Chaotic analysis of gyroscope control system via differential transformation method

This paper presents an analysis of PSIuIeTR type model, which are used to study the transmission dynamics of typhoid fever diseases in a population. Basic idea of typhoid fever disease transmission using compartmental modeling is discussed. Differential Transformation Method (DTM) is discussed in detail, which is used to compute the series solution of the non-linear system of differential equation governing the model equations. The validity of the (DTM) in solving the proposed model is established by classical fourth-order Runge-Kutta method which is implemented in Maple 18. Graphical results confirm that (DTM) is in good agreement with RK-4 and this produced correctly same behaviour of the model, thus validating the efficiency and accuracy of (DTM) in finding the series solution of an epidemic model.

In recent years, there has been a tendency towards renewable energy sources considering the damages caused by non-renewable energy resources to nature and humans. One of the renewable energy sources is wind and energy is obtained with the help of wind turbines. To determine the behavior of wind turbines under earthquake loads, dynamic characteristics are required. In this study, the differential transformation method is proposed to determine the free vibration analysis of wind turbines with a variable cross-section. The wind turbine is modeled as an equivalent variable continuous flexural beam and blade weight is considered as a point mass at the top of the structures. The differential equation representing the free vibration of the wind turbine is transformed into an algebraic equation with the help of differential transformation method and the angular frequencies and the mode shapes of the wind turbine are obtained by the help of the differential transformation method. In the study, a sample taken from the literature was solved with the presented method and the suitability of the method was investigated. The same wind turbine example also modeled by finite element modelling software, ABAQUS. Results of the finite element model and differential transformation method are compared with each other and the results are in good agreement.

As a nonlinear dynamic system, the dynamic equation of triboelectric nanogenerator system is not only difficult to accurately establish, but also difficult to solve. The output signals of triboelectric nanogenerator system depends on the initial conditions of the system, which reflects the dynamic behavior of the system. Currently, there is little research on using the output signals to study the dynamic behavior of triboelectric nanogenerator system. Therefore, a dynamic simulation model of a freestanding triboelectric nanogenerator (F-TENG) system was established using COMSOL software. The mathematical model of the F-TENG system was fitted. Using chaos theory and its phase-space reconstruction technology, the phase-space trajectories of the F-TENG system were reconstructed through voltage signals under external factors, and the dynamic behavior of the F-TENG system was discussed and studied. Finally, validation experiments were carried out. The research shows that with the variation of internal factors and operating frequency, the mathematical model of the F-TENG system remains basically unchanged, and the variation law of output characteristics also remains basically unchanged. The F-TENG system is in periodic motion; with the variation of external factors such as resistance and friction layer spacing, the mathematical model of the F-TENG system will evolve in a complex direction, and the output characteristics will gradually distort. The F-TENG system will evolve from periodic motion to chaotic motion. The research results not only reveal the dynamic behavior and enrich the nonlinear dynamic theory of triboelectric nanogenerator system, but also provide theoretical guidance for increasing the working bandwidth and stability of triboelectric nanogenerator system.

Heat transfer analysis for the Falkner–Skan wedge flow by the differential transformation method

In this paper the Modified Equations of Emden type (MEE), χ+αχχ+βχ 3 is solved numerically by the differential transform method. This technique doesn’t require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. The current results of this paper are in excellent agreement with those provided by Chandrasekar et al. [1] and thereby illustrate the reliability and the performance of the differential transform method. We have also compared the results with the classical Runge-Kutta 4 (RK4) Method.

Aero-engine rotor system contains a variety of connectors such as bolts and bearings. However, the dynamic behaviors of the connection are complex in actual engineering. In this paper, the dynamic behavior of the rotor-bearing system with bolted joints and the influence of the parameters of the joints on it are clarified. The parameters include the bearing clearance and the deflection caused by the uneven bolt preload. A rotor-bearing model with bolted joints is established by using Lagrange equations. The bifurcation diagram is solved to explore the tendency of vibration of the system at different speeds by considering the Hertz contact force. Furthermore, the phase diagram, Poincaré map, time-domain steady-state response curve, and spectrum diagram are used to discuss the dynamic behaviors of the system specifically. The influence of the bearing clearance change and uneven bolt preload on the dynamic behaviors is investigated by using the bifurcation diagram. The result shows that the motion of the system is extremely complex, which includes chaotic motion. The Lyapunov exponent is then calculated to verify whether the system enters chaos. The correctness of the model is verified by comparing the result in this work with those in the literature. The model can clarify the dynamic behaviors of the system well and has high accuracy, which can provide the theoretical guidance for the design of rotor-bearing systems with bolted joints.

Modeling and analysis of dynamic behaviors for a flexible rotor system supported in tilting pad aerodynamic bearings

Simulation of settling velocity and motion of particles in drilling operation

In this work, the multi-step differential transformation method (MSDTM) is applied to approximate a solution of the hyperchaotic Rossler system. MSDTM is adapted from the differential transformation method (DTM). In this method, DTM is implemented in each subinterval. Results are compared with a fourth-order Runge Kutta method and a standard DTM. The results show that the MSDTM is an efficient and powerful technique for solving hyperchaotic Rossler systems and this method is more accurate than DTM.

A multi-step differential transform method and application to non-chaotic or chaotic systems

Problem statement: In this study, a continuous solution for Genesio system was considered using Differential Transformation Method (DTM). Approach: Numerical results were compared to those obtained by the Runge-Kutta method to evaluate the performance of the suggested method. Results: The accuracy of the DTM was tested as the chaotic Genesio system. Conclusion/Recommendations: It was shown that the DTM was robust, accurate and easy to apply and gave analytical solution on each subinterval, which was not possible in the purely numerical method.

Analyzing the free vibrations of a plate using finite difference and differential transformation method

Application of the differential transformation method to vibration analysis of pipes conveying fluid

The application of differential transformation method to nonlinear equations arising in heat transfer

Predator-prey models are the building blocks of the ecosystems as biomasses are grown out of their resource masses. Different relationships exist between these models as different interacting species compete, metamorphosis occurs and migrate strategically aiming for resources to sustain their struggle to exist. To numerically investigate these assumptions, ordinary differential equations are formulated, and a variety of methods are used to obtain and compare approximate solutions against exact solutions, although most numerical methods often require heavy computations that are time-consuming. In this paper, the traditional differential transform (DTM) method is implemented to obtain a numerical approximate solution to prey-predator models. The solution obtained with DTM is convergent locally within a small domain. The multi-step differential transform method (MSDTM) is a technique that improves DTM in the sense that it increases its interval of convergence of the series expansion. One predator-one prey and two-predator-one prey models are considered with a quadratic term which signifies other food sources for its feeding. The result obtained numerically and graphically shows point DTM diverges. The advantage of the new algorithm is that the obtained series solution converges for wide time regions and the solutions obtained from DTM and MSDTM are compared with solutions obtained using the classical Runge-Kutta method of order four. The results demonstrated is that MSDTM computes fast, is reliable and gives good results compared to the solutions obtained using the classical Runge-Kutta method.