New limit cycles of dry friction oscillators under harmonic load

We consider a system composed of two masses connected by linear springs. One of the masses is in contact with a driving belt moving at a constant velocity. Friction force, with Coulomb’s characteristics, acts between the mass and the belt. Moreover, the mass is also subjected to a harmonic external force. Several periodic orbits including stick phases and slip phases are obtained. In particular, the existence of periodic orbits including a part where the mass in contact with the belt moves in the same direction at a higher speed than the belt itself is proved. Non-sticking orbits are also found for a non-moving belt. We prove that this kind of solution is symmetric in space and in time.


Introduction
This paper is a continuation of several investigations [5][6][7], related to vibrating systems excited by dry friction. These systems are frequently encountered in many industrial applications. One of the most popular models of stick-slip oscillators consists of several masses connected by linear springs, one (or more) of the masses is in contact with a driving belt moving at a constant velocity. In the past, several authors investigated the behavior of this system, with different friction laws and with or without external actions and damping [1,4], mainly via numerical approach.
However, assuming Coulomb's laws of dry friction, the corresponding dynamical model is a piecewise linear system, and even for multi-degree-of-freedom cases, some analytical results about the existence and the stability of periodic orbits including stick-slip phases have been obtained [5][6][7].
One interesting phenomenon is the existence, inside periodic orbits with stick and slip parts, of an "overshooting" slip phase. During this part of the orbit, the mass in contact with the belt moves in the same direction at a higher speed than the belt itself.
Up to now [9], this phenomenon has been observed only for more complex friction characteristics than Coulomb's ones. In [7], a self-excited stick-slip oscillator including two degrees of freedom is considered. Assuming Coulomb's laws of dry friction, a set of periodic orbits including an overshooting part is obtained using analytical methods.
In this work, we consider the same model of dry friction oscillator subjected to a harmonic external force. Several periodic orbits containing stick phases and slip phases are obtained. In particular, the existence of periodic orbits including an overshooting part is proved. The system (Fig. 1) is composed of two masses m 1 , m 2 connected by two linear springs of stiffness k 1 , k 2 . The second mass is in contact with a belt moving at a constant velocity v 0 . A friction forceF acts between the mass and the belt. Moreover, the second mass is also subjected to an external forceR given byR =p cos ωt + ϕ (p,ω, ϕ are constant parameters) (1) The equations of motion related to this system are written as x 1 , x 2 are the displacements of the masses, The dry friction force u is obtained from Coulomb's laws: u r is the friction force at rest (sticking), u s is the slipping friction force.

Prediction of the oscillations exhibited by the system
The dynamical behavior of this oscillator includes several phases of slip and stick motion of m 2 . For each kind of motion, a close form solution is available.

Slip motion of m 2 with x 2 < V
The solution is obtained from a modal analysis of (2) where u = u s The two by two matrices H i (t) (i = 1, 2, 3) and the natural frequencies (ω 1 , ω 2 ) are obtained in analytical form (see Appendix 1).

Slip motion of
The solution is obtained from (2) where u = −u s This motion is related to the following dynamical system: The solution [5] is given by The two by two matrices Γ i (t) (i = 1, 2, 3) are given in Appendix 1. Moreover, during all this kind of motion, the following constraint holds:

Symmetrical periodic solutions
A first set of periodic orbits of period Θ = 2π/ω is obtained. These motions involve for each period first a slip motion of m 2 with x 2 < V followed by a phase of stick motion of the mass (x 2 = V ). At the beginning (t = 0) of the slip motion, the initial conditions are given by For 0 < t < τ , the system undergoes a slip motion defined by (5). At the end (t = τ ) of the slip motion, the following condition is assumed: The time lag ϕ of the external force is given by ϕ = (π − ωτ )/2. From (12) and (13), we deduce: For τ < t < τ + T , the system motion is a sticking motion given by A periodic solution of period Θ is obtained if the following relation is fulfilled: Taking into account the properties (Appendix 1) of the matrices H (t), Γ (t), the conditions for the existence of this periodic solution are given by the following system of four scalar equations: τ and hence the time duration T = Θ − τ of the stick motion, together with the initial conditions z(0), z (0) are computed. As in the case of the self-excited dry friction oscillator considered in [5], an interesting property of symmetry is proved for these orbits (see Appendix 2): A numerical validation is made for the following set of data: The other parameters related to this orbit are computed: The constraints derived from (11) during the stick motion:

Periodic orbits including an overshooting part
A second set of periodic orbits of period Θ = 2π/ω is obtained. For each period, the motion is composed of three parts. The first one is a slip motion of m 2 with x 2 < V for 0 < t < τ ; the next part (0 < t − τ < τ 1 ) is an overshooting slip motion of the mass ( At the beginning of the motion for t = 0, the conditions (12) are fulfilled. If at t = τ , instead of (13), the following conditions: + η(u r + u s ) < 0 are fulfilled, we get an overshooting motion for t > τ . This motion ends at t = τ + τ 1 if, at this time: For τ + τ 1 < t < τ + τ 1 + T , the system undergoes a sticking motion. A periodic solution of period τ + τ 1 + T = Θ is obtained if: Taking into account the constraints deduced from (18), (19) and (20), the solution is defined by five linear equations with respect to (z 10 , z 20 , z 10 ): 1, 2, 3), the values of (z 10 , z 20 , z 10 ) are obtained: The parameters (τ, τ 1 ) are the roots of the compatibility conditions: Assuming that (χ, η, V , u s ) are given data, u r is deduced from the relation:

Non-sticking periodic solutions
In industrial applications, avoiding sticking phases of motion is sometimes necessary. In the past, several authors [2,3,8] investigated the existence of periodic non-sticking solutions of a one-degree-of-freedom oscillator subjected to simple harmonic loading. The mass is in contact with a fixed surface and a dry friction force acts between the mass and the surface. The aim of these works is to obtain some estimates about the minimum external force amplitude needed to prevent this sticking motion. The non-sticking orbit involves for each period a slipping motion with a negative mass velocity, and a slipping motion with a positive mass velocity (overshooting motion). Moreover, the authors assumed that the motion is symmetric in space [2,3,8] and time [3,8].
In the following, this problem is revisited for the two-degree-of-freedom oscillator considered in this work.
Let us consider the system described in Fig. 1, and assume the following initial conditions: For 0 < t < τ , the system undergoes a slipping motion with z 2 < V , given by (5).
Let us assume that τ = π/ω = Θ/2 and that For t > τ the system undergoes an overshooting slipping motion (z 2 > V ). This motion is given by A periodic motion of period Θ = 2π/ω is obtained if From (28) we deduce: From (29), we obtain the following results are obtained This last condition and the relation z 20 ≡ z 2B = V lead to V = 0.
In conclusion, non-sticking periodic orbits are obtained only for V = 0, and the motion is symmetric in space and time (see Appendix 4).
The initial conditions and the time lag ϕ of the external force are deduced from (33): z 10 = q 1 cos ϕ − d 01 , z 20 = q 2 cos ϕ − d 02 , z 10 = −q 1 ω sin ϕ − a 1 d 01 − a 2 d 02 The constraints deduced from (25) and (26) lead to the same condition: and from (36), a condition about the minimum value of the external force amplitude needed to avoid a sticking motion is obtained: A numerical validation is performed for the following values of the parameters: The corresponding values of the initial conditions and of the time lag ϕ are obtained: x 10 = 1.5608, x 20 = 3.3295,

Conclusion
In this work, the steady state response of a two-degreeof-freedom oscillator subjected to dry friction and harmonic load is considered. Assuming Coulomb's laws of dry friction, the existence of several interesting periodic orbits, including stick and slip phases, is proved. In particular, periodic solutions with a phase during which the mass in contact with the belt moves faster The natural frequencies (ω 1 , ω 2 ) are the roots of the characteristic equation: The eigenvectors ψ j = 1 λ j , (j = 1, 2) are defined by (K − I ω 2 j )ψ j = 0. These matrices fulfil the following property: The matrices Γ i (t) fulfil also the property:

Appendix 2
For τ/2 < t < τ , the periodic solution is defined by From the identities: the first relation (17) is deduced. For T /2 < t 1 < T , the solution is defined by From the identities: the last relation (17) follows.

Appendix 5
Let us consider the same kind of non-sticking periodic orbits as in Sect. 6, with V = 0 but without the assumption τ = /2. For 0 < t < τ , the motion is given by (5), while for τ < t < , the motion is described bȳ