GEOMETRICALLY NON-LINEAR FREE VIBRATION OF FULLY CLAMPED SYMMETRICALLY LAMINATED RECTANGULAR COMPOSITE PLATES

The geometrically non-linear free vibration of thin composite laminated plates is investigated by using a theoretical model based on Hamilton's principle and spectral analysis previously applied to obtain the non-linear mode shapes and resonance frequencies of thin straight structures, such as beams, plates and shells (Benamar et al. 1991Journal of Sound and Vibration149 , 179-195; 1993, 164, 295-316; 1990 Proceedings of the Fourth International Conference on Recent Advances in Structural Dynamics, Southampton; Moussaoui et al. 2000 Journal of Sound and Vibration232, 917-943 [1-4]). The von Kármán non-linear strain-displacement relationships have been employed. In the formulation, the transverse displacement W of the plate mid-plane has been taken into account and the in-plane displacements U and V have been neglected in the non-linear strain energy expressions. This assumption, quite often made in the literature has been adopted in reference [2] and (El Kadiri et al. 1999 Journal of Sound and Vibration228, 333-358 [5]), in the isotropic case and has been mentioned here because the results obtained have been found to be in very good agreement with those based on the hierarchical finite element method (HFEM). In a previous study, it was assumed, based on the analogy with the isotropic case, that the fundamental carbon fibre reinforced plastic (CFRP) plate non-linear mode shape could be well estimated, by using nine plate functions, obtained as products of clamped-clamped beam functions in the x and y directions, symmetric in both the length U001and width directions [3]. In the present work, a convergence study has been performed and has shown that, although such an assumption may yield a good estimate for the non-linear resonance frequency, 18 plate functions should be taken into account instead of nine in the first non-linear mode shape and associated bending stress patterns calculations. This allows the anisotropy induced by the fibre orientations to be taken into account. Results are given for the fundamental mode of fully clamped CFRP rectangular plates, for various plate aspect ratios and vibration amplitudes. The non-linear mode shows a higher bending stress near the clamps at large deflection, compared with that predicted by linear theory. Some experimental measurements are presented which are in good qualitative agreement with the theory.     

NORMAL MODES OF A NON-LINEAR CLAMPED-CLAMPED BEAM

Non-linear modal analysis approach based on invariant manifold method proposed earlier by Shaw and Pierre (Journal of Sound and Vibration164, 85-124) is utilized here to obtain the non-linear normal modes of a clamped-clamped beam for large amplitude displacements. The results obtained for the fundamental normal mode are compared with the corresponding reported experimental and theoretical studies. The effects of modal coupling are examined in greater detail. The limitation of the present method for analyzing non-linear behavior is highlighted.     

EXISTENCE OF NATURAL FREQUENCIES OF SYSTEMS WITH ARTIFICIAL RESTRAINTS AND THEIR CONVERGENCE IN ASYMPTOTIC MODELLING

A major limitation of the Rayleigh-Ritz method for determining the natural frequencies of a system is the need to choose admissible functions that do not violate the geometric constraints of that system (Courant 1943 Bulletin of the American Mathematical Society49, 1-23). Several researchers have attempted to overcome this problem by asymptotically modelling the rigid constraints with artificial (imaginary) restraints of very large stiffness (Courant 1943Bulletin of the American Mathematical Society49 , 1-23; Warburton and Edney 1984 Journal of Sound and Vibration95, 537-552; Gorman 1989 Journal of Applied Mechanics56, 893-899; Kim et al. 1990 Journal of Sound and Vibration143, 379-394; Yuan and Dickinson 1992 Journal of Sound and Vibration153, 203-216; Yuan and Dickinson 1992 Journal of Sound and Vibration159, 39-55; Cheng and Nicolas 1992 Journal of Sound and Vibration155, 231-247; Yuan and Dickinson 1994Computers and Structures53 , 327-334; Lee and Ng 1994 Applied Acoustics42, 151-163; Amabili and Garziera 1999 Journal of Sound and Vibration224, 519-539; Amabili and Garziera 2000 Journal of Fluids and Structures14, 669-690). While the numerical results thus obtained for the systems considered in the literature were in close agreement with exact values for the natural frequencies corresponding to the first few modes, sample calculations show that the error introduced by the asymptotic modelling increases with mode number and therefore to obtain accurate results for higher modes the magnitude of stiffness should also be increased. In any event, the error due to the asymptotic modelling would remain uncertain, except when the correct frequency values are known. However, the use of artificial restraints with negative stiffness, a new concept which was introduced in a recent publication (Ilanko and Dickinson 1999 Journal of Sound and Vibration219, 370-378) paves the way for estimating the error due to asymptotic modelling. This is possible since in this work, the Rayleigh-Ritz frequencies of the constrained system were found to be bracketed by the frequencies of the asymptotic models with positive and negative restraints. However, the use of artificial restraints with negative stiffness has raised some important questions: would a system with a large negative restraint become unstable, and if so what is the guarantee that the frequencies of the asymptotic model would converge to that of the constrained system? This paper is the result of the author's attempt to answer these questions and gives a proof of existence of natural frequencies for systems with artificial restraints (springs) having positive or negative stiffness coefficients, and their convergence towards constrained systems. Based on Rayleigh's theorem of separation, it has been shown that a vibratory system obtained by the addition of h restraints to an n -degree-of-freedom (d.o.f.) system, where h<n, will have at least (n÷h) natural frequencies and modes and that as the magnitude of the stiffness of the added restraints becomes very large, these (n÷h) natural frequencies will converge to the (n÷h) natural frequencies of a constrained system in which the displacements restrained by the springs are effectively constrained.     

A SEMI-ANALYTICAL APPROACH TO THE NON-LINEAR DYNAMIC RESPONSE PROBLEM OF BEAMS AT LARGE VIBRATION AMPLITUDES, PART II: MULTIMODE APPROACH TO THE STEADY STATE FORCED PERIODIC RESPONSE

The semi-analytical approach to the non-linear dynamic response of beams based on multimode analysis has been presented in Part I of this series of papers (Azrar et al., 1999 Journal of Sound and Vibration224, 183-207 [1]). The mathematical formulation of the problem and single mode analysis have been studied. The objective of this paper is to take advantage of applying this semi-analytical approach to the large amplitude forced vibrations of beams. Various types of excitation forces such as harmonic distributed and concentrated loads are considered. The governing equation of motion is obtained and can be considered as a multi-dimensional form of the Duffing equation. Using the harmonic balance method, the equation of motion is converted into non-linear algebraic form. Techniques of solution based on iterative-incremental procedures are presented. The non-linear frequency and the non-linear modes are determined at large amplitudes of vibration. The basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated. The percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution. Also, the percentage contributions of the higher modes to the bending moment near to the clamps are given, in order to determine accurately the error introduced in the non-linear bending stress estimated by different approximations. Solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency. The non-linear deflection shapes in various regions of the solution, the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes. The numerical results obtained here for the non-linear forced response are compared with those from the linear theory, with available non-linear results, based on various approaches, and with the single mode analysis.     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.     

IDENTIFICATION OF DAMPING: PART 4, ERROR ANALYSIS

In previous papers (S. ADHIKARI and J. WOODHOUSE 2001 Journal of Sound and Vibration243, 43-61; 63-88; S. ADHIKARI and J. WOODHOUSE 2002 Journal of Sound and Vibration251, 477-490) methods were proposed to obtain the coefficient matrix for a viscous damping model or a non-viscous damping model with an exponential relaxation function, from measured complex natural frequencies and modes. In all these works, it has been assumed that exact complex natural frequencies and complex modes are known. In reality, this will not be the case. The purpose of this paper is to analyze the sensitivity of the identified damping matrices to measurement errors. By using numerical and analytical studies it is shown that the proposed methods can indeed be expected to give useful results from moderately noisy data provided a correct damping model is selected for fitting. Indications are also given of what level of noise in the measured modal properties is needed to mask the true physical behaviour.     

IDENTIFICATION OF DAMPING: PART 3, SYMMETRY-PRESERVING METHODS

In two recent papers (Adhikari and Woodhouse 2001 Journal of Sound and Vibration243, 43-61; 63-88), methods were proposed to identify viscous and non-viscous damping models for vibrating systems using measured complex frequencies and mode shapes. In many cases, the identified damping matrix becomes asymmetric, a non-physical result. Methods are presented here to identify damping models which preserve symmetry of the system. Both viscous and non-viscous models are considered. The procedure is based on a constrained error minimization approach and uses only experimentally identified complex modes and complex natural frequencies together with, for the non-viscous model, the mass matrix of the system. The methods are illustrated by numerical examples.     

TOWARDS DEEP AND SIMPLE UNDERSTANDING OF THE TRANSCENDENTAL EIGENPROBLEM OF STRUCTURAL VIBRATIONS

When using exact methods for undamped free vibration problems the generalized linear eigenvalue problem (K−ω²M) D=0 of approximate methods, e.g., finite elements, is replaced by the transcendental eigenvalue problem K (ω) D=0. Here ω is the circular frequency; D is the displacement amplitude vector; M and K are the mass and static stiffness matrices; and K (ω) is the dynamic stiffness matrix, with coefficients which include trigonometric and hyperbolic functions involving ω and mass because elements (for example, bars or beams) are analyzed exactly by solving their governing differential equations. The natural frequencies of this transcendental eigenvalue problem are generally found by the Wittrick-Williams algorithm which gives the number of natural frequencies below ω_t, a trial value of ω, as ∑J_m+s{K (ω_t)} wheres {} denotes the readily computed sign count property of K (ω) and the summation is over the clamped-clamped natural frequencies of all elements of the structure. Understanding the alternative solution forms of the transcendental eigenvalue problem is important both to accelerate convergence to natural frequencies, e.g., by plotting ∣K (ω)∣, and to improve the mode calculations, which lack the complete reliability of natural frequencies obtained by using the Wittrick-Williams algorithm. The three solution forms are: ∣K (ω)∣=0; D=0 with ∣K (ω)∣→∞; and ∣K (ω)∣≠0 with D≠0. The literature covers the first two forms thoroughly but the third form has been almost totally ignored. Therefore, it is now examined thoroughly, principally by analytical studies of simple bar structures and also by confirmatory numerical results for a rigidly jointed plane frame. Although structures are unlikely to have exactly the properties giving this form, it needs to be understood, particularly because ill-conditioning can occur for structures approximating those for which it occurs.     

EXPERIMENTAL STUDY ON THE VORTEX-INDUCED VIBRATION OF TOWED PIPES

We experimentally attempted to understand the vibration characteristics of a flexible pipe excited by vortex shedding. This has been extensively studied in the previous decades (for example, see Sarpkaya 1979 Journal of Applied Mechanics46, 241-258; Price et al. 1989 Eighth International Conference on Offshore Mechanics and Arctic Engineering, The Hague-March 19 -23, 447-454; Yoerger et al. 1991 Journal of Offshore Mechanics and Arctic Engineering, Transaction of Engineers113, 117-127; Grosenbaugh et al. 1991Journal of Offshore Mechanics and Arctic Engineering, Transaction of Engineers113 , 199-204; Brika and Laneville 1992 Journal of Fluid Mechanics250, 481-508; Chakrabarti et al. 1993 Ocean Engineering20, 135-162; Jong 1983 Ph.D. Dissertation, Department of Ocean Engineering, M. I. T.; Kimet al. 1986 Journal of Energy Resources Technology, Transactions of American Society of Mechanical Engineers108, 77-83). However, there are still areas that need more study. One of them is the relation between spatial characteristics of a flow-induced vibrating pipe, such as its length, the distribution of wave number, and frequency responses. A non-linear mechanism between the responses of in-line and cross-flow directions is also an area of interest, if the pipe is relatively long so that structural modal density is reasonably high. In order to investigate such areas, two kinds of instrumented pipe were designed. The instrumented pipes, of which the lengths are equally 6 m, are wound with rubber and silicon tape in different ways, having different vortex-shedding conditions. One has uniform cross-section of diameter of 26·7 mm, and the other has equally spaced four sub-sections, which are composed of different diameters of 75·9, 61·1, 45·6 and 26·7 mm. Both pipes are towed in a water tank (200 m×16 m×7 m) so that they experienced different vortex-shedding excitations. Various measures were obtained from the towing experiment, including frequency responses, the time-domain tracing of in-line and cross-flow responses, and Wigner-Ville distributions. The experimental results analyzed by using these measures exhibit several valuable features. One of them is that the natural frequencies and their corresponding strain mode shapes dominate the strain response of the uniform pipe. However for those of non-uniform pipe, the responses are more likely local and many modes participate in it.     

Performance of a slow positron beam using a hybrid lens design

The University of Hong Kong positron beam employs conventional magnetic field transport to the target, but has a special hybrid lens design around the positron moderator that allows the beam to be focused to millimeter spot sizes at the target. The good focusing capabilities of the beam are made possible by extracting work-function positrons from the moderator in a magnetic field free region using a conventional Soa lens thus minimizing beam canonical angular momentum. An Einzel lens is used to focus the positrons into the magnetic funnel at the end of transportation magnetic field while at the same time bringing up the beam energy to the intermediate value of 7.5 keV. The beam is E × B filtered at this intermediate energy. The final beam energy is obtained by floating the Soa-Einzel system, E × B filter and flight tube, and accelerating the positrons just before the target. External beam steering saddle coils fine tune the position, and the magnetic field around the target chamber is adjusted so as to keep one of the beam foci always on the target. The system is fully computer controlled. Variable energy-Doppler broadened annihilation radiation (VEDBAR) data for a GaN sample are shown which demonstrate the performance of the positron beam system.     

Organic radical molecular solids based on [(TCNQ)n] (n=1 or 2): Syntheses, crystal structures, magnetic properties and DFT analyses

Four molecular solids consisting of the 7,7,8,8-tetracyanoquinodimethane (TCNQ) radical and benzylpyridinium or benzylquinolinium derivatives with molar ratios of 1:1 (1-3) and 2:1 (4) have been prepared and characterized. In the crystals of 1 and 3, TCNQ⁻ monoanions and the corresponding cations form segregated stacks, which are regular in 1 but irregular in 3. Instead of segregated stacks, TCNQ⁻ monoanions in 2 form isolated π-dimers. In the crystals of 4, two crystallographic independent TCNQ species possess almost equal fractional negative charge (ca. −0.5). Two types of TCNQ species form a tetrad, these tetrads make a TCNQ stack with the pattern …BAAB…BAAB… along the crystallographic a-b direction. The magnetisms for 1-4 can be simply explained by the formation of singlet spin state. A broken symmetry approach in a density functional theory framework at the ub3lyp/6-31 g level was used to calculate the magnetic exchange constants in 1-4. The results qualitatively demonstrate the observed magnetic properties.     

Photophysical properties of a series of high luminescent europium complexes with fluorinated ligands

A series of high luminescent europium complexes have been synthesized, such as Eu(TFNB)₃phen (1), Eu(PFNP)₃phen (2), Eu(HFNH)₃phen (3) and Eu(PFND)₃phen (4), which have β-diketone ligands containing fluorinated alkyl chains with different lengths and conjugated naphthyl groups, i.e., 4,4,4-trifluoro-1-(2-naphthyl)butane-1,3-dione (TFNB); 4,4,5,5,5-pentafluoro-1-(2-naphthyl)pentane-1,3-dione (PFNP); 4,4,5,5,6,6,6-heptafluoro-1-(2-naphthyl)hexane-1,3-dione (HFNH) and 4,4,5,5,6,6,7,7,8,8,9,9,10,10,10-pentadecafluoro-1-(2-naphthyl)decane-1,3-dione (PFND). And 10-phenanthroline (phen) is coordinated as the neutral second ligand in 1-4. The crystal structures of 1 and 2 have been studied, which are typical and similar to that of 3. The results of TGA-DTA suggest that these Eu complexes have good thermal stabilities. By means of absorption and (time resolved) emission spectroscopy including determination of luminescence quantum yields, energy transfer dynamics and so on, the following results have been obtained: first, these Eu complexes show characteristic pure red color photoluminescence emission with high quantum efficiencies from the central Eu³⁺ ions through the excitation of the ligands; secondly, photophysical properties of 1, 2, 3 and 4, especially the lifetimes of excited states ⁵D₀ of Eu³⁺ ions and quantum efficiencies are influenced by the different lengths of fluorinated alkyl chains, though the singlets (S₁) and triplets (T₁) of the fluorinated ligands are almost the same.     

A NOVEL APPROACH FOR THE ANALYSIS OF HIGH-FREQUENCY VIBRATIONS

Despite much effort in the past few decades, the numerical prediction of high-frequency vibrations remains a challenging task to the engineering and scientific communities due to the numerical instability of existing computational methods. However, such prediction is of crucial importance to certain problems of pressing practical concern, as pointed out by Langley and Bardell (1998 The Aeronautical Journal102, 287-297). This paper introduces the discrete singular convolution (DSC) algorithm for the prediction and analysis of high-frequency vibration of structures. Both a beam and two-span plates are employed as test examples to demonstrate the capability of the DSC algorithm for high-frequency vibration analysis. A completely independent approach, the Levy method, is employed to provide exact solutions for a cross validation of the proposed method. The reliability of the DSC results is also validated by convergence studies. Remarkably, extremely accurate and stable results are obtained in this work, e.g., the relative DSC errors for the first 7100 modes of the beam and the first 4500 modes of the two-span plates are all <1%. No numerical instability is encountered in the present study.     

Interaction between 1-[p-(dimethylamino) benzoyl]-4′-phenyl-semicarbazide and Cu

A new compound, 1-[p-(dimethylamino)benzoyl]-4′-phenyl-semicarbazide (1) was synthesized and showed highly selective response to Cu²⁺ over other metal ions such as Pb²⁺, Mg²⁺, Fe²⁺, Co²⁺, Zn²⁺, Cd²⁺, Hg²⁺, Ni²⁺, Ca²⁺, Ag⁺, Na⁺, K⁺, and Li⁺. The control compound, 1-[p-(dimethylamino)benzoyl]-4-phenyl-thiosemicarbazide (2), showed different fluorescence spectral response to Cu²⁺. A 1:1 complex between Cu²⁺ and 1 was formed while 1:1 and 1:2 complexes between Cu²⁺ and 2 were formed. The binding model between the receptor (1 or 2) and Cu²⁺ was supported by IR spectra, mass spectra, and computation model. 1 possessed higher selectivity towards Cu²⁺ compared with 2 owing to the difference of complexation ability between urea and thiourea groups.     

The molecular vibration-rotation kinetic-energy operator for general internal coordinates

The molecular kinetic-energy operator for general internal coordinates is formulated in terms of simple generalisations of the matrices A, B, α, and β of Crawford. A new matrix γ of the gradients of the translational and rotational constraints facilitates the calculation of the β matrix. It is shown that the kinetic pseudo-potential U is most conveniently calculated as an atomic sum, and results are given for valence coordinates in various types of molecules.     

Eu(III) and Tb(III) luminescent lanthanide systems derived from DTPA-bisamide and DTTA-based ligands incorporating bipyridine antenna

Four new polycarboxylate ligands H₃Lⁿ have been synthesized by the attachment of two or one 2,2′-bipyridine subunits onto a diethylenetriamine pentacarboxylic acid (DTPA-bisamide derivatives: H₃L¹, H₃L²) or a diethylenetriamine tricarboxylic acid (DTTA derivatives: H₃L³, H₃L⁴) core. The neutral Eu^III and Tb^III complexes of these chelates have been prepared and studied from their UV-vis and luminescence data. The main photophysical characteristics of these complexes, i.e. the absorption and luminescence spectra, the metal-centred lifetimes and the overall luminescence yields (Φ) were measured in buffered aqueous solutions. In addition the role played by non-radiative paths (vibrational energy transfer involving coordinated water molecules, involvement of ligand-to-metal charge-transfer excited states, or metal→ligand back-transfer) was investigated. In all complexes, we found that the bidentate bipyridine chromophore is not coordinated to the lanthanide ion, allowing one (LnL¹, LnL²) or two (LnL³, LnL⁴) water molecules to penetrate the first coordination sphere of the metal. Although the bipyridine chromophore behaves as remote (from the binding site) light-harvesting unit for the lanthanide ion in these systems, a sizeable sensitization of the Eu- and Tb-centred luminescence can be effective (LnL², LnL³, Φ=16-19% in aerated D₂O solutions). Our photophysical investigations show that overall non-radiative deactivation is not dependant of thermally activated non-radiative channels but the efficiency of the ligand→Ln intramolecular energy transfer has to be taken into account to explain the obtained results.     

Photoluminescent properties of heteroleptic cyclometalated platinum(II) complexes bearing 1,3-bis(3,4-dibutoxyphenyl)propane-1,3-dionate as an ancillary ligand

A new series of heteroleptic cyclometalated platinum(II) complexes Pt-1a-f was synthesized, employing 2-arylpyridine (or 1-arylisoquinoline) (HC^∧N-1) and 1,3-bis(3,4-dibutoxyphenyl)propane-1,3-dione (HO^∧O-1) for cyclometalation and as ancillary ligands, respectively, and photoluminescent properties were investigated. Focusing on red-shifted phosphorescence, C^∧N ligands containing π-extended aromatics and electron-rich heterocycles were examined. All obtained complexes exhibited photoluminescence at ambient temperature, and the emission maxima ranged from green (λ_PL=518 nm) to far red (λ_PL=708 nm). The large Stokes shifts of more than 100 nm and sub-microsecond or microsecond emission lifetimes revealed that these complexes are phosphorescent emissive. The quantum yield of Pt-1 ranged from 0.02 to 0.59 at ambient temperature and decreased as the emission maximum was red-shifted. In comparison with the reference platinum(II) complexes, Pt-2 bearing an aliphatic ancillary ligand, such as 2,2,6,6-tetramethylheptane-3,5-dionate (O^∧O-2), the ligand O^∧O-1 did not significantly affect the photoluminescence emission maxima, indicating that the energy gap between the singlet ground state and the triplet level was predominantly dependent on the C^∧N ligand.     

Magnetic and electronic properties of Cr- and Mn-doped SnO2: ab initio calculations

The ab initio calculations, based on the Korringa–Kohn–Rostoker (KKR) approximation method combined with the coherent potential approximation (CPA), indicated as KKR–CPA, have been used to study the stability of ferromagnetic and ferrimagnetic states, for systems that are SnO₂ doped and co-doped with two transition metals, that is, chromium and manganese. Our results indicate that the ferromagnetic state is more stable than the spin-glass state for the (Sn_1−xCr_xO₂; x = 0.07, 0.09, 0.12 and 0.15)-doped system, while the spin-glass state is more stable than the ferromagnetic state for the (Sn_1−xMn_xO₂; x = 0.02 and 0.05)-doped system. However, the ferromagnetic and/or the ferrimagnetic states are stable for the (Sn_0.98−xMn_0.02Cr_xO₂; x = 0.05, 0.09 and 0.13)-doped system depending on the Cr concentration. Moreover, we estimated the Curie temperature (T_c) for the Cr-doped tin dioxide (SnO₂), and we explained the origin of magnetic behaviour through the total density of states for different doped and co-doped SnO₂ systems.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

Synthesis, X-ray structure, EPR and optical properties of a ferrocene substituted polychlorotriphenylmethyl radical

The optical, magnetic and electrochemical properties of the octamethylferrocene aldehyde substituted polychlorotriphenylmethyl radical 1 are reported. Radical 1 is prepared in a three step synthetic route starting with a Wittig-Horner reaction to yield (E)-1-formyl-1′-{2-{4-[bis(2,3,4,5,6-pentachlorophenyl)methyl]-2,3,5,6-tetrachloro phenyl}ethen-1-yl}-2,2′,3,3′,4,4′,5,5′-octamethyl ferrocene (6), which is subsequently deprotonated to yield the corresponding anion 7 and finally oxidized to (E)-4-[2-(1′-formyl-2,2′,3,3′,4,4′,5,5′-octamethylferrocen)ethen-1-yl]-2,3,5,6-tetrachlorophenyl-bis(2,3,4,5,6-pentachlorophenyl)methyl radical (1). Radical 1 exhibits a charge-transfer band transition in the near infrared region which is associated with an intramolecular electron transfer from the ferrocene unit (donor) to the radical unit (acceptor) of this dyad molecule. The X-ray crystal structure of [K⁺(18-crown-6)] (E)-[4-[2-(1′-formyl-2,2′,3,3′,4,4′,5,5′-octamethylferrocen)ethen-1-yl]-2,3,5,6-tetrachlorophenyl-bis(2,3,4,5,6-pentachlorophenyl) methide] (7) has been determined. This organic salt shows an interesting one-dimensional polymeric structure formed by the coordination of the K⁺ cation with several atoms of the organic carbanion.