Continuous and Discrete Spectra

This chapter explores the spectra of audio signals first from a continuous time and continuous frequency perspective. It starts by reviewing Fourier's theorem and then moves on to put it into its most general form, the Fourier transform. The spectra of simple signals are explored and determined. The operation of convolution is introduced, and through its discrete form, it is applied to the concepts of spectrum and waveform as mediated by the Fourier transform. The chapter is completed with a study of the discrete spectra of classic synthesis waveforms. A revised notion of spectrum is presented at the conclusion.

Camshaft Loosening Diagnosis on the Basis of Generalised Force Recognition at the Centre of Gravity of an Engine

A valve mechanism supports the working process of an engine cylinder, and a camshaft is a key component required to open and close a valve. When a camshaft loosens, the balance of the engine disrupts. In the meanwhile, the generalised force at its centre of gravity (CG) alters. This study proposed a novel technique to detect camshaft loosening based on recognising the generalised force at the CG of the engine. We conducted Hanning windowed interpolation of discrete spectra to extract the precise phase and amplitude by utilising the acceleration signals at the engine cylinder and mounts and cylinder head. We then accurately computed the generalised force at the CG. Finally, we accurately extracted the camshaft loosening features by analysing the main harmonic orders for the generalised force. As indicated by simulations, our method can be used to effectively detect combustion engine faults involving camshaft loosening.

Nonperturbative Quantization Approach for QED on the Hopf Bundle

We consider the Dirac equation and Maxwell’s electrodynamics in R×S3 spacetime, where a three-dimensional sphere is the Hopf bundle S3→S2. In both cases, discrete spectra of classical solutions are obtained. Based on the solutions obtained, the quantization of free, noninteracting Dirac and Maxwell fields is carried out. The method of nonperturbative quantization of interacting Dirac and Maxwell fields is suggested. The corresponding operator equations and the infinite set of the Schwinger–Dyson equations for Green’s functions is written down. We write a simplified set of equations describing some physical situations to illustrate the suggested scheme of nonperturbative quantization. Additionally, we discuss the properties of quantum states and operators of interacting fields.

Discrete spectra for some complex infinite band matrices

Under suitable assumptions the eigenvalues for an unbounded discrete operator \(A\) in \(l_2\), given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let \[\Lambda (A)=\{\lambda \in {\rm Lim}_{n\to \infty} \lambda _n : \lambda _n \text{ is an eigenvalue of } A_n \text{ for } n \geq 1 \},\] where \({\rm Lim}_{n\to \infty} \lambda_n\) is the set of all limit points of the sequence \((\lambda_n)\) and \(A_n\) is a finite dimensional orthogonal truncation of \(A\). The aim of this article is to provide the conditions that are sufficient for the relations \(\sigma(A) \subset \Lambda(A)\) or \(\Lambda (A) \subset \sigma (A)\) to be satisfied for the band operator \(A\).

Chaotic dynamics of cylindrical roller bearing supported by unbalanced rotor due to localized defects

This article presents a nonlinear vibration signature study of high-speed defective cylindrical roller bearings under unbalance rotor conditions. Qualitative analysis is conducted considering a spall defect of a specific size on major elements such as outer race, inner race, and rollers. A spring-mass model with nonlinear stiffness and damping is formulated to study the dynamic behavior of the rotor-bearing model. The set of nonlinear differential equations are solved using the fourth-order Runge–Kutta method to predict the characteristics of the discrete spectra and analyze the stability of the system. The results show that higher impulsive forces are generated because of outer race defects than defects in the inner race and roller. This can be explained as every time the roller passes through the defect in the outer race during rotation, the energy is released. However, in the case of both the roller and inner race defects, the impulsive force generated in the load zone is averaged because of the force generated in the unloading zone. The route to chaos from periodic to quasiperiodic response has been observed and analyzed that vibration signature is very much sensitive not only to the defects of bearing components but also to the rotor speed.

On the B-discrete spectrum

In this paper, we introduce the B-discrete spectrum of an unbounded closed operator and we prove that a closed operator has a purely B-discrete spectrum if and only if it has a meromorphic resolvent. After that, we study the stability of the B-discrete spectrum under several type of perturbations and we establish that two closed invertible linear operators having quasisimilar totally paranormal inverses have equal spectra and B-discrete spectra.