Irrational Term Research Articles

Study of different theories of thermoelasticity under the propagation of Rayleigh waves in thermoelastic medium

The present research is on the propagation of Rayleigh waves in a homogenous thermoelastic solid half-space by considering the compact form of six different theories of thermoelasticity. The medium is subjected to an insulated boundary surface that is free from normal stress, tangential stress, and a temperature gradient normal to the surface. After developing a mathematical model, a dispersion equation is obtained with irrational terms. To apply the algebraic method, this equation must be converted into a rational polynomial equation. From this, only those roots are filtered out, which has satisfied both of the above equations for the propagation of waves decaying with depth. With the help of these roots, different characteristics are computed numerically, like phase velocity, attenuation coefficient, and path of particles. Various particular cases are compared graphically by using phase velocity and attenuation coefficient. The elliptic path of surface particles in Rayleigh wave propagation is also presented for the different theories using physical constants of copper material for different depths and thermal conductivity.

Feedforward PID Controller Design For a Parabolic Trough Solar Collector

Distributed parameter systems (DPS) are widely recognised in the process industry. An innovative feedforward control strategy for the DPS of a parabolic trough collector (PTC) has been developed with the aim of rejecting disturbances. This feedforward is designed on a transfer function based on a semi-physical model. This transfer function includes an irrational term that models the resonance dynamics in the system response. For validation, this approach is compared with a proportional-integral (PI) design and other existing feedforward designs in literature. The results show that the proposed approach significantly improves the system performance in the presence of disturbances addressing system dynamics at medium and high frequencies.

The oscillating periodic solutions of a classical pendulum system with smooth and discontinuous dynamics

The oscillating periodic solutions of a classical pendulum system with an irrational and fractional nonlinear restoring force are studied both theoretically and numerically under sufficiently small perturbations of a viscous damping and a harmonic excitation. The most salient feature of this pendulum system is to exhibit both smooth and discontinuous dynamics depending on the value of a geometrical parameter. In order to precisely describe the local dynamics of small-angle oscillations, we introduce a simplified approximate system which not only successfully retains the non-smooth characteristics but also completely reflects the local feature of the complex restoring force, especially the equilibrium bifurcation. Compared with the cubic and quintic polynomial systems derived by Taylor expansion, the application range of the simplified approximate system is enlarged within same margin of absolute error. With the help of the simplified approximate system, the periodic oscillatory solution around a stable equilibrium is examined analytically by using the averaging method in both smooth and discontinuous cases. Numerical simulations are carried out to verify the theoretical analysis and demonstrate the predicted periodic motions. The contribution of this study is to present an effective approximation to precisely describe the local dynamics of a classical pendulum system with smooth and discontinuous dynamics in terms of the qualitative analysis and quantitative calculation, which is also helpful for exploring the local dynamics of the nonlinear dynamical system containing a coupling of the irrational term and trigonometric function.

Efficient Planning and Solving Algorithm of S -Shape Acceleration and Deceleration

S -shape acceleration and deceleration are the most widely used flexible acceleration and deceleration method in the current CNC system, but its velocity solution equation contains irrational terms, which create a more complicated solution process. When analyzing the solution process of the S -shape acceleration and deceleration directly, using a traditional numerical solution method, the phenomenon of “solving the interval jump” arises, which is the main reason for low efficiency and poor stability of the solution. According to the S -curve profile and solution, the concept of separating the curve profile recognition from the velocity solution was proposed, and a method of quickly identifying the interval of the solution location was introduced. Through the method mentioned above, the complete acceleration and deceleration curve parameters can be obtained through a one-time plan and a one-time solution, and the solution efficiency and stability are guaranteed; solving the Newton problem depends too much on the initial value of Newton velocity, which not only retains the speed advantage of the Newton method but also uses the downhill factor to ensure its convergence. Through the simulation comparison and analysis, the efficiency, stability, and universality of the method are verified.

The Variational Approach Coupled with an Ancient Chinese Mathematical Method to the Relativistic Oscillator

This paper applies the variational approach to the relativistic oscillator. In order to effectively deal with the irrational term, an ancient Chinese mathematics is introduced. Comparison of the obtained result with the numerical one elucidates the efficiency of the present treatment.

Higher Accuracy Approximate Solution for Oscillations of a Mass Attached to a Stretched Elastic Wire by Rational Harmonic Balance Method

A second-order modified rational harmonic balance method is used for approximately solve the nonlinear differential equation that governs the oscillations of a system typified as a mass attached to a stretched elastic wire for which the restoring force for this oscillator has an irrational term with a parameter λ that characterizes the system. A frequency-amplitude relation is constructed and this frequency is valid for the complete range of oscillation amplitudes A and parameter λ, and excellent agreement of the approximate frequencies with the exact one is demonstrated and discussed. The discrepancy between the approximate frequency and the exact one never exceed 0.12%. This error corresponds to λ = 1, while for λ<1 the relative error is much lower. For example, its value is lower than 0.017% for λ = 0.5.

Stochastic shape functions and stochastic strain–displacement matrix for a stochastic finite element stiffness matrix

For conventional finite element problems, element geometry is adequate to determine shape functions. However, to account for secondary effects due to material randomness, conventional shape functions need to be modified according to the spatial fluctuation of constitutive variables in each Monte Carlo sample. This paper develops a method to compute stochastic shape functions based on local equilibrium criteria when each simulated sample complies with the same order of accuracy as designated for the associated deterministic problem. The resulting stochastic stiffness matrix is then calculated via the stochastic strain–displacement matrix based on those stochastic shape functions. In order to attain high accuracy, which is the characteristic of the boundary element method, rational polynomial shape functions are used in this paper. The proposed formulation is indispensable when secondary effects (due to nano size and time scale in modern technology, fiber randomness in composites, thermodynamic interactions in biological tissues, to name a few) demand a high accuracy finite element formulation. The elasto-plastic deformation that introduces concavity motivated the numerical example elaborated here. An example of a concave quadrilateral element with spatial randomness for the modulus of elasticity is illustrated. Since isoparametric shape functions for concave quadrilaterals do not exist, the Wachspress rational polynomial shape functions with irrational terms are used. The computer algebra environment Mathematica is employed here.

Approximate solutions of a nonlinear oscillator typified as a mass attached to a stretched elastic wire by the homotopy perturbation method

The homotopy perturbation method is used to solve the nonlinear differential equation that governs the nonlinear oscillations of a system typified as a mass attached to a stretched elastic wire. The restoring force for this oscillator has an irrational term with a parameter λ that characterizes the system (0 ⩽ λ ⩽ 1). For λ = 1 and small values of x, the restoring force does not have a dominant term proportional to x. We find this perturbation method works very well for the whole range of parameters involved, and excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions and the maximal relative error for the approximate frequency is less than 2.2% for small and large values of oscillation amplitude. This error corresponds to λ = 1, while for λ < 1 the relative error is much lower. For example, its value is as low as 0.062% for λ = 0.5.

Active vibration isolation by cancelling bending waves

Active forces applied to a vibrating beam structure may be driven to cancel propagating and evanescent waves entering the region beyond their point of application, thereby preventing those waves from carrying the effects of an external disturbance into that region. The application of this concept to the problem of vibration isolation has great potential; a system designed to detect bending waves as they pass a sensor and then inject an antidote further downstream can achieve total isolation of the entire structure beyond the point of application of just two point forces. The feedforward configuration ensures that the effectiveness of active isolation is limited only by the accuracy of implementation of the control system. Synthesis of the controller in terms of flexural waves suggests that the compensation required would contain irrational terms resulting from the dispersive nature of the waves; the inaccessibility of individual wave components to any measurement system in a finite structure, where vibrations are the superposition of waves in both directions, would complicate implementation and make accurate identification of the system impractical. These difficulties were removed by including the effects of waves generated by the active forces and detected at the controller input in the control system synthesis, and a simple, stable, active isolation system emerged. We show that the compensation required may be expressed in a rational polynomial form, and that identification of the optimal compensation in a noisy environment may be made using input-output response measurements on the structure to be isolated. The same response data can be used to predict limits imposed on the achievable isolation by inaccuracies in implementation. This work culminated in the application of these techniques to the problem of isolating a cantilever beam from vibrations caused by a random, broadband disturbing force near the fixed end. Although component inaccuracies in the particular apparatus used in the experiment allowed only 6 dB of attenuation to be achieved over a bandwidth of 200 Hz, good agreement between predicted and experi­mental results was shown.

Advances in Ophthalmology Vol 14

This annual review consists of three papers. The first is an account of hemorrhagic by Sibylle Kruger of Basel. In an attempt to retain this irrational term the author restricts its meaning to that glaucoma which follows central retinal vein thrombosis and secondary rubeosis iridis. The account is a detailed one, but since pathogenesis is unknown and treatment is notably ineffective the overall result is not impressive. The second paper is an account of the experimental and spontaneous retinal degenerations of animals—a useful summary in itself. Curiously, the effect of glutamate poisoning on the retina is overlooked. The third review, and the last 194 pages of the book, deals with the literature on the retina from 1952 to 1959. The exhaustive nature of this review is impressive. The cutoff date four years before publication leaves many problems inadequately discussed in the light of present knowledge.