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Helical polymers present some interesting and distinctive properties, and one of the most distinguished applications of them is the chiral recognition and resolution of enantiomers. In this work, star-shaped hybrid helical poly (phenyl isocyanide) (PPI) with polyhedral oligomeric silsesquioxanes (POSS) as the core was designed and synthesized by "grafting to" strategy. The homoarm star-shaped hybrid POSS-(PPI)8 was first obtained by the click reaction between azide-modified POSS (POSS-(N3 )8 ) and alkynyl-modified PPI (PPI-Alkynyl). The hybrid POSS-(PPI)8 was with predominated left-handed helical conformation and exhibited excellent ability in the enantioselective crystallization of racemic compounds. In the meantime, heteroarm star-shaped hybrid (PEG)4 -POSS-(PPI)4 was prepared by the click reaction of POSS-(N3 )8 with PPI-Alkynyl and alkynyl-modified poly (ethylene glycol) (PEG-Alkynyl). The hybrid (PEG)4 -POSS-(PPI)4 was amphiphilic, and it could self-assemble to form spherical micelles in aqueous solutions.

Helical polymers have attracted a great deal of attention and been extensively investigated due to their various applications. One of the most important applications of helical polymers is chiral recognition and resolution of enantiomers for the reason that a pair of enantiomers is commonly with different physiological and toxicological behaviors in biological systems. Helical polymers usually present unexpected high chiral recognition ability to a variety of racemic compounds. What’s more, the chiral recognition and resolution abilities of the system are dependent on the highly ordered helical structures of the helical polymers. This mini review mainly focuses on the recent progress in chiral recognition and resolution based on helical polymers. The synthetic methodology for helical polymers is firstly discussed briefly. Then recent advances of chiral recognition and resolution systems based on helical polymers, especially polyacetylenes and polyisocyanides, are described. We hope this mini review will inspire more interest in developing helical polymers and encourage further advances in chiral-related disciplines.

Polyhedral oligomeric silsesquioxane (POSS) based starlike hybrid helical poly(phenyl isocyanide)s (PPIs) with well-defined structures were designed and prepared in this contribution. The POSS modified with Pd(II) complexes (POSS-Pd(II)) was firstly obtained. Then well-defined starlike hybrid helical PPIs with POSS cores were synthesized through the polymerization of phenyl isocyanide monomers using POSS-Pd(II) as the initiator. The polymerization proceeded in a living and controlled manner. Meanwhile, polymerization of chiral phenyl isocyanides with decyl chain modified pendant L- or D-alanine afforded optically active hybrid starlike helical PPIs with preferred handedness. What's more, amphiphilic starlike hybrid helical PPI block copolymers showing thermoresponses were easily obtained, and their self-assembly and thermo-responsive behaviours were studied. In addition, cross-linked hybrid helical PPIs were obtained via the transesterification reaction between pentafluorophenyl ester and cystamine. Benefitting from the essential chirality of the handed helical PPIs, the cross-linked hybrid PPIs showed excellent ability in the enantioselective crystallization of racemic compounds. The enantiomeric excess (ee) of the induced crystal of threonine could be up to 95%.

A dumbbell-shaped copolymer (PDMAEMA)4-PCL-SS-PCL-(PDMAEMA)4 with a lower critical solution temperature (LCST) and redox responses was first synthesized by the combination of ring opening polymerization (ROP), click chemistry and atom transfer radical polymerization (ATRP). Then a dumbbell-shaped copolymer (PDMAPS)4-PCL-SS-PCL-(PDMAPS)4 with an upper critical solution temperature (UCST) and redox responses was obtained through the quaternization reaction between DMAEMA and excess 1,3-propane sultone. Both (PDMAEMA)4-PCL-SS-PCL-(PDMAEMA)4 and (PDMAPS)4-PCL-SS-PCL-(PDMAPS)4 were amphiphilic, and they could self-assemble into spherical micelles under certain conditions. Benefitting from the LCST-type thermoresponse of PDMAEMA and UCST-type thermoresponse of PDMAPS, both dumbbell-shaped copolymers (PDMAEMA)4-PCL-SS-PCL-(PDMAEMA)4 and (PDMAPS)4-PCL-SS-PCL-(PDMAPS)4 presented temperature-responsive properties. The morphology and size of their assemblies would be changed when the temperature is altered. Meanwhile, due to the redox-response of disulfide bonds, both (PDMAEMA)4-PCL-SS-PCL-(PDMAEMA)4 and (PDMAPS)4-PCL-SS-PCL-(PDMAPS)4 showed redox-responsive properties. When DL-dithiothreitol (DTT) was added into the systems, disulfide bonds were broken, causing the change of (PDMAEMA)4-PCL-SS-PCL-(PDMAEMA)4 to PCL-(PDMAEMA)4, while (PDMAPS)4-PCL-SS-PCL-(PDMAPS)4 changed into PCL-(PDMAPS)4. The distributions of Rh for both (PDMAEMA)4-PCL-SS-PCL-(PDMAEMA)4 and (PDMAPS)4-PCL-SS-PCL-(PDMAPS)4 assemblies broadened with asymmetric peaks, giving rise to curves with double peaks. After 24 h, DLS curves became symmetric with a single peak, and the Rh was about half that of the original assemblies. In the meantime, dispersed spherical micelles could be observed by TEM.

An improved component modal synthesis-based nonproportional mistuning method (ICMS-NPMM) is proposed to investigate mistuned turbine blisks (MTBs) since the high-fidelity finite element models (HFEMs) involve large number of computations, which leads to low calculation efficiency. To reduce degrees of freedom and suppress the flutter of MTB, it is divided into mistuned blade structure and tuned disk structure, and the intentional mistuning is considered. Furthermore, the mistuned parameters, nonproportional mistuning, and complex loads are also considered. Firstly, the basic theory of ICMS-NPMM is investigated; secondly, the model of MTB is established via ICMS-NPMM; finally, the intentionally mistuned design of modal shape amplitudes (MSAs) is investigated via ICMS-NPMM. The results indicate that the calculation efficiency is enhanced via ICMS-NPMM relative to that of via HFEM. In addition, the sensitivity and the flutter are decreased; meanwhile, the amplitude fluctuations of MSAs are distinctly decreased and become comparatively smooth. This investigation provides an important guidance for the vibration characteristic study of complex mechanical structures in engineering practice.

Aeroengine is a complex mechanical equipment, and it works at high temperature, pressure, rotational-speed, and severe loads. One of the core problems is that the vibration and mistuning of bladed disk lead to failure and affect the safety and reliability of aeroengine. Previously, one sector taken as the research object is not suitable; the integrally mistuned bladed disk (blisk) is taken as the research object is very necessary; however, the computational efficiency of mistuned blisk is very low. Therefore, a reduced-order model approach, i.e., an improved dynamic substructure finite element model-based state-space technique (IDSFEM-SST), is proposed to investigate the mistuned blisk. Firstly, the reduced-order substructure finite element model is established by this method, and then, the modal frequencies and modal strain energy amplitudes are investigated. Secondly, the maximum displacement responses are analyzed. Finally, the computational efficiency and accuracy of mistuned blisk via IDSFEM-SST is compared with that of the classical dynamic substructure finite element model and the high-fidelity finite element model to verify the effectiveness of this approach. This study has significance to the dynamic research and engineering practices for complex mechanical structures.

Nonlinear dynamical systems with irrational and fractional nonlinear restoring forces often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational and fractional nonlinear restoring forces avoiding the conventional Taylor's expansion to retain the natural characteristics of the system. By introducing a particular dimensionless representation and a series of transformations, the two-degree-of-freedom system can be transformed into a perturbed Hamiltonian system. The extended Melnikov method is directly used to detect the chaotic threshold of the perturbed system theoretically, which overcomes the barrier caused by solving theoretical solution for the homoclinic orbit of the unperturbed system. The numerical simulations are carried out to demonstrate the complicated dynamics of the nonlinear spring-pendulum system, which show the efficiency of the criteria for chaotic motion in the system.

The smooth-and-discontinuous oscillator under constant excitations (CSD) is an unsymmetrical system with an irrational restoring force, which leads to a barrier to detect the chaotic threshold using conventional nonlinear techniques. The goal of the present work is to overcome the trouble raised by constant excitations and irrational nonlinearity to investigate the necessary and insufficient condition for chaos. A smooth approximate system, whose behaviors are topologically equivalent to the ones of the original system, is proposed to investigate the chaotic boundary. The Hamiltonian system of the approximate system is analyzed and the homoclinic orbits in analytical form are obtained. The Melnikov method is employed to determine the distance between the stable and unstable manifolds under the perturbation of damping and external forcing. Bifurcation diagrams and numerical simulations are used to reveal the motion of chaos and the period for approximate system which is in good agreement with the original system. It is worth recalling that the approach for constructing the approximate function of smoothness is presented, which puts forward a step toward the solution of the irrational nonlinearity system.

The effects of constant excitation on the recently proposed smooth-and-discontinuous (SD) oscillator are investigated, which may lead to the variation of equilibrium and the property of phase portrait. By solving a quartic algebraic equation, the transition set and bifurcation for SD oscillator under constant excitation (CSD) are presented, while the number of equilibria depends on the values of the smoothness parameter and the constant excitation. Complicated structures of Kolmogorov-Arnold-Moser (KAM) structures on the Poincare section are depicted for the driven system without dissipation. Chaotic behaviour is also demonstrated numerically for the system perturbed by both viscous-damping and external excitation. The results show that CSD is an unsymmetrical system that displays different dynamical behaviours from an SD oscillator and will enrich the range of the SD oscillator in research and application.

Nonlinear dynamical systems with an irrational restoring force often occur in both science and engineering, and always lead to a barrier for conventional nonlinear techniques. In this paper, we have investigated the global bifurcations and the chaos directly for a nonlinear system with irrational nonlinearity avoiding the conventional Taylor's expansion to retain the natural characteristics of the system. A series of transformations are proposed to convert the homoclinic orbits of the unperturbed system to the heteroclinic orbits in the new coordinate, which can be transformed back to the analytical expressions of the homoclinic orbits. Melnikov's method is employed to obtain the criteria for chaotic motion, which implies that the existence of homoclinic orbits to chaos arose from the breaking of homoclinic orbits under the perturbation of damping and external forcing. The efficiency of the criteria for chaotic motion obtained in this paper is verified via bifurcation diagrams, Lyapunov exponents, and numerical simulations. It is worthwhile noting that our study is an attempt to make a step toward the solution of the problem proposed by Cao Q J et al. (Cao Q J, Wiercigroch M, Pavlovskaia E E, Thompson J M T and Grebogi C 2008 Phil. Trans. R. Soc. A 366 635).