In this work, the Duffing’s type analytical frequency–amplitude relationship for nonlinear oscillators is derived by using Hés formulation and Jacobi elliptic functions. Comparison of the numerical ...results obtained from the derived analytical expression using Jacobi elliptic functions with respect to the exact ones is performed by considering weak and strong Duffing’s nonlinear oscillators.
In this paper, the approximate frequency-amplitude response of the fractal equation that describes the non-Gaussian polymer chains’ motion is derived using a two-scale dimension transform and the ...ancient Chinese algorithm Ying Bu Zu Shu. Theoretical predictions show that the oscillation amplitude of the polymer chains varies since the response curves shift to the left (
α
<
1
) and to the right (
α
>
1
) from those curves plotted for a fractal parameter value
α
=
1
. This system response suggests that the magnitude of the molecular chain stretching is modified during motion due to the polymer fractal molecular structure, which indicates the mobility and flexibility of the polymer molecules that unveil possible molecular structure defects and intermolecular cohesion.
In this work, a previously developed mathematical model to predict bulk density of SLMed (produced via Selective Laser Melting) component is enhanced by taking laser power, scanning speed, hatch ...spacing, powder's thermal conductivity and specific heat capacity as independent variables. Experimental data and manufacturing conditions for the selective laser melting (SLM) of metallic materials (which include aluminum, steel, titanium, copper, tungsten and nickel alloys) are adapted from the literature and used to evaluate the validity of the proposed enhanced model. A strong relation between dependent and independent dimensionless products is observed throughout the studied materials. The proposed enhanced mathematical model shows to be highly accurate since the computed root-mean-square-error values (RMSE) does not exceed 5 × 10
. Furthermore, an analytical expression for the prediction of bulk density of SLMed components was developed. From this, an expression for determining the needed scanning speed, with respect to laser power, to achieve highly dense components produced via SLM, is derived.
This paper aims to apply a transformation method that replaces the elastic forces of the original equation of motion with a power-form elastic term. The accuracy obtained from the derived equivalent ...equations of motion is evaluated by studying the finite-amplitude damped, forced vibration of a vertically suspended load body supported by incompressible, homogeneous, and isotropic viscohyperelastic elastomer materials. Numerical integrations of the original equations of two oscillators described by neo-Hookean and Mooney–Rivlin viscohyperelastic elastomer material models, and their equivalent equations of motion, are compared to the frequency–amplitude steady-state solutions obtained from the harmonic balance and the averaging methods. It is shown from numerical integrations and approximate steady-state solutions that the equivalent equations predict well the original system dynamic response despite having higher system nonlinearities.
This paper introduces a novel methodology to determine the frequency-amplitude relationship of fractal-order viscoelastic polymer materials using the two-scale fractal dimension transform, the ...equivalent power-form representation of the conservative restoring forces, and a simple coordinate transformation to eliminate viscoelastic effects. Then, the ancient Chinese algorithm Ying Bu Zu Shu and He’s formulation are used for obtaining the frequency-amplitude relationship. Simulation results obtained from the derived expressions exhibit good agreement when compared to numerical integration solutions. This article elucidates how the molecular structure of polymer chains influences the relaxation oscillations as a function of the fractal parameter values.
In this study, the kinetic parameters belonging to the cross-linking process of a modified epoxy resin, Aerotuf 275-34™, were investigated. Resin curing kinetics are crucial to understanding the ...structure-property-processing relationship for manufacturing high-performance carbon-fiber-reinforced polymer composites (CFRPCs). The parameters were obtained using differential scanning calorimetry (DSC) measurements and the Flynn-Wall-Ozawa, Kissinger, Borchardt-Daniels, and Friedman approaches. The DSC thermograms show two exothermic peaks that were deconvoluted as two separate reactions that follow autocatalytic models. Furthermore, the mechanical properties of produced carbon fiber/Aerotuf 275-34™ laminates using thermosetting polymers such as epoxies, phenolics, and cyanate esters were evaluated as a function of the conversion degree, and a close correlation was found between the degree of curing and the ultimate tensile strength (UTS). We found that when the composite material is cured at 160 °C for 15 min, it reaches a conversion degree of 0.97 and a UTS value that accounts for 95% of the maximum value obtained at 200 °C (180 MPa). Thus, the application of such processing conditions could be enough to achieve good mechanical properties of the composite laminates. These results suggest the possibility for the development of strategies towards manufacturing high-performance materials based on the modified epoxy resin (Aerotuf 275-34™) through the curing process.
In this work, dimensional analysis is used to develop a general mathematical model to predict bulk density of SLMed components taking volumetric energy density, scanning speed, powder's thermal ...conductivity, specific heat capacity, and average grain diameter as independent variables. Strong relation between dependent and independent dimensionless products is observed. Inconel 718 samples were additively manufactured and a particular expression, in the form of a power-law polynomial, for its bulk density, in the working domain of the independent dimensionless product, was obtained. It is found that with longer laser exposure time, and lower scanning speed, better densification is attained. Likewise, volumetric energy density has a positive influence on bulk density. The negative effect of laser power in bulk density is attributed to improper process conditions leading to powder particle sublimation and ejection. A maximum error percentage between experimental and predicted bulk density of 3.7119% is achieved, which corroborates the accuracy of our proposed model. A general expression for determining the scanning speed, with respect to laser power, needed to achieve highly dense components, was derived. The model's applicability was further validated considering SLMed samples produced by AlSi10Mg and Ti6Al4V alloys. This article elucidates how to tune relevant manufacturing parameters to produce highly dense SLM parts using mathematical expressions derived from Buckingham's π- theorem.
ABSTRACTWe introduce the innovative concept of Sustainable Laser Energy Consumption (SLEC, ε) that promotes sustainable metal additive manufacturing practices while maintaining productivity and ...tailored densification in the Selective Laser Melting (SLM) process. This study focuses on deriving analytical expressions through dimensional analysis and Buckingham’s π theorem to predict bulk density with high precision across studied materials: In718, W, AlSi10Mg, Ti6Al4V, and SS316L. This approach provides valuable insights and allows the setting of an appropriate combination of scanning speed and hatch distance for a range of laser power, enabling sustainable manufacturing of SLMed components with tailored density. Furthermore, the mathematical expression introduced as SLEC facilitates the tradeoff between energy consumption and productivity, promoting a greener and more sustainable future of metal additive manufacturing. Furthermore, the underlying fractal nature of SLM is explored through the investigation of the influence of fractal structures, present in the powder bed, on densification that provides information on how powder morphology and homogeneity (i.e. fractal dimension and lacunarity) of the powder bed are linked to a sustainable SLM process.
This paper uses the two-scale fractal dimension transform and He’s formula derived from the ancient Chinese algorithm Ying Bu Zu Shu to find the approximate frequency–amplitude expression of the ...fractal and forced anharmonic oscillator that can be used to study the nonlinear oscillations produced by the plasma physics fractal structures. The results show how the electron frequency and wavelength change as a function of the plasma physics fractal structure. In fact, if the value of the fractal parameter is decreased, the wavelength increases, and consequently, the system frequency decreases. The introduced solution procedure sheds a bright light on the easy-to-follow steps to obtain an accurate steady-state analytical solution of fractal anharmonic nonlinear oscillators.
Dimensional analysis through the Buckingham Pi theorem was confirmed as an efficient mathematical tool to model the otherwise non-linear high order ultrasonic micro-injection molding process (UMIM). ...Several combinations of processing conditions were evaluated to obtain experimental measurements and validate the derived equations. UMIM processing parameters, output variable energy consumption, and final specimen’s Young modulus were arranged in dimensionless groups and formulated as functional relationships, which lead to dimensionless equations that predict output variables as a function of the user-specified processing parameters and known material properties.
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