Forward and inverse dynamics of nonholonomic mechanical systems

This paper deals with the forward and the inverse dynamic problems of mechanical systems subjected to nonholonomic constraints. The intrinsically dual nature of these two problems is identified and utilised to develop a systematic approach to formulate and solve them according to an unified framework. The proposed methodology is based on the fundamental equations of constrained motion which derive from Gauss’s principle of least constraint. The main advantage arising from using the fundamental equations of constrained motion is that they represent an effective method capable to derive the generalised acceleration of a mechanical system, constrained in general by a set of nonholonomic constraints, together with the generalized constraint forces (forward dynamics). When the constraint equations are used to represent the desired behaviour of the mechanical system under study, the generalised constraint forces deriving from the fundamental equations of constrained motion provide the control actions which reproduce the specified motion for the system (inverse dynamics). This approach is systematically extended to underactuated mechanical systems introducing a new method named underactuation equivalence principle. The underactuation equivalence principle is founded on the key idea that the underactuation property of a mechanical system can be mathematically represented using a particular set of nonholonomic constraint equations. Two simple case-studies are reported to exemplify the proposed methodology. In the first case-study the computation of the generalised constraint forces relative to the revolute joint constraints of a physical pendulum is illustrated. In the second case-study the calculation of the control action which solves the swing-up problem for an inverted pendulum is described.     

Copolymerization of ethylene with isoprene promoted by titanium complexes containing a tetradentate [OSSO]‐type bis(phenolato) ligand

Copolymerization of ethylene with isoprene (IP) catalyzed by 1,4‐dithabutanediyl‐linked bis(phenolato) titanium complexes 1 and 2 and methylaluminoxane (MAO) produced exclusively ethylene‐IP copolymers with good activity. The copolymer microstructure can be varied by changing the ratio between the monomers in the copolymerization feed, affording copolymers with IP content ～60%. The copolymer microstructure was fully elucidated by ¹³C‐NMR spectroscopy of the copolymers with various IP content revealing a strong tendency to the alternating microstructure. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 4200–4206, 2010     

Copolymerization of ethylene with styrene catalyzed by a linked bis(phenolato) titanium catalyst

Copolymerization of ethylene with styrene, catalyzed by 1,4‐dithiabutanediyl‐linked bis(phenolato) titanium complex and methylaluminoxane, produced exclusively ethylene–styrene copolymers with high activity. Copolymerization parameters were calculated to be r_E = 1.2 for ethylene and r_S = 0.031 for styrene, with r_E r_S = 0.037 indicating preference for alternating copolymerization. The copolymer microstructure can be varied by changing the ratio between the monomers in the copolymerization feed, affording copolymers with styrene content up to 68%. The copolymer microstructure was fully elucidated by ¹³C NMR spectroscopy revealing, in the copolymers with styrene content higher than 50%, the presence of long styrene–styrene homosequences, occasionally interrupted by isolated ethylene units. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 1908–1913, 2006     

Synthesis of branched polyethylene by ethylene homopolymerization using titanium catalysts that contain a bridged bis(phenolate) ligand

The synthesis of branched polyethylene from single ethylene feed has been achieved by using a methylaluminoxane‐activated titanium complex bearing a tetradentate bis(phenolate) ligand with a 1,4‐dithiabutanediyl bridge 1 . This catalyst produces polyethylene with activities up to 6200 kg polymer/mol h bar. As evidenced by ¹³C NMR analyses, the polyethylenes contain ethyl, n‐butyl, and long‐chain (n‐hexyl or longer) branches in a range variable from 0.2 to 2.0%, depending on the experimental parameters. NMR and gas chromatography/mass spectrometry analyses suggest that such polymer microstructure arises from the in situ production of oligomers and their subsequent incorporation into the growing polyethylene chain. The broad molecular weight distribution of these polyethylenes indicates the presence of different catalytic species. The related catalyst system 2 bearing a longer 1,5‐dithiapentanediyl bridge produces linear polyethylene with moderate activity. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2815–2822, 2004     

“Push‐pull” supramolecular chromophores supported on cyclopolymers

Well‐defined macromolecules have been obtained through free‐radical cyclopolymerization and cyclocopolymerization of difunctional and acrylic‐like monomers, which contained “push‐pull” supramolecular chromophores, able to form 1:1 complexes with Eu³⁺ ions in solution. The monomeric molecular modules are built around bismalonate crown ethers in a convergent fashion, in which one of the malonate moiety is derivatized as the ylidene malonate push‐pull fragment, and the other malonate moiety is elaborated to introduce two polymerizable and acrylic‐like substituents. The free‐radical induced cyclopolymerization of these monomers, or their cyclocopolymerization with UV/Vis “silent” but structurally related monomers, afforded macromolecular architectures characterized by GPC, NMR and DSC techniques. UV/Vis titration studies, performed with Eu(OTf)₃ as the supramolecular probe, revealed how adjacent chromophores within the polymeric backbone are virtually independent from each other, and how the binding ability towards the probe of these multivalent, highly packed cyclopolymeric architectures, although reduced, is still clearly detectable. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 5202–5213, 2008     

Bui's path-independent integral in finite elasticity

A path-independent integral has been stated by Bui in the presence of a straight crack in a two-dimensional deformation field. Such an integral isdual to the Rice integral in the sense that it is based on the complementary stress energy density. Here we establish a boundary-independent integral in finite elasticity from which Bui's result follows as a particular case.

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Sommario Un integrale indipendente dal cammino intorno al vertice di una frattura in un campo di deformazione bi-dimensionale è stato stabilito da Bui. Tale integrale èduale all'integrale di Rice, nel senso che si basa sulla densità di energia complementare o degli sforzi. Qui si propone un integrale invariante in un continuo tridimensionale soggetto a deformazioni finite. Si mostra che il risultato di Bui segue come caseo particolare.

     

Mechanical characterization of hyperelastic materials with fringe projection and optimization techniques

This paper presents a new hybrid technique for mechanical characterization of hyperelastic materials. The research is motivated by the fact that standard identification procedures based on the fitting of strain–stress curves determined experimentally from planar biaxial tests may be inaccurate for non-uniform states of deformation. Therefore, we propose an alternative approach where the difference Ω between the displacement field measured with Projection Moiré and its counterpart predicted by FEM is minimized using non-linear optimization algorithms that finally find unknown material properties.In order to check the feasibility of the new procedure, we considered a thin latex membrane modelling it as a two-parameter Mooney–Rivlin (MR) hyperelastic material. The Ω function is minimized either using optimization routines available in a commercial finite element package and by implementing a global optimizer able to deal with non-linearity and non-convexity included in the identification process. In order to check accuracy of optimization results, target values of MR constants for the latex specimen tested have previously been determined by fitting experimental stress–strain data gathered from a standard planar biaxial tension test.Results indicate that the present hybrid identification procedure can determine accurately properties of the hyperelastic material under investigation. In fact, the average residual error on displacements was less than 1% while the difference between the MR constants found with optimization and their target values was less than 3.5%.