Non-classical Modes Research Articles

Timoshenko Beam-Column with Generalized End Conditions and Nonclassical Modes of Vibration of Shear Beams

The main objective of this publication is to derive, in a classic manner, the characteristic equations for the undamped natural frequencies and the corresponding modes of vibration of a two-dimensional (2D) Timoshenko beam–column with generalized support conditions (i.e., with semirigid flexural restraints and lateral bracings as well as lumped masses at both ends) and subjected to a constant axial load along its span. The model includes the simultaneous effects (or couplings) of bending and shear deformations, translational and rotational inertias of all masses considered. The proposed model is general, showing that the natural frequencies and the corresponding modes of vibration of 2D beam–columns are highly sensitive to the coupling effects just mentioned. This is particularly true in members with low shear stiffness and with the end flexural restraints and lateral bracing approaching those of free–free and pinned–free conditions. A second objective of this paper is to show that the obtained solution reproduce, as a special case, the nonclassical modes of shearbeams, including the inversion of modes of vibration (i.e., higher modes crossing lower modes) in shear beams with pinned–free and free–free end conditions, and the phenomena of double frequencies at certain values of beam slenderness (r/L).

Function and mode of action of nuclear receptors: Estrogen, progesterone, and vitamin D

Estrogens and progestins play important roles in regulating growth and differentiation of a wide range of cell types, in both reproductive and nonreproductive tissues. Not surprisingly, therefore, endocrine active substances that mimic the actions of these steroid hormones have been demonstrated to have untoward effects on the reproductive function of a variety of animals. Although direct links between exposure to endocrine active substances of this class and reproductive abnormalities in humans have been difficult to establish, the potential for harm by this class of chemicals warrants further investigation. This chapter summarizes our current understanding of the molecular pharmacology of the estrogen and progesterone receptors, ER and PR, beginning with a historical perspective of ER pharmacology and ending with a comparison of these receptors to the vitamin D receptor (VDR), a nonsteroid hormone nuclear receptor involved in regulating growth and development in nonreproductive tissues. Major topics discussed include receptor subtypes (or isoforms), receptor functional domains, ligand-binding characteristics, receptor structure, cofactor binding, effects of phosphorylation, and nonclassical modes of action. This discussion will demonstrate the need for developing novel screens for potential endocrine disruptors that incorporate our current understanding of nuclear receptor pharmacology.

Nonclassical Modes of Unrestrained Shear Beams

This paper considers the effect that a rotational motion has on the normal modes of a shear beam that is free to rotate, either because it is free in space or it is pivoted at one end. It is shown that the classical solutions for these two cases violate the principle of conservation of angular momentum, and that this is true even when the rotational inertia of the beam vanishes or is neglected.

Estimates of errors in the frequency response of non-classically damped systems

Abstract This paper examines the vibration behaviour of non-classically damped systems in the frequency domain. Errors in the frequency response incurred by neglecting off-diagonal entries of a diagonally dominant modal damping matrix are assessed. The exact response is shown to be equivalent to an infinite Taylor’s series, and it is demonstrated that the commonly employed approximation of ignoring off-diagonal entries of the modal damping matrix is equivalent to retaining one term of the series. The frequency dependence of coupling of the non-classical modes is obtained.

Optimal operating conditions for the production of some fission products in nuclear reactors

The above examples confirm the validity of the analysis of optimal operating conditions for producing certain nuclides-fission products-in the framework of the maximum principle and also confirm the validity of the resulting conclusions concerning the form of the optimal mode. When fission products without burnup are produced, the nonclassical modes are optimal: they consist of alternating phases of minimum and maximum control. When neutron capture takes place in the intermediate links, classical control may appear in the optimal mode.

Use of higher order displacement functions in the free vibration analysis of shells of revolution having meridional singularities

A finite element technique in which higher order displacement functions are used in the analysis of vibration of shells of revolution with joints and poles is described. The joints typically arise where a meridional curve is made up from a number of different curves, as in the case of a cone-toroid cooling tower or a cone-cylinder chimney. A technique for overcoming the difficulties associated with the representation of the stress discontinuities at the joints when higher order displacement functions are used is discussed. A number of typical shells with joints are analysed by using this technique, and the results are found to compare very well with existing theoretical and experimental data. A pole of a shell of revolution is a point where the meridian intersects the axis of revolution, such as the apex of a spherical dome or the apex of a conical shell. It is shown that, in analysing such shells, it is not necessary to exclude the pole by introducing a small hole around it or to develop special elements. A cone-cylinder combination with an apex is analysed by using a combination of the techniques for analysing jointed shells and shells with poles. The calculated frequencies and mode shapes are shown to agree well with experimental results. It is demonstrated that the cone-cylinder combination with the apex can be idealized accurately by using a curved shell element with four nodal points. The “non-classical modes” of vibration reported in the experiment could not be predicted theoretically, even though the curved shell element was used and the value of the circumferential wave number n was not prescribed.

Free vibration, stability and “non-classical modes” of cooling tower shells

A ring and a shell finite element for the analysis of free vibration of shells of revolution and cooling tower shells are described. In the formulation of the ring element the effects of initial in-plane stresses are included so that both vibration and stability of the shells can be studied. For both elements, a variation of the shell thickness along the meridian is considered. The two elements are used to analyse a hyperboloid and a tower similar to those at Didcot Power Station. The results agree well with those obtained by other methods for the case when no singularities (such as legs) are introduced. It is found that no “non-classical” modes of vibration exist when the shell is completely rotationally symmetric without any singularities. The effects introduced by the legs of the cooling tower are studied by omitting parts of the shell near the base. It is shown that the natural frequencies are lower than for the complete shell, and that there are “non-classical” modes of vibration. It is shown that increasing the thickness in the lower part of the tower raises the natural frequencies, and that the self load stresses have only a marginal effect on the vibration and stability of the towers. It is also shown that lack of complete restraint at the base of a shell can greatly reduce the critical stress.

Vibration Phase Measurement by Rotation-Strobe Holography

A new holography technique that permits direct visual observation of the relative phases of all parts of a vibrating surface is described. A small rotation is introduced between exposures of a stroboscopic hologram of a vibrating surface to produce a series of vertical fringes on the surface. The vertical fringes are distorted to the right or left by the surface deformation of the object under observation; the distortions form a record of the relative phases of all parts of the surface. The ambiguities existing in the interpretation of the shape of nonclassical modes from a conventional time-averaged hologram are, thereby, resolved.

Karyokinesis of somatic nuclei of Neurospora crassa. IV. The occurrence of the juvenile ring phase during division IV of ascosporogenesis.

Introduction T h e observation (Weijer , 1961; Weijer et al., 1963, 1961, 1965), that different modes of lcaryokinesis are involved in the differentiation of specific n~orphological struct;res of the vegetative cycle of Nez~rosporrr crassa; raises the immediate question whether anv of these non-classical modes of nuclear division extend h t o the reproductive cycle of the mould. Although Singleton (1953) in his description of the chromosome cycle in the ascus of N . cmssn states that as the spore matures after division IV , its cytoplasm, originally rather uni forn~ly granular o r finely in appearance, beconles highly vacuolate (p. 137), it was felt that this information on the last interphase of a scos~oro~enes i s n7as too incomplete to be useful for a critical assessment of the prob/em Gated above. ~ h e r e f A r e , a detailed study was made of the late telophase-interphase nuclei of division I V in the devel6ping ascus. Weijer (1964) and Weijer et al., (1964, 1965) observed in the juvenile cycle of N . cmssn ring shaped nuclear configurations, which are usually diffuse in character early in Icaryolcinesis, whereas in more advanced stages, the nuclear material is separated into eight Feulgen stainable bodies (seven chromosomes and a centriole) surrounding a non-stainable nucleolus. Similar ring shaped nuclei were described b y Balcerspigel (1958, 1959) in Gelclsi~~ospovn tetrrrsperma Dowd., and in Endogolze spl~ag7zophila Atlc., and b y Robinow (1957, 1962) in il4ucor hie7nnlis and i M . fragilis. Hall (1963), in his studies concerning the cytology of iMo7zili7tin fructicolrr (Wint . ) Honey., did not observe ring shaped nuclei, although the published photograph of interphase nuclei is suggestive f o r their occurrence. Since the ring shaped nucleus is an easily recognizable structure of the juvenile cvclc, a search for the occurrence of this nuclear configuration was initiated in the developing ascus. Material and lMethods Crosses \vere made between strain F G S C 262 and F G S C 16 on liquid crossing medium (Stanford Neurospora A~lcthods, 1964). Half-pint milk bottles containing 70 ml. of crossing medium \irere inoculated with hvphal material of strain F G S C 16 on a partly submerged filter paper square (7 s 8.~111.) and incubated at 23°C for 6-7 days to allow the develop~nent of protoperithecia. Conidia of strain F G S C 262 werk then dusted on to the resulting growth. T h e cultures \i.erc then reincubated a t 23°C fo r 128 to 130 hours, after which the perithecia \.ere removed and the asci wcre dissected on to a covers l i~ . After 1 drying at 60°C, the preparations \irere stained according to the Feulgen technique as outlincd bv h/IcDonald and Weijer (1965, in preparation). Observations and microphoto~raphs \irere made with a Leitz Ortholux I1 microscope fitted wit11 a Lcitz plano objective (100 x N.A. 1.32) and a Leitz Orthomat 35 mm. automatic camera. Balzers interference filters (540 and 576 mu) wcre used.