Fengxia Wang

Bio: Fengxia Wang is an academic researcher from Southern Illinois University Edwardsville. The author has contributed to research in topics: Nonlinear system & Energy harvesting. The author has an hindex of 10, co-authored 33 publications receiving 451 citations. Previous affiliations of Fengxia Wang include Purdue University & Beijing University of Technology.

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

Abstract: This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.

Global bifurcations and chaos for a rotor-active magnetic bearing system with time-varying stiffness

Abstract: In this paper, we investigate the global bifurcations and chaotic dynamics in the rotor-active magnetic bearings (AMB) system with 8-pole legs and time-varying stiffness. From the averaged equation obtained in another paper, the theory of normal form is applied in this paper to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Since the normal form obtained here is not the simplest one, the methods of choosing other complementary space and utilizing the inner product are presented to further reduce the normal form and obtain a simpler normal form. Based on simpler the normal form obtained above, a global perturbation method is utilized for the analysis of global bifurcations and chaotic dynamics of the rotor-AMB system. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation of the rotor-AMB system with the time-varying stiffness. These results indicate that the chaotic motions can occur in the rotor-AMB system with time-varying stiffness. Numerical simulations verify the analytical predictions. The jumping and catastrophic phenomena of the amplitude for the chaotic oscillations in the system are also found by using numerical simulations.

Nonlinear Normal Modes and Their Bifurcations for an Inertially Coupled Nonlinear Conservative System

Abstract: This work concerns the nonlinear normal modes (NNMs) of a 2 degree-of-freedom autonomous conservative spring–mass–pendulum system, a system that exhibits inertial coupling between the two generalized coordinates and quadratic (even) nonlinearities. Several general methods introduced in the literature to calculate the NNMs of conservative systems are reviewed, and then applied to the spring–mass–pendulum system. These include the invariant manifold method, the multiple scales method, the asymptotic perturbation method and the method of harmonic balance. Then, an efficient numerical methodology is developed to calculate the exact NNMs, and this method is further used to analyze and follow the bifurcations of the NNMs as a function of linear frequency ratio p and total energy h. The bifurcations in NNMs, when near 1:2 and 1:1 resonances arise in the two linear modes, is investigated by perturbation techniques and the results are compared with those predicted by the exact numerical solutions. By using the method of multiple time scales (MTS), not only the bifurcation diagrams but also the low energy global dynamics of the system is obtained. The numerical method gives reliable results for the high-energy case. These bifurcation analyses provide a significant glimpse into the complex dynamics of the system. It is shown that when the total energy is sufficiently high, varying p, the ratio of the spring and the pendulum linear frequencies, results in the system undergoing an order–chaos–order sequence. This phenomenon is also presented and discussed.

Introduction to Functional Analysis. By Angus E. Taylor. Pp. xvi. 423. $12.50. 1958. (John Wiley and Sons)

Abstract: This is lecture notes for several courses on Functional Analysis at School of Mathematics of University of Leeds. They are based on the notes of Dr. Matt Daws, Prof. Jonathan R. Partington and Dr. David Salinger used in the previous years. Some sections are borrowed from the textbooks, which I used since being a student myself. However all misprints, omissions, and errors are only my responsibility. I am very grateful to Filipa Soares de Almeida, Eric Borgnet, Pasc Gavruta for pointing out some of them. Please let me know if you find more. The notes are available also for download in PDF. The suggested textbooks are [1,6,8,9]. The other nice books with many interesting problems are [3, 7]. Exercises with stars are not a part of mandatory material but are nevertheless worth to hear about. And they are not necessarily difficult, try to solve them! CONTENTS List of Figures 3 Notations and Assumptions 4 Integrability conditions 4 1. Motivating Example: Fourier Series 4 1.1. Fourier series: basic notions 4 1.2. The vibrating string 8 1.3. Historic: Joseph Fourier 10 2. Basics of Linear Spaces 11 2.1. Banach spaces (basic definitions only) 12 2.2. Hilbert spaces 14 2.3. Subspaces 16 2.4. Linear spans 19 3. Orthogonality 20 3.1. Orthogonal System in Hilbert Space 21 3.2. Bessel’s inequality 23 3.3. The Riesz–Fischer theorem 25 3.4. Construction of Orthonormal Sequences 26 3.5. Orthogonal complements 28 4. Fourier Analysis 29 Date: 16th October 2017. 1

3D Printing Technologies for Flexible Tactile Sensors toward Wearable Electronics and Electronic Skin.

Abstract: 3D printing has attracted a lot of attention in recent years. Over the past three decades, various 3D printing technologies have been developed including photopolymerization-based, materials extrusion-based, sheet lamination-based, binder jetting-based, power bed fusion-based and direct energy deposition-based processes. 3D printing offers unparalleled flexibility and simplicity in the fabrication of highly complex 3D objects. Tactile sensors that emulate human tactile perceptions are used to translate mechanical signals such as force, pressure, strain, shear, torsion, bend, vibration, etc. into electrical signals and play a crucial role toward the realization of wearable electronics and electronic skin. To date, many types of 3D printing technologies have been applied in the manufacturing of various types of tactile sensors including piezoresistive, capacitive and piezoelectric sensors. This review attempts to summarize the current state-of-the-art 3D printing technologies and their applications in tactile sensors for wearable electronics and electronic skin. The applications are categorized into five aspects: 3D-printed molds for microstructuring substrate, electrodes and sensing element; 3D-printed flexible sensor substrate and sensor body for tactile sensors; 3D-printed sensing element; 3D-printed flexible and stretchable electrodes for tactile sensors; and fully 3D-printed tactile sensors. Latest advances in the fabrication of tactile sensors by 3D printing are reviewed and the advantages and limitations of various 3D printing technologies and printable materials are discussed. Finally, future development of 3D-printed tactile sensors is discussed.