Vol. 1, No. 1 (2017): Regular Issue

The experimental free vibrations of a laminated glass beam are inves- tigated with the aim of extracting the nonlinear characteristics of the dynamical behaviour, by an appropriate post-processing of data en- suing from the tests. An updated version of the Fitting Time History (FTH) technique is used. It is based on the least square approx- imation of the measured damped free vibrations of the laminated glass, and provides the optimal values of the natural frequencies and damping coefficients. While in a previous work attention was mainly devoted to the determination of the linear dynamical properties, here the focus is on the nonlinear behaviour, in particular on the nonlinear relationship between the excitation amplitude and: (i) the natural frequencies, a fact that is commonly encountered in nonlinear dy- namics and known as ‘backbone curve’; (ii) the damping coefficient, a fact that is somehow unexpected and commonly not reported in the literature.

The workpiece and tool vibrations on a lathe are experimentally stud- ied to establish improved understanding of cutting dynamics that would support efforts in exceeding the current limits of the turning process. A Keyence laser displacement sensor is employed to monitor the workpiece and tool vibrations during chatter-free and chatter cut- ting. A procedure is developed that utilizes instantaneous frequency (IF) to identify the modes related to measurement noise and those innate of the cutting process. It is found that IF thoroughly charac- terizes the underlying turning dynamics and identifies the exact time when chatter is fully developed. That IF provides the needed reso- lution for identifying the onset of chatter suggests that the stability of the process should be monitored in the time-frequency domain to effectively detect and characterize machining instability. It is deter- mined that for the cutting tests performed chatter of the workpiece and tool are associated with the changing of the spectral components and more specifically period-doubling bifurcation.

A comprehensive three-dimensional turning model is validated by comparing numerical simulations to the experimental data obtained in Part 1 using instantaneous frequency. Comparison of chatter-free cutting demonstrates that the model effectively captures the work- piece natural frequency, tool natural frequency, a nonlinear mode, and the spindle speed, which are main components of the underlying dynamics observed experimentally. The model accurately simulates chatter vibrations characterized as increased vibration amplitude and the appearance of coupled tool-workpiece vibrations at a chatter frequency. The stability diagram constructed by running the model at various spindle speeds and depth-of-cuts demonstrates a general increase in the chatter-free critical depth-of-cut as the spindle speed increased. This chatter-free limit levels out as the spindle speeds exceeded 1500 rpm. At high spindle speeds the workpiece motions dominate the cutting dynamics, resulting in cases of excessive work- piece vibration amplitude and highly nonlinear frequencies which af- fect the efficiency of the process. The excessive workpiece amplitude cases create a second stability limit to be considered as a result of imbalance and configuration of the workpiece. Thus, workpiece dy- namics should not be neglected in mathematical and experimental analyses for the design of machine tools and robust cutting control law.

In light of Approximate Inertial Manifolds stemmed from nonlinear dynamics and Mode Analysis Technique in linear structural dynam- ics, a combined method is presented to reduce the second order in time nonlinear dissipative autonomous dynamic systems with higher dimension or many degrees-of-freedom to lower dimensional systems, and the ensuing error estimate is investigated theoretically. By this method, the system is studied in the phase space with a distance definition which can describe the distance between the original and reduced systems, and the solution of the original system is then pro- jected onto the complete space spanned by the eigenvectors of the linear operator of the governing equations. With the introduction of an Approximate Inertial Manifolds, the interaction between lower and higher modes or the influence of higher modes on the long-term behaviors, which will be ignored if the traditional Galerkin proce- dure is used to approach the governing equation, is considered to improve the distance between the original and reduced systems. Ad- ditionally, the error estimate for the approximation to the attractor is presented, and an explicit iterative scheme is then proposed to ap- proach the Approximate Inertial Manifolds. Finally, a comparison between the traditional Galerkin method and the method presented has been given and discussed. The results show that the method presented can provide a better and acceptable approximation to the long-term behaviors of the second order in time nonlinear dissipative autonomous dynamic systems with many degrees-of-freedom, espe- cially for the numerical analysis of complex bifurcation and chaos in complicated dynamical systems.

To control a multi-joints bipedal robot, it is necessary to choose a proper law of control that ensures the safe and stable motions of the robot. Even for the simplest postures, such as sitting and stand- ing, there are constraints to respect so that the inverse geometric model (MGI) would not have in nitive solutions. In this paper, a particular task of sitting-to-standing is simulated for a bipedal robot using MATLAB. The articulation evolution and the variation of the shoulder position are evaluated using typical control and hierarchical redundant control in which the cases of with and without constraints are both considered. The simulation results show that the hierarchi- cal redundant control has a better effect than the typical one, and when considering the constraints, the articulations' angular velocities vary more smoothly so that the motor is better protected.

In this paper, time-delay effects on periodic motions in a periodi- cally forced, time-delayed, hardening Duffing oscillator are discussed. One often considers the time-delay interval is very small compared to the oscillation period. In engineering application, the time-delay interval is often very large. Bifurcation trees of periodic motions to chaos varying with time-delay are presented for such a time-delayed, Duffing oscillator. Using the nite discrete Fourier series, harmonic amplitude varying with time-delay for stable and unstable solutions of period-1 to period-4 motions are developed. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, time-delay effects on period-1 motions to chaos in non- linear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes.