The CRONE Suspension: Modelling and Stability Analysis

The CRONE suspension, French acronym of “suspension a` Comportement Robuste d’Ordre Non Entier”, results from a traditional suspension system whose spring and damper are replaced by a mechanical and hydropneumatic system defined by a fractional (so-called non-integer) order force-displacement transfer function. Modelling, frequency-domain robust control design methodology and internal stability analysis are presented in this paper.

Nonlinear Dynamics of One-Way Clutches in Belt-Pulley Systems

One-way clutches are frequently used in the serpentine belt accessory drives of automobiles and heavy vehicles. The clutch plays a role similar to a vibration absorber in order to reduce belt/pulley vibration and noise and increase belt life. This paper analyzes a two-pulley system where the driven pulley has a one-way clutch between the pulley and accessory shaft that engages only for positive relative displacement between these components. The belt is modeled with linear springs that transmit torque from the driving pulley to the accessory pulley. The one-way clutch is modeled as a piecewise linear spring with discontinuous stiffness that separates the driven pulley into two degrees of freedom (DOF). The harmonic balance method (HBM) combined with arc-length continuation is employed to illustrate the nonlinear dynamic behavior of the one-way clutch. HBM with arc-length continuation yields the stable and unstable periodic solutions for given parameters. These solutions are examined across a range of excitation frequencies. The results are confirmed by numerical integration and the widely used bifurcation software AUTO. At the first primary resonance, most of the responses are aperiodic, including quasiperiodic and chaotic solutions. At the second primary resonance, the peak bends to the left with classical softening nonlinearity because clutch disengagement decouples the pulley and the accessory over portions of the response period. The dependence on system parameters such as clutch stiffness, excitation amplitude, and inertia ratio between the pulley and accessory is studied to characterize the nonlinear dynamics across a range of conditions.

Nonlinear Dynamics of a System of Coupled Oscillators With Essential Stiffness Nonlinearities

We study the resonant dynamics of a two-degree-of-freedom system composed a linear oscillator weakly coupled to a strongly nonlinear one, with an essential (nonlinearizable) cubic stiffness nonlinearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (nonlinear normal modes — NNMs), as well as, asynchronous periodic motions (elliptic orbits — EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive nonlinear energy pumping phenomena from the linear to the nonlinear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.

Asperity Spring Friction Model With Application to Belt-Drives

An asperity spring friction model that uses a variable anchor point spring along with a velocity dependent force is presented. The model is incorporated in an explicit timeintegration finite element code. The friction model is used along with a penalty-based normal contact model to simulate the dynamic response of a two-pulley belt-drive system. It is shown that the present friction model accurately captures the stick-slip behavior between the belt and the pulleys using a much larger time-step than a pure velocity-dependent approximate Coulomb friction model.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

On the Generalized Navier-Stokes Equations

In this paper, we present a general Inodel of the classical Navier-Stokes equations. With the help of Laplace, Fourier Sine transforms, finite Fourier Sine transforms, and finite Hankel transforms, an exact solutions for three different special cases have been obtained.

Internal Variable Models of Viscoelasticity With Fractional Relaxation Laws

Internal parameters are used primarily as a numerically convenient surrogate of hereditary formulation of dispersive media. They are however also helpful in combining hereditary and nonlinear effects. Geometrical nonlinearity and the structure of strain invariants imply that nonlinearity has to be contemplated at all levels, including the relaxation equations satisfied by the internal parameters. An obvious application is viscosity of rubber and polymers.

Coupling of an Aircraft Landing Gear Simulation Strategy to Parameter Identification Techniques for Enhanced Fidelity

Brake-induced vibration often involves strong coupling with the aircraft structure, thereby necessitating a system level understanding. However, testing a full system is both costly and impossible until late in the development cycle. To overcome these issues, it becomes necessary to utilize simulation tools to assess the system behavior at an earlier stage. This paper demonstrates a methodology which implements such simulations to guide appropriate brake component tests that are more tractable for the brake supplier. As a result of this combined simulation and testing, a reduction in actual testing can be achieved. However, the ultimate success of the methodology depends on how well the components and the various parameters are estimated. Consequently, means to improve these estimations are also addressed in this paper.

Fractional Order Dynamics in the Trajectory Planning of Redundant and Hyper-Redundant Manipulators

Redundant manipulators have some advantages when compared with classical arms because they allow the trajectory optimization, both on the free space and on the presence of obstacles, and the resolution of singularities. For this type of arms the proposed kinematic control algorithms adopt generalized inverse matrices but, in general, the corresponding trajectory planning schemes show important limitations. Motivated by these problems this paper studies the pseudoinverse-based trajectory planning algorithms, using the theory of fractional calculus.

Rotordynamics Analysis: Experimental and Numerical Investigations

This paper presents a research devoted to the study of vibration problems in turbofan application. Several numerical and experimental tools have been developed. An experimental test rig that simulates the vibrational behavior of a turbofan engine is presented. Moreover, a finite element model is used in order to predict the non-linear dynamic behavior of rotating machines and to predict the first critical speed of engineering machine. A comparison between the experimental tests and the numerical model is conducted in order to evaluate the critical speed of the rotating structure and to update the finite element model.