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24 pages, 12147 KiB

Open AccessArticle

Analysis of Nonlinear Vibration Characteristics and Whirl Behavior of Dual-Rotor Systems with Inter-Shaft Rub Impact

by Zhi Wang, Rui Sun, Yu Liu, Yudong Yao and Jing Tian

Mathematics 2024, 12(10), 1436; https://doi.org/10.3390/math12101436 - 7 May 2024

Viewed by 916

Previous studies on rub-impact faults have mainly focused on the rub-impact between rotors and stators, with less research on inter-rotor rub impact. The impact of inter-rotor rub impact on rotor nonlinear vibration is particularly significant. This study investigates the effects of inter-shaft rub [...] Read more.

Previous studies on rub-impact faults have mainly focused on the rub-impact between rotors and stators, with less research on inter-rotor rub impact. The impact of inter-rotor rub impact on rotor nonlinear vibration is particularly significant. This study investigates the effects of inter-shaft rub impact on the vibration characteristics and whirl behavior of dual-rotor systems. Initially, a dual-rotor model with inter-shaft bearings is established using the finite element method, and inter-shaft rub-impact forces are derived based on contact mechanics. Next, the system response is solved using the Newmark method. Vibration characteristics are analyzed through Campbell diagrams, 3D waterfall plots, time-frequency domain plots, and steady-state rub-impact force plots. Finally, the influence of inter-shaft rub impact on the whirl behavior of the dual-rotor system is studied based on the theory of full-spectrum analysis. The study concludes that inter-shaft rub-impact faults shift the system’s resonance points backward, increase harmonic and combination frequency components, and significantly affect the system response under dual-rotor co-rotation. Excessive friction can lead to self-excited vibrations and sudden amplitude increases, particularly in the LP rotor frequency. Additionally, inter-shaft rub impact primarily affects the whirl behavior of the LP-compressor disk1, showing multiple cycles of forward and backward whirl alternation during acceleration due to combined unbalanced and rub-impact excitations. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

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15 pages, 4964 KiB

Open AccessArticle

A Method for Reducing Sub-Divisional Errors in Open-Type Optical Linear Encoders with Angle Shift Pattern Main Scale

by Xinji Lu, Fan Yang and Artūras Kilikevičius

Mathematics 2024, 12(3), 474; https://doi.org/10.3390/math12030474 - 1 Feb 2024

Cited by 1 | Viewed by 1066

Abstract

In this research, a novel approach is presented to enhance the precision of open-type optical linear encoders, focusing on reducing subdivisional errors (SDEs). Optical linear encoders are crucial in high-precision machinery. The overall error in optical linear encoders encompasses baseline error, SDE, and [...] Read more.

In this research, a novel approach is presented to enhance the precision of open-type optical linear encoders, focusing on reducing subdivisional errors (SDEs). Optical linear encoders are crucial in high-precision machinery. The overall error in optical linear encoders encompasses baseline error, SDE, and position noise. This study concentrates on mitigating SDEs, which are recurrent errors within each pitch period and arise from various contributing factors. A novel method is introduced to improve the quality of sinusoidal signals in open-type optical linear encoders by incorporating specially designed angle shift patterns on the main scale. The proposed method effectively suppresses the third order harmonics, resulting in enhanced accuracy without significant increases in production costs. Experimental results indicate a substantial reduction in SDEs compared to traditional methods, emphasizing the potential for cost-effective, high-precision optical linear encoders. This paper also discusses the correlation between harmonic suppression and SDE reduction, emphasizing the significance of this method in achieving higher resolutions in optical linear encoders. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

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26 pages, 550 KiB

Open AccessArticle

Approximate Closed-Form Solutions for a Class of 3D Dynamical Systems Involving a Hamilton–Poisson Part

by Remus-Daniel Ene and Nicolina Pop

Mathematics 2023, 11(23), 4811; https://doi.org/10.3390/math11234811 - 28 Nov 2023

Cited by 2 | Viewed by 774

Abstract

The goal of this paper is to build some approximate closed-form solutions for a class of dynamical systems involving a Hamilton–Poisson part. The chaotic behaviors are neglected. These solutions are obtained by means of a new version of the optimal parametric iteration method [...] Read more.

The goal of this paper is to build some approximate closed-form solutions for a class of dynamical systems involving a Hamilton–Poisson part. The chaotic behaviors are neglected. These solutions are obtained by means of a new version of the optimal parametric iteration method (OPIM), namely, the modified optimal parametric iteration method (mOPIM). The effect of the physical parameters is investigated. The Hamilton–Poisson part of the dynamical systems is reduced to a second-order nonlinear differential equation, which is analytically solved by the mOPIM procedure. A comparison between the approximate analytical solution obtained with mOPIM, the analytical solution obtained with the iterative method, and the corresponding numerical solution is presented. The mOPIM technique has more advantages, such as the convergence control (in the sense that the residual functions are smaller than 1), the efficiency, the writing of the solutions in an effective form, and the nonexistence of small parameters. The accuracy of the analytical and corresponding numerical results is illustrated by graphical and tabular representations. The same procedure could be successfully applied to more dynamical systems. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

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14 pages, 299 KiB

Open AccessArticle

Oscillation Analysis Algorithm for Nonlinear Second-Order Neutral Differential Equations

by Liang Song, Shaodong Chen and Guoxin Wang

Mathematics 2023, 11(16), 3478; https://doi.org/10.3390/math11163478 - 11 Aug 2023

Viewed by 1138

Abstract

Differential equations are useful mathematical tools for solving complex problems. Differential equations include ordinary and partial differential equations. Nonlinear equations can express the nonlinear relationship between dependent and independent variables. The nonlinear second-order neutral differential equations studied in this paper are a class [...] Read more.

Differential equations are useful mathematical tools for solving complex problems. Differential equations include ordinary and partial differential equations. Nonlinear equations can express the nonlinear relationship between dependent and independent variables. The nonlinear second-order neutral differential equations studied in this paper are a class of quadratic differentiable equations that include delay terms. According to the t-value interval in the differential equation function, a basis is needed for selecting the initial values of the differential equations. The initial value of the differential equation is calculated with the initial value calculation formula, and the existence of the solution of the nonlinear second-order neutral differential equation is determined using the condensation mapping fixed-point theorem. Thus, the oscillation analysis of nonlinear differential equations is realized. The experimental results indicate that the nonlinear neutral differential equation can analyze the oscillation behavior of the circuit in the Colpitts oscillator by constructing a solution equation for the oscillation frequency and optimizing the circuit design. It provides a more accurate control for practical applications. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

17 pages, 6012 KiB

Open AccessArticle

An Advancement in Truck-Tire–Road Interaction Using the Finite Element Analysis

by Haniyeh Fathi, Mehran Khosravi, Zeinab El-Sayegh and Moustafa El-Gindy

Mathematics 2023, 11(11), 2462; https://doi.org/10.3390/math11112462 - 26 May 2023

Cited by 7 | Viewed by 2174

Abstract

This paper aimed to investigate the cornering characteristics of a Regional Haul Steer II, RHS 315/80 R22.5 truck tire traveling on a dry, hard surface using the Finite element analysis (FEA). This research was carried out using commercial Finite Element software and Pam-Crash [...] Read more.

This paper aimed to investigate the cornering characteristics of a Regional Haul Steer II, RHS 315/80 R22.5 truck tire traveling on a dry, hard surface using the Finite element analysis (FEA). This research was carried out using commercial Finite Element software and Pam-Crash in an Explicit Environment. A finite element truck tire model was developed to apply the tire terrain cornering condition. The concentrated loads and boundary conditions for the rim and wheel were applied to the model. The rubber material was defined using the Mooney–Rivlin model. The truck tire cornering operating conditions, including three different speeds with respect to various positive slip angles, were investigated. Several simulations were repeated at various operating conditions, including three different inflation pressures and three different vertical loads. Subsequently, the tire lateral force was computed using the local and global frame coordinates. Additionally, the self-aligning moment was extracted from the tire cross-section at each operating condition. Finally, a comparison between the simulation results showed that the tire lateral force was highly sensitive to the variation of the slip angles at the higher domain, and also that the tire inflation pressure, regardless of the speed, was considered to be one of the main parameters directly affecting the tire-cornering properties. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

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28 pages, 1010 KiB

Open AccessArticle

Partial Slip Effects for Thermally Radiative Convective Nanofluid Flow

by Remus-Daniel Ene, Nicolina Pop and Rodica Badarau

Mathematics 2023, 11(9), 2199; https://doi.org/10.3390/math11092199 - 6 May 2023

Viewed by 1494

Abstract

The partial slip effects for radiative convective nanofluid flow over a stretching sheet in porous medium are analytically explored in this work. The Navier–Stokes equations, the momentum and the energy equations are converted into a set of non-linear ODEs by the similarity transformation. [...] Read more.

The partial slip effects for radiative convective nanofluid flow over a stretching sheet in porous medium are analytically explored in this work. The Navier–Stokes equations, the momentum and the energy equations are converted into a set of non-linear ODEs by the similarity transformation. Using the modified optimal homotopy asymptotic method (OHAM), the resulting non-linear ODEs are analytically approximately solved. The impact of various parameters, such as: the velocity exponential factor n, the wall thickness parameter

γ

, the dimensionless velocity slip parameter

δ_{1}

, the Prandtl number

P r

, the radiation parameter R, and the dimensionless temperature jump parameter

δ_{2}

, on the behaviour of the mass and heat transfer is presented. The influence of these parameters is tabular and graphically presented. An excellent agreement between the approximate analytical solution and the corresponding numerical solution is highlighted. The results obtained confirm that modified OHAM is a useful and competitive mathematical tool to explore a large class of non-linear problems with applications in various fields of science and engineering. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

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24 pages, 10894 KiB

Open AccessArticle

Analysis of the Vibro-Impact Nonlinear Damped and Forced Oscillator in the Dynamics of the Electromagnetic Actuation

by Nicolae Herisanu, Bogdan Marinca, Livija Cveticanin and Vasile Marinca

Mathematics 2023, 11(9), 2194; https://doi.org/10.3390/math11092194 - 6 May 2023

Cited by 1 | Viewed by 1404

Abstract

In this work, the effect of vibro-impact nonlinear, forced, and damped oscillator on the dynamics of the electromagnetic actuation (EA) near primary resonance is studied. The vibro-impact regime is given by the presence of the Hertzian contact. The EA is supplied by a [...] Read more.

In this work, the effect of vibro-impact nonlinear, forced, and damped oscillator on the dynamics of the electromagnetic actuation (EA) near primary resonance is studied. The vibro-impact regime is given by the presence of the Hertzian contact. The EA is supplied by a constant current generating a static force and by an actuation generating a fast alternative force. The deformations between the solids in contact are supposed to be elastic and the contact is maintained. In this study, a single degree of freedom nonlinear damped oscillator under a static normal load is considered. An analytical approximate solution of this problem is obtained using the Optimal Auxiliary Functions Method (OAFM). By means of some auxiliary functions and introducing so-called convergence-control parameters, a very accurate approximate solution of the governing equation can be obtained. We need only the first iteration for this technique, applying a rigorous mathematical procedure in finding the optimal values of the convergence-control parameters. Local stability by means of the Routh-Hurwitz criteria and global stability using the Lyapunov function are also studied. It should be emphasized that the amplitude of AC excitation voltage is not considered much lower than bias voltage (in contrast to other studies). Also, the Hertzian contact coupled with EA is analytically studied for the first time in the present work. The approximate analytical solution is determined with a high accuracy on two domains. Local stability is established in five cases with some cases depending on the trace of the Jacobian matrix and of the discriminant of the characteristic equation. In the study of global stability, the estimate parameters which are components of the Lyapunov function are given in a closed form and a graphical form and therefore the Lyapunov function is well-determined. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

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14 pages, 13230 KiB

Open AccessArticle

Nonlinear Constitutive and Mechanical Properties of an Auxetic Honeycomb Structure

by Qian Ma and Junhua Zhang

Mathematics 2023, 11(9), 2062; https://doi.org/10.3390/math11092062 - 26 Apr 2023

Cited by 2 | Viewed by 1621

Abstract

Auxetic honeycomb has unique mechanical properties such as good energy absorption capacity, tensile strength and fracture toughness, etc. Therefore, honeycomb with a negative Poisson’s ratio is used widely in medical, biological, aerospace and other fields. This honeycomb has large deformations in energy absorption [...] Read more.

Auxetic honeycomb has unique mechanical properties such as good energy absorption capacity, tensile strength and fracture toughness, etc. Therefore, honeycomb with a negative Poisson’s ratio is used widely in medical, biological, aerospace and other fields. This honeycomb has large deformations in energy absorption and vibration reduction. It is very important to study the nonlinear constitutive of the honeycomb structure. Therefore, this paper establishes the nonlinear constitutive relationship of the auxetic honeycomb structure under large deformations. This constitutive relation includes the in-plane stress, in-plane strain, Young’s modulus and Poisson’s ratio of the negative Poisson’s ratio honeycomb. The finite element model of the negative Poisson’s ratio honeycomb cells is established, and the calculated results of finite element model are compared with that of the theoretical calculation results. On this basis, the influence of the geometric parameters on the mechanical properties of the structure is studied. The results of this paper will provide a theoretical basis for the further study of the auxetic honeycomb sandwich structure and provide a basis for the engineering application of honeycomb structures. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

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11 pages, 2754 KiB

Open AccessArticle

Exact Closed-Form Solution for the Oscillator with a New Type of Mixed Nonlinear Restitution Force

by Livija Cveticanin

Mathematics 2023, 11(3), 596; https://doi.org/10.3390/math11030596 - 23 Jan 2023

Cited by 1 | Viewed by 1227

Abstract

This paper shows an oscillator with a spring made of material where the stress is a function not only of strain but also strain rate. The corresponding restitution force is of strong nonlinear monomial type and is the product of displacement and velocity [...] Read more.

This paper shows an oscillator with a spring made of material where the stress is a function not only of strain but also strain rate. The corresponding restitution force is of strong nonlinear monomial type and is the product of displacement and velocity of any order. The mathematical model of the oscillator is a homogenous strong nonlinear second-order differential equation with an integer- or non-integer-order mixed term. In the paper, an analytical procedure for solving this new type of strong nonlinear equation is developed. The approximate solution is assumed as the perturbed version of the exact solution in the form of a sine Ateb function. As a result, it is obtained that the amplitude, period, and frequency of vibration depend not only on the coefficient and order of nonlinearity, but also on the initial velocity. The procedure is tested on two examples: oscillator perturbed with small linear damping and small linear displacement functions. The analytically obtained results are compared with the exact numerical ones and show good agreement. It is concluded that the mathematical model and also the procedure developed in the paper would be convenient for prediction of motion for this type of oscillator without necessary experimental testing. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

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11 pages, 1781 KiB

Open AccessArticle

Nonlinear Vibration of Electrostatically Actuated Microbeam

by Gamal M. Ismail, Md. Alal Hosen, Mostafa Mohammadian, Maha M. El-Moshneb and Mahmoud Bayat

Mathematics 2022, 10(24), 4762; https://doi.org/10.3390/math10244762 - 15 Dec 2022

Cited by 3 | Viewed by 1606

Abstract

In this paper, an analytical technique based on the global residue harmonic balance method (GRHBM) is applied in order to obtain higher-order approximate analytical solutions of an electrostatically actuated micro-beam. To illustrate the applicability and accuracy of the method, a high level of [...] Read more.

In this paper, an analytical technique based on the global residue harmonic balance method (GRHBM) is applied in order to obtain higher-order approximate analytical solutions of an electrostatically actuated micro-beam. To illustrate the applicability and accuracy of the method, a high level of accuracy was established for the analytical solutions by comparing the results of the solutions with the numerical solution as well as the already published literature, such as the variational approach (VA), Hamiltonian approach (HA), energy balance method (EBM), and homotopy analysis method (HAM). It is shown that the GRHB method can be easily applied to nonlinear problems and provides solutions with a higher precision than existing methods. The obtained analytical expressions are employed to study the effects of axial force, initial gape, and electrostatic load on nonlinear frequency. Full article

(This article belongs to the Special Issue Nonlinear Vibration Theory and Mechanical Dynamics)

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