Effect of static deflection on natural frequency of non-linear spring mass system by direct linearization method

An exact expression for the frequency of a non-linear cubic spring mass system is obtained considering the effect of static deflection. An alternative expression for the approximate frequency is also obtained by the direct linearization procedure; it is shown that this is very accurate as compared with the exact method. This approximate frequency equation is used to explain a “dual behaviour” of the frequency amplitude curves.

Response spectrum of non-linear spring mass system subjected to transient disturbances

The transient response spectrum of a cubic spring mass system subjected to a step function input is obtained. An approximate method is adopted where non-linear restoring force characteristic is replaced by two linear segments, so that the mean square error between them is a minimum. The effect of viscous damping on the peak response is also discussed for various values of the damping constant and the non-linearity restoring force parameter.

Non-Linear vibration of an elastic string

Equations proposed in previous work on the non-linear motion of a string show a basic disagreement, which is here traced to an assumption about the longitudinal displacement u. It is shown that it is neither necessary nor justifiable to assume that u is zero; and also that the velocity of propagation of u disturbances in a string is different from that in an infinite medium, although this difference is usually negligible. After formulating the exact equations of motion for the string, a systematic procedure is described for obtaining approximations to these equations to any order, making only the assumption that the strain in the material of the string is small. The lowest order equations in this scheme are non-linear, and are used to describe the response of a string near resonance. Finally, it is shown that in the absence of damping, planar motion of a string is always unstable at sufficiently high amplitudes, the critical amplitude falling to zero at the natural frequency and its subharmonics. The effect of slight damping on this instability is also discussed.

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the type y″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.

Non-linear couette flow with heat transfer using the B-G-K model

In recent years a large number of investigators have devoted their efforts to the study of flow and heat transfer in rarefied gases, using the BGK [1] model or the Boltzmann kinetic equation. The velocity moment method which is based on an expansion of the distribution function as a series of orthogonal polynomials in velocity space, has been applied to the linearized problem of shear flow and heat transfer by Mott-Smith [2] and Wang Chang and Uhlenbeck [3]. Gross, Jackson and Ziering [4] have improved greatly upon this technique by expressing the distribution function in terms of half-range functions and it is this feature which leads to the rapid convergence of the method. The full-range moments method [4] has been modified by Bhatnagar [5] and then applied to plane Couette flow using the B-G-K model. Bhatnagar and Srivastava [6] have also studied the heat transfer in plane Couette flow using the linearized B-G-K equation. On the other hand, the half-range moments method has been applied by Gross and Ziering [7] to heat transfer between parallel plates using Boltzmann equation for hard sphere molecules and by Ziering [83 to shear and heat flow using Maxwell molecular model. Along different lines, a moment method has been applied by Lees and Liu [9] to heat transfer in Couette flow using Maxwell's transfer equation rather than the Boltzmann equation for distribution function. An iteration method has been developed by Willis [10] to apply it to non-linear heat transfer problems using the B-G-K model, with the zeroth iteration being taken as the solution of the collisionless kinetic equation. Krook [11] has also used the moment method to formulate the equivalent continuum equations and has pointed out that if the effects of molecular collisions are described by the B-G-K model, exact numerical solutions of many rarefied gas-dynamic problems can be obtained. Recently, these numerical solutions have been obtained by Anderson [12] for the non-linear heat transfer in Couette flow,

Optimal non-linear reinforcement schemes for stochastic automata

Two optimal non-linear reinforcement schemes—the Reward-Inaction and the Penalty-Inaction—for the two-state automaton functioning in a stationary random environment are considered. Very simple conditions of symmetry of the non-linear function figuring in the reinforcement scheme are shown to be necessary and sufficient for optimality. General expressions for the variance and rate of learning are derived. These schemes are compared with the already existing optimal linear schemes in the light of average variance and average rate of learning.

Natural modes of a coupled non-linear system

The natural modes of a non-linear system with two degrees of freedom are investigated. The system, which may contain either hard or soft springs, is shown to possess three modes of vibration one of which does not have any counterpart in the linear theory. The stability analysis indicates the existence of seven different modal stability patterns depending on the values of two parameters of non-linearity.

Scalable non-linear Support Vector Machine using hierarchical clustering

This paper discusses a method for scaling SVM with Gaussian kernel function to handle large data sets by using a selective sampling strategy for the training set. It employs a scalable hierarchical clustering algorithm to construct cluster indexing structures of the training data in the kernel induced feature space. These are then used for selective sampling of the training data for SVM to impart scalability to the training process. Empirical studies made on real world data sets show that the proposed strategy performs well on large data sets.

Inhomogeneity in sintered ZnO based non-linear resistors

Investigations have been made on the inhomogeneous characteristics of sintered ZnO based non-linear resistors caused by localized particle growth on the surface and by internal flaws. The presence of Sb2O3 was responsible for the observed particle growth. A part of the Bi2O3 on the surface was found to be in the reduced state. Two kinds of failure mode, cracking and puncturing, were observed when the samples were subjected to high-energy pulses. The puncture mode is caused by local melting around the regions of high current density, whereas the cracking mode results from thermally induced stresses.

Field-dependent changeover from current-to voltage-limiting characteristics by non-linear n-type BaTiO3 ceramics

Non-linear resistors having current-limiting capabilities at lower field strengths, and voltage-limiting characteristics (varistors) at higher field strengths, were prepared from sintered polycrystalline ceramics of (Ba0.6Sr0.4)(Ti0.97Zr0.03)O3+0.3 at % La, and reannealed after painting with low-melting mixtures of Bi2O3 + PbO +B2O3. These types of non-linear characteristics were found to depend upon the non-uniform diffusion of lead and the consequent distribution of Curie points (T c) in these perovskites, resulting in diffuse phase transitions. Tunnelling of electrons across the asymmetric barrier at tetragonak-cubic interfaces changes to tunnelling across the symmetric barrier as the cubic phase is fully stabilized through Joule heating at high field strengths. Therefore the current-limiting characteristics switch over to voltage-limiting behaviour because tunnelling to acceptor-type mid-bandgap states gives way to band-to-band tunnelling.

Non-Linear effects of membrane fluctuations in the dilute lamellar phase

The static structure factor of the dilute sterically stabilised lamellar phase is calculated and found to have an Ornstein-Zernike form with a correlation length that diverges at infinite dilution. The relaxation time for concentration fluctuations at large wave number q is shown to go as q-3 with a coefficient independent of the membrane bending rigidity. The membrane fluctuations also give rise to strongly frequency-dependent viscosities at high frequencies.

Soft-chemical synthesis of new non-linear optical materials, K0.5M0.5Ti0.5OPO4(M = Nb, Ta),related to KTiOPO4

Soft-chemical oxidation of KTiOPO4-like KM(0.5)(V)Ti(0.5)(III)OPO(4) (M = Nb, Ta) using chlorine in CHCl3 is accompanied by partial deintercalation of potassium, yielding K(0.5)MV(0.5)Ti(0.5)(IV)OPO(4) compounds which are new non-linear optical materials that exhibit efficient second-harmonic generation of 1064 nm radiation, as does KTiOPO4.

Non-linear optical response of rutile-related oxides, LiM(V)M(VI)O(6), and their derivatives obtained by ion-exchange and intercalation

Polycrystalline samples of oxides of the general formula LiM(V)M(VI)O(6) (M(V) = Nb, Ta; M(VI) = Mo, W), crystallizing in a non-centrosymmetric (space group P (4) over bar 2(1)m) trirutile structure, exhibit second harmonic generation (SHG) of 1064 nm radiation with efficiencies 15-45 times that of alpha-quartz; interestingly, the SHG response is retained by the protonated derivatives HM(V)M(VI)O(6) . xH(2)O, and their n-alkylamine intercalates as well.