Periodic motions of coupled oscillators excited by dry friction and harmonic force

International audienceVibrating systems excited by dry friction are frequently encountered in technical applications. These systems are strongly nonlinear, and they are usually modeled as spring-mass oscillators. 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. In this work, we consider a system composed of two masses connected by linear springs. One of the mass 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, it is assumed that the mass is also subjected to a harmonic external force. Several periodic orbits including stick phases and slip phases are obtained in closed form. In particular, the existence of periodic orbits including an overshooting part is proved. In the case of a nonmoving belt, a set of nonsticking periodic solutions is obtained, and we prove that these orbits are symmetrical in space and in time

Sensory uncertainty and stick balancing at the fingertip

The effects of sensory input uncertainty, ε , on the stability of time-delayed human motor control are investigated by calculating the minimum stick length, ℓcrit , that can be stabilized in the inverted position for a given time delay, τ . Five control strategies often discussed in the context of human motor control are examined: three time-invariant controllers [proportional–derivative, proportional–derivative–acceleration (PDA), model predictive (MP) controllers] and two time-varying controllers [act-and-wait (AAW) and intermittent predictive controllers]. The uncertainties of the sensory input are modeled as a multiplicative term in the system output. Estimates based on the variability of neural spike trains and neural population responses suggest that ε≈7 –13 %. It is found that for this range of uncertainty, a tapped delay-line type of MP controller is the most robust controller. In particular, this controller can stabilize inverted sticks of the length balanced by expert stick balancers (0.25–0.5 m when τ≈0.08 s). However, a PDA controller becomes more effective when ε>15% . A comparison between ℓcrit for human stick balancing at the fingertip and balancing on the rubberized surface of a table tennis racket suggest that friction likely plays a role in balance control. Measurements of ℓcrit,τ , and a variability of the fluctuations in the vertical displacement angle, an estimate of ε , may make it possible to study the changes in control strategy as motor skill develops