Equivalent Electronic Circuit of a System of Oscillators Connected With Periodically Variable Stiffness

In this paper, we have shown the electronic circuit equivalence of a mechanical system consists of two oscillators coupled with each other. The mechanical design has the effects of the magnetic, resistance forces and the spring constant of the system is periodically varying. We have shown that the system’s state variables, such as the displacements and the velocities, under the effects of different forces, lead to some nonlinear behaviors, like a transition from the fixed point attractor to the chaotic attractor through the periodic and quasi-periodic attractors. We have constructed the equivalent electronic circuit of this mechanical system and have verified the numerically obtained behaviors using the electronic circuit.

Soft-Transition Sub-PCA Monitoring of Batch Processes

Batch or semi-batch processes have been utilized to produce high-value-added products in the biological, food, semi-conductor industries. Batch process, such as fermentation, polymerization, and pharmacy, is highly sensitive to the abnormal changes in operating condition. Monitoring of such processes is extremely important in order to get higher productivity. However, it is more difficult to develop an exact monitoring model of batch processes than that of continuous processes, due to the common natures of batch process: non-steady, time-varying, finite duration, and nonlinear behaviors. The lack of exact monitoring model in most batch processes leads that an operator cannot identify the faults when they occurred. Therefore, effective techniques for monitoring batch process exactly are necessary in order to remind the operator to take some corrective actions before the situation becomes more dangerous.

Assessment of Complex System Dynamics via Harmonic Mapping in a Multifractal Paradigm

In the present paper, nonlinear behaviors of complex system dynamics from a multifractal perspective of motion are analyzed. In the framework of scale relativity theory, by analyzing the dynamics of complex system entities based on continuous but non-differentiable curves (multifractal curves), both the Schrödinger and Madelung scenarios on the holographic implementations of dynamics are functional and complementary. In the Madelung scenario, the holographic implementation of dynamics (i.e., free of any external or internal constraints) has some important consequences explicated by means of various operational procedures. The selected procedures involve synchronous modes through SL (2R) transformation group based on a hidden symmetry, coherence domains through Riemann manifold embedded with a Poincaré metric based on a parallel transport of direction (in a Levi Civita sense). Other procedures used here relate to the stationary-non-stationary dynamics transition through harmonic mapping from the usual space to the hyperbolic one manifested as cellular and channel type self-structuring. Finally, the Madelung scenario on the holographic implementations of dynamics are discussed with respect to laser-produced plasma dynamics.

Chaos Control and Stabilization of a PID Controlled Buck Converter Using the Spotted Hyena Optimizer

The voltage controlled buck converter by constant-frequency pulse-width modulation in continuous conduction mode gives rise to a variety of nonlinear behaviors depending on the circuit parameters values, which complicate their analysis and control. In this paper, a description of the DC/DC buck converter and an overview of some of its chaotic dynamics is presented. A solution based on the optimized PID controller is suggested to eliminate the observed nonlinear phenomena and to enhance the dynamics of the converter. The parameters of the controller are optimized with the Spotted Hyena Optimizer (SHO) which uses the sum of the error between the reference voltage and the output voltage as well as the error between the values of the inductor current in every switch opening instant to determine the fitness of each solution. The simulations results in MATLAB proved the efficiency of the proposed solution.

Study on Damage Evolution Model of Sandstone under Triaxial Loading and Postpeak Unloading Considering Nonlinear Behaviors

Conventional triaxial loading and unloading tests were carried out on sandstone samples in the Zigong area, of Sichuan Province, China. The changes in the elastic modulus of the unloading curves under different confining pressures were calculated, and the evolution law of the nonlinear properties of rock was analyzed. The results show that the rock is subjected to nonlinear damage during initial compaction, the elastic phase, destruction, and postpeak unloading. Moreover, the nonlinear behaviors of rock are restrained by the confining pressures. On this basis, a nonlinear stress-strain relationship affected by the average stress is proposed to describe nonlinear behaviors in the initial compaction stage. According to the test data, the evolution laws of various energies inside the rock during loading and unloading cycles are obtained. The results show that the external work is transformed into elastic energy and damage dissipated energy. Based on the energy analysis, the energy balance equation is established according to the law of energy conservation. By deriving the energy balance equation, the damage evolution equation of sandstone under triaxial loading is solved to establish a continuous constitutive model. The calculation results of the model are compared with the test results from two aspects of loading and postpeak unloading. The comparison results show that the proposed model, which reflects the whole stress-strain process and nonlinear properties of rock, could also describe the stress-strain relationship at the postpeak unloading stage to some extent.

Nonlinear dynamics of a spur gear pair with tooth root crack based on an amplitude modulation function

PurposeGear transmissions are widely utilized in practice. This paper aims to uncouple the crack feature from the cracked time-varying mesh stiffness (TVMS) and investigate the effects of the crack on the nonlinear dynamics of a spur gear pair.Design/methodology/approachAn approximate method to simulate the cracked TVMS is proposed by using an amplitude modulation function. The ratio of mesh stiffness loss is introduced to estimate the TVMS with different crack depths and angles. The dynamic responses are obtained by solving a torsional model which takes the non-loaded static transmission error, the backlash and the cracked TVMS into account. By using the bifurcation diagram, the largest Lyapunov exponent (LLE) and dynamic mesh force, the influences of crack on nonlinear behaviors are examined. The dynamic characteristics are identified from the phase diagram, Poincaré map, dynamic mesh force, time series and FFT spectra.FindingsThe comparison between the healthy and cracked gear pairs indicates that the crack affects the system motions, such as the obvious changes of impact force and unpredictable instability. Besides, the additive and difference combination frequencies can be found in periodic-1 and -2 motions, but they are covered in periodic-3 and chaotic motions. Deeper crack is an important determinant of the nonlinear behaviors at a higher speed.Originality/valueThe research provides an interesting perspective on cracked TVMS and reveals the connection between crack and nonlinear behaviors of the gear pairs.

Non-planar Dynamics of a CFRP Cable with an External 1/3-order Subharmonic Resonance

Based on the non-planar vibration equations of a cable made of carbon fiber reinforced polymer (CFRP), the nonlinear behaviors of the cable are studied. The one-to-one internal resonance of the lowest in-plane and out-of-plane modes of the cable is investigated. Three different cases, namely, 1/3-order subharmonic resonance of the in-plane mode, 1/3-order subharmonic resonance of the out-of-plane mode and 1/3-order subharmonic resonance of both of the in-plane and out-of-plane modes are examined. The vibration equations of the cable are discretized by using Galerkin’s method. In this way, the ordinary differential equations (ODEs) are obtained. The multiple time scale method is employed to solve the ODEs and the corresponding modulation equations are derived. Then, the response curves of the cable are obtained by using Newton-Raphson method and pseudo arclength algorithm. Meanwhile, the response curves of the CFRP cable are compared with those of the steel cable to explore the differences in nonlinear behaviors of the cables made of different materials. The results show that the CFRP cable has potential advantages over the steel cable from a nonlinear point of view.