Links in this page: (The Skeleton method in Matlab) (The Original Maxwell-Slip applet) (The GMS applet) (Equivalent Dynamic Parameters applet)
Links in this page: (The Skeleton method in Matlab) (The Original Maxwell-Slip applet) (The GMS applet) (Equivalent Dynamic Parameters applet)
I. Backlash Identification
The Frequency Response Function (FRF) method using an experimental analysis such as free vibration with shock excitation or forced vibration with step or chirp excitation has proven to be a most efficient way to identify the modal parameters of mechanical structures. However, there is a limitation that only linear dynamic systems can be tested through these methods. The problem becomes more complex when nonlinear systems have to be identified. If the nonlinear system is ‘well-behaved’, i.e. if it shows periodic response to a periodic excitation, ‘skeleton’ identification techniques may be used to estimate the modal parameters, in function of the amplitude and frequency of excitation. However, under certain excitation conditions, chaotic behaviour might occur so that the response is aperiodic. In that case, chaos quantification techniques, such as Lyapunov exponent, are proposed in the literature. This paper deals with the application of the aforementioned nonlinear identification techniques to an experimental mechanical system with backlash. It compares and contrasts Hilbert transforms with Wavelet analysis in case of skeleton identification showing their possibilities and limitations. Chaotic response, which appears under certain excitation conditions and could be used as backlash signature, is dealt with both by a simulation study and by experimental signal analysis after application of appropriate filtration techniques.
In the first place, this part of the research study the application and implementation of the skeleton method in the mechanical system comprising a backlash component. This research also offers an improvement on the estimation of the instantaneous frequency and amplitude of the input and output signal as well by implementing the Wavelet analysis, and it gives promising results. An illustration of this simulation technique (in Simulink and Matlab v.7) can be downloaded here.
Secondly, this research proposed a correlation between a chaos quantification to a nonlinear parameter in a mechanical system, in particular backlash element, if the corresponding system gives nonperiodic responses for periodic input, namely chaotic response. In short, although quite difficult to perform in practice, chaos quantification could be used as a quantitative mechanical signature of a backlash component.
II. Friction Identification
The nonlinear dependence of friction shows a very complex behavior. The most recent model of friction, namely the Generalized Maxwell-Slip friction [1] has been proven to be able to capture the unique behavior of the friction satisfactorily. This research includes the study of an efficient and accurate method for identifying the parameters of the model and also a new contol approach for compensating friction.
Friction models
Friction in presliding regime, as can be represented by N parallel connection of the original Maxwell-Slip (link contains Java applet) elements, appears as a hysteretic function of the displacement only. In particular, the function is not only depending on the displacement at one time instant but also on the history of the input (displacement) and output (friction force) as well.
At the beginning, when an element subjected on this hysteretic spring moves, the force follows a certain even function of displacement so called a virgin curve y(x) (see figure below). If the motion is reversed after some time instance (let say at xm), the trajectory of the friction force follows the double-stretched profile of the virgin curve (see the path from 1-2). If the motion is reversed again (after it arrives at point 2), the trajectory will follow the same double-stretched profile of the virgin curve (see the path from 2-3), and the trajectory will behave in the same way if the motion is reversed again. But, on the contrary, if the motion is not reversed after it arrives in the previous reversal point (1) from point 2, instead of following the continuation of the double-stretched virgin curve from point 2 to 3, it will follow the continuation of the original virgin curve y(x). This unique behavior can also be seen when the motion is reversed at point 3 and we keep the movement to the left after it arrives at point 4. Instead of following the dashed-line as the continuation of the path 3-4, it will follow the previous trajectory from 4 to 5.
This behavior of friction in presliding regime also appears in the gross-sliding friction of rolling elements.
Each Maxwell-Slip element behaves as a linear spring at the beginning until it reaches a maximum force it can afford.
Beyond this maximum limit, the force of this element will become constant in displacement, until the motion is reversed.
When it reverses, the element will behave as a linear spring again until it reaches the maximum force at the opposite direction.
However, when the friction force appears in two contacting surface (dry friction), at sliding regime, it will not behave as a friction in rolling element. In other words, at sliding regime, the friction behavior in presliding regime will not persist. When the motion is 'far enough' from the reversal point, instead of becoming a constant, the friction will appear as a function of the velocity (state-rate) of the motion, as can be seen in the figure below.
The Generalized Maxwell-Slip (GMS) (link contains Java applet) merges two separate models of the friction in both regimes in order to have a complete and compact model. The GMS friction is a qualitatively new formulation of the rate-state approach of the LuGre and the Leuven models. The GMS model retains the original Maxwell-slip model structure, which is a parallel connection of different elementary slip-blocks and springs, but replaces the simple Coulomb law governing each block, by another state equation to account for sliding dynamics. Thus, the friction force is given as the summation of the outputs of the N elementary state models.
Friction identification
Two sets of points containing the friction and displacement data were collected for identification purpose. The first set were used as a training and the other were used as testing set. As a rough estimation of the parameters in the model, a Genetic Algorithm strategy was used. In order to refine the estimated parameters, the Nelder-Mead Simplex algorithm was then be used.
The conclusion shows that friction identification using a single experiment is possible to conduct. Friction identification utilizing the most recent GMS model, which incorporates two regimes of friction, was also possible to conduct using a single experiment. However, selection of the excitation signal plays an important role for identification using single experiment.
Friction compensation
Despite of its complexity, the equivalent dynamic parameters of the system consisting of a mass subjected to the hysteretic frictional forces can be formulated. By using a classical approach of describing function to linearize the system, the formulation is deducted. This method replaces the nonlinear element by an element, which has a monotonic output at the fundamental frequency component if the nonlinear element is excited by a sinusoidal input. Splitting the equation of the hysteresis friction force for different direction of motion and assuming that the hysteretic spring is excited by x = A cos q, the equivalent dynamic parameters (link contains Java applet) can be formulated as follows [2]:
where ke is the equivalent stiffness, ce is the equivalent damping, w is the frequency excitation and A is the amplitude of the motion.
This research proposed and study a gain scheduling controller technique based on the corresponding equivalent dynamic properties. This controller consists of two main control modes. The first mode is dealing with the gross-sliding friction when the distance to the desired position is larger than the sticking distance. In this mode, the system uses a linear compensation scheme together with equivalent Coulomb friction compensation. When the motion is reversed and when the distance to the desired position is within the sticking limit, a second mode of the controller is activated until the motion leaves the sticking region. In this mode, the controller uses scheduled gains, which are designed based on the equivalent dynamic properties. The controller scheme is shown below:
As a conclusion of this research, the dynamic parameters based gain scheduling controller is proven to be an effective way to deal with the system with frictional force, in particular for the case of point-to-point motion. It offers faster response without sacrificing the steady state error. However, for the gain scheduling controller, it has to be mentioned that high sensitivity of the position sensor is an essential requisite in this controller strategy, and the application of a fine velocity sensor in the output side is expected to improve the result. Less sensitivity of the position sensor requires narrower band pass filter, which causes slower response of the controller.
III. Harmonic Drive Modelling
Invented by Walton Musser in 1955, primarily for aerospace applications, harmonic drives are high-ratio, compact torque transmission systems. As shown in the figure below, this nascent mechanical transmission, occasionally labeled ‘strain-wave gearing’, employs a continuous deflection wave along a non-rigid gear, the so-called ‘flexspline’ (the red part in the figure), to allow gradual engagement of gear teeth. Besides a thin-walled flexible cap with small external gear of the flexspline, a harmonic drive also contains two other important components, namely a wave-generator (the yellow part), which is a ball-bearing assembly with a rigid elliptical inner-race, and a circular-spline (the blue part), a rigid ring with internal teeth machined along a slightly larger pitch diameter than that of the flexspline. When properly assembled, the wave-generator is nested inside the flexspline, causing the flexible gear-tooth circumference on the flexspline to adopt the elliptical profile of the wave-generator. While the wave-generator is rotated, the engagement of the external teeth of the flexspline to the internal teeth of the circular spline will cause highly reduced rotation of the circular spline. Through this unconventional mechanism, gear ratios up to 500:1 can be achieved in a single transmission step.
Under ideal assumptions, a harmonic drive transmission is treated as a perfectly rigid gear reduction. However, due to the relatively low torsional stiffness of harmonic drives, a more detailed understanding of the transmission flexibility is often required for accurate modeling. As described in a manufacturer’s catalog [3], the typical shape of the stiffness curve consists of two characteristic properties: increasing stiffness with displacement and hysteresis loss. To capture this nonlinear stiffness behaviour, the manufacturers suggest using piecewise linear approximations.
Apparently, from this research, it is found that the shape of the hysteresis in the torsional stiffness, can be attributed to the friction behaviour in microscopic relative motion between teeth contact area, and then can be modeled as a parallel connection of the original Maxwell-slip elements. Therefore, the overall torsional stiffness model of the harmonic drive is represented as a parallel connection between a piecewise linear and Maxwell-slip elements.
IV. Planar and Rotary Piezoactuator Drives
Piezoactuators are gaining popularity due to their precise positioning ability. However, piezoelectric actuators can only deliver a limited stroke. Some linear piezoactuator drives are manufactured and available nowadays originated from the inchworm (travelling wave) principle, which is based on the simple concept of incrementally summing the relatively small displacements produced by three piezoelectric ceramics that arranged together to allow a worm-like motion. By designing a special structure, the stroke can be amplified and manipulated to deliver unlimited (planar and rotary) motions. Video
V. Route Planning Optimization
Route planning sometimes plays a crucial factor for enchancing the operational efficiency. A Travelling Salesman Problem (TSP) is a typical problem, where the route has to be optimized in order to achieve the best cost function value. This optimization problem can be solved with many algorithms. Among those, the Genetic Algorithm (GA) appears to be one of the potential candidates. However, in some cases, such as a flight planning problem, GA cannot be applied straightforward. A number of chromosome - in this case is the waypoint to visit - is not fixed. This research focuses on the development of the 'aneuploidic' genetic algorithm.
Aneuploidy, litterally, refers to a type of chromosome abnormality that cause an extra or missing chromosome. Some cancer cells sometime also have abnormal number of chromosomes.
References