c) η = 5/3; (d) η = 7/3.

curves are lying to the right of the diagram. As the friction force increases, the stick motion is suppressed in _{0} in the ratio η so that the big values of η diminish significantly the bandwidth of stick-slip. Using this parameter map the system behaviour can be characterized for any set of parameters within the plotted range.

Using the same parameters system as in _{1} is mostly nil or outside the n− u_{0} plane. If n ∊ [_{0} if the frequency w₁ is less than w₀. The features of high amplitude and high n-factor are clearly recognized.

In the case of Stribeck friction,_{1} greater than w_{0},

another parameter map showing the influence of the excitation amplitude u_{0} is represented and the stick-slip phenomenon disappear partially in

The first main purpose of this paper was to investigate the influence of the frequencies, in external excitation with two harmonic driving forces in the case of a moving base. The second main purpose was to show the influence of various friction characteristics and parameter maps on the system response. Numerical and analytical predictions of the scenarios varying with the ratios of the two excitation frequencies and amplitudes are carried out, and the parameter maps for specific motions are presented. This study suggests that, the transition between each of the motions strongly depends on the parameters (frequencies and amplitudes) of the two driving forces; this was appreciated in the analytical result. The applying two driving forces were also found to give rise to stick-slip modulation. We observed the amplitude modulation of the velocity during slip phase. These and other results contribute to the general understanding of how friction properties may change under the action of the two vibrating forces. If two harmonics of almost periodic excitation are interacted, modulated motions take place. But we must control the ratio η to control the dynamic of the system. In all mechanical system, Stick-Slip motion is considered as harmful effects. So that too many researches are concentrated to diminish amplitude of vibration which can directly affect stresses and thus the life of the system and our results have potentially an equally wide range of applications in engineering.