Dimensionless Equations Of Motion Research Articles

Effect of Angular Speed Variations on the Nonlinear Vibrations of a Rotational Spring-Mass System

A rotating spring-mass system is considered using polar coordinates. The system contains a cubic nonlinear spring with damping. The angular velocity harmonically fluctuates about a mean velocity. The dimensionless equations of motion are derived first. The velocity fluctuations resulted in a direct and parametric forcing terms. Approximate analytical solutions are sought using the Method of Multiple Scales, a perturbation technique. The primary resonance and the principal parametric resonance cases are investigated. The amplitude and frequency modulation equations are derived for each case. By considering the steady state solutions, the frequency response relations are derived. The bifurcation points are discussed for the problems. It is found that speed fluctuations may have substantial effects on the dynamics of the problem and the fluctuation frequency and amplitude can be adjusted as passive control parameters to maintain the desired responses.

Scaling of attraction force and rolling resistance in DEM with reduced particle stiffness

In Discrete Element Method (DEM), the stiffness of particles is often reduced artificially for calculation speed-up. It is known that the attraction force and rolling resistance should be scaled to counter-balance the excessive energy dissipation caused by prolonged contact duration with reduced stiffness. The present study theoretically derives the scaling laws from dimensionless equations of motion in a generic form, which are for both particle translation and rotation and with multi-body interactions. It is verified that the proposed scaling laws keep the same particle motion in the dimensionless time and space during the contact of more than two particles. It is also demonstrated that the behaviour of the original (stiff) particles in Couette flow can be replicated well in various conditions in general. Some deviation to the original system could be observed in certain conditions, which is likely related to the reduction of contact frequency.

Size-dependent vibrations of thermally pre/post-buckled FG porous micro-tubes based on modified couple stress theory

Based on the modified couple stress theory, an analytical approach is presented to study vibrations of thermally pre/post-buckled functionally graded (FG) tubes. The case of simply supported FG micro-tubes with uniform distributed porosity surrounded by nonlinear elastic medium in uniform temperature field is analyzed. Thermomechanical properties are functionally graded across the radius of cross-section of the tube by means of a power law function. The traction free boundary conditions on the inner and outer surfaces of the structure are satisfied. Basic formulations are constructed using the higher-order shear deformation tube theory and the von Kármán type of nonlinear strain-displacement relationships. The governing equations of motion are derived using the Hamilton principle. The dimensionless equations of motion are solved analytically by employing the two-step perturbation technique and the Galerkin procedure. An explicit closed-form solution is obtained to analyze the size-dependent linear and nonlinear free vibrations of micro-tubes in thermal environment. The numerical results are verified by comparison with the existing data in literature for tubes without couple stress effect. Novel numerical results are revealed in the parametric studies to show the large amplitude vibration responses of thermally pre/post-buckled FG porous micro-tubes. The effects of microstructural length scale parameter, functionally graded patterns, porosity distribution parameter, nonlinear elastic foundation, temperature dependence, and geometrical properties of the structure are investigated.

Analysis and design of a torsional vibration isolator for rotating shafts

A torsional vibration isolator which connects input and output rotating shafts was proposed in this study. For the design of the torsional vibration isolator, two dimensionless performance indices that quantify vibration isolation and response agility were defined. They were calculated using the dynamic responses of the input and output shafts connected by the torsional vibration isolator. To obtain dynamic responses of the shafts including the vibration isolator, dimensionless equations of motion were derived. By choosing proper dimensionless design parameters, the vibration isolation performance could be sufficiently improved while response agility performance maintained within an acceptable range. The proposed performance analysis procedure could be effectively employed for the design of the torsional vibration isolator.

Self-Oscillating Mode of a Nanoresonator

The paper proposes a new graphene resonator circuit which operates on the principle of a self-oscillator and has no drawbacks typical of nanoresonators as mass detectors and associated with their law quality factor, eigenfrequency errors (measurements from resonance curves), and dependence of quench frequency on oscillation frequency (curves with quenching for nonlinear systems). The proposed circuit represents a self-oscillator comprising an amplifier, a graphene resonator, and a positive feedback loop with a graphene oscillation transducer, and its major advantage is in self-tuning to resonance frequency at slowly varying resonator parameters, compared to oscillation periods. The graphene layer with a conducting substrate beneath it forms a capacitor which is recharged by a dc voltage source as its capacitance varies due to graphene deformation, and the recharge current is an oscillation- dependent signal transmitted from the transducer to the amplifier input. The graphene layer is placed in a magnetic field and is deformed when a current from the amplifier output is passed through. By properly choosing the magnetic field direction and the amplifier gain, it is possible to provide swinging oscillation whose amplitude is limited by the amplifier nonlinearity. For the proposed system we present an electromechanical model, dimensionless equations of motion, and numerical data demonstrating the generation of steady-state oscillations with eigenfrequency. Also presented is an analysis showing that the system can have only one limit cycle and that this cycle is always stable. The proposed resonator circuit can be used as a mass detector which determines the added mass from a change in self-oscillation frequency.

Hydrodynamic structure of the boundary layers in a rotating cylindrical cavity with radial inflow

A flow model is formulated to investigate the hydrodynamic structure of the boundary layers of incompressible fluid in a rotating cylindrical cavity with steady radial inflow. The model considers mass and momentum transfer coupled between boundary layers and an inviscid core region. Dimensionless equations of motion are solved using integral methods and a space-marching technique. As the fluid moves radially inward, entraining boundary layers develop which can either meet or become non-entraining. Pressure and wall shear stress distributions, as well as velocity profiles predicted by the model, are compared to numerical simulations using the software OpenFOAM. Hydrodynamic structure of the boundary layers is governed by a Reynolds number, Re, a Rossby number, Ro, and the dimensionless radial velocity component at the periphery of the cavity, Uo. Results show that boundary layers merge for Re < < 10 and Ro > > 0.1, and boundary layers become predominantly non-entraining for low Ro, low Re, and high Uo. Results may contribute to improve the design of technology, such as heat exchange devices, and turbomachinery.

Dynamics and chaos control of asymmetric gyrostat satellites

The motion of a free gyrostat consisting of a platform with a triaxial ellipsoid of inertia and a rotor with a slight asymmetry with respect to the axis of rotation is considered. Dimensionless equations of motion for a system with perturbations caused by the small asymmetries of the rotor are written in Andoyer-Deprit variables. These perturbations result in a chaotic layer in the separatrix vicinity. Heteroclinic and homoclinic trajectories are written in analytical form for gyrostats with different ratios of their moments of inertia. These trajectories are used to construct a modified Melnikov function, and to produce control that eliminates separatrix chaos. The Poincare sections and Melnikov function are constructed via numerical modeling that demonstrates the effectiveness of control.

Nanoparticle delivery via stocky single-walled carbon nanotubes: A nonlinear-nonlocal continuum-based scrutiny

Excellent mechanical properties plus to the good biocompatibility of single-walled carbon nanotubes (SWCNTs) are offering them as efficient nanodevices for delivering of nanoparticles. Due to the friction between the outer surface of the nanoparticle and the inner surface of the SWCNTs, mass weight of the nanoparticle, and the interactional van der Waals (vdW) forces between the constitutive atoms of the nanoparticle and those of the SWCNT, both longitudinal and transverse waves propagate within the SWCNT. Herein, such vibrations in stocky SWCNTs are of concern in the context of nonlinear-nonlocal continuum theory of Eringen. Based on the Rayleigh, Timoshenko, and higher-order beam theories, the dimensionless equations of motion are constructed and then numerically solved. The effects of the mass weight and velocity of the nanoparticle, radius and length of the SWCNT, the abovementioned vdW forces, and the small-scale parameter on the maximum displacements and forces within the SWCNT are exclusively explored. Additionally, the limitations of both the linear and local analyses for the problem under study are discussed.

Revisiting the Theory of Transverse Vibrations of Plates with Shear Deformation

Introduction. The theory of plates with shear deformation has been developed for several decades beginning from the studies [13, 17] (first-order shear deformation theory or Reissner–Mindlin theory [2–4, 20]). The dynamics of plates with shear compliance and rotary inertia was for the first time studied in [6]. Later, extensive studies in this area were performed in [1, 5, 8, 10–12, 16, etc.], where the effect of shear compliance on the natural frequencies was numerically analyzed in detail using various theories. The following circumstance is noteworthy. The first studies on the dynamics of plates and rods with shear deformation ([6], etc.) discovered two spectra of natural frequencies (two very different (low and high) natural frequencies correspond to the same total deflection mode shape). However, the sense of these two branches is still unclear. It is usually stated that the first branch is related to flexural vibrations and waves, whereas the second branch to shear vibrations and waves. Since this is about the ratio between the bending and shear components, which are present in both spectra, the difference between the two spectra appears to be purely quantitative. Some authors cast doubt on the physical sense of the second spectrum for both Timoshenko beams and for plates with shear compliance and proposed to reject the associated results [7, 17, 18]. A literature review indicates that there has yet been no consensus on the sense of the second eigenvalue spectrum. An important fact is pointed out in [19]: for the first spectrum of a Timoshenko beam, the shear and bending angles have the same phase and are added to produce the total deflection angle, whereas for the second spectrum, the shear and bending angles are opposite in phase, and the total angle is equal to their difference. This observation for beams was later confirmed in [9, 17]. As far as we know, such statements, even as illustrative calculations, have not been made for plates. Here we will use dimensionless equations of motion of a plate (in first-order shear deformation theory) that incorporate only one generalized dimensionless shear compliance parameter to rigorously substantiate the qualitative difference between the two natural frequency spectra (for a hinged plate as an example) and carry out a generalized parametrical analysis. 1. Problem Formulation. Basic Equations. There are various ways of deriving the basic equations of the first-order shear deformation theory of plates (discussed in [2, 3, 20, etc.]). The simplest method is given below. Consider an isotropic plate of constant thickness h with the coordinates x and y in the midsurface and the coordinate z running along the normal. Let w x y ( , ) denote the total deflection (resulting from bending and shear), x and y the angles of rotation of normal sections due to bending, and x and y the shear angles (averaged over the thickness of the plate). The total angles of rotation of the normal to the mid-surface in two planes are given by International Applied Mechanics, Vol. 50, No. 2, March, 2014

Vibrations of Biaxially Tensioned-embedded Nanoplates for Nanoparticle Delivery

It is aimed to investigate vibrations of an embedded nanoplate subjected to biaxially applied loads and a moving nanoparticle. To this end, the nanoplate and the moving nanoparticle in order are modeled via nonlocal Kirchhoff plate theory and a rigid body. The mass weight of the moving nanoparticle, and its friction with the upper surface of the nanoplate are taken into account in the proposed model. For a more general study of the problem, the dimensionless equations of motion are obtained. The in- and out-of-plane displacements of the nanoplate are discretized in the space and time using assumed mode and generalized Newmark-β methods, respectively. The roles of the moving nanoparticle velocity, small-scale parameter, biaxially tension forces, and the lateral stiffness of the surrounding medium on the displacements of the nanoplate are addressed and discussed.

Vibrations of Biaxially Tensioned-embedded Nanoplates for Nanoparticle Delivery

It is aimed to investigate vibrations of an embedded nanoplate subjected to biaxially applied loads and a moving nanoparticle. To this end, the nanoplate and the moving nanoparticle in order are modeled via nonlocal Kirchhoff plate theory and a rigid body. The mass weight of the moving nanoparticle, and its friction with the upper surface of the nanoplate are taken into account in the proposed model. For a more general study of the problem, the dimensionless equations of motion are obtained. The in- and out-of-plane displacements of the nanoplate are discretized in the space and time using assumed mode and generalized Newmark-β methods, respectively. The roles of the moving nanoparticle velocity, small-scale parameter, biaxially tension forces, and the lateral stiffness of the surrounding medium on the displacements of the nanoplate are addressed and discussed.

Stability Analyses of a Moving Rectangular Plate by the Element Free Galerkin Method

The element-free Galerkin method is proposed to solve the stability of the moving rectangular plates. Utilizing the extended Hamilton’s principle for the elastic dynamics system, the variational expression of the moving thin plate are established. The dimensionless equations of motion of the moving thin plate are obtained by the element-free Galerkin method, and the complex eigenvalue equation is presented. Via numerical calculation, the variation relationship between the first three complex frequencies of the system and the moving speed is obtained. The effects of dimensionless moving speed on the stability and critical load of the thin plates are analyzed.

Swing-Up Control of a Three-Link Underactuated Manipulator by High-Frequency Horizontal Excitation

In the present paper, we consider a three-link underactuated manipulator, the first joint of which is active and the second and third joints of which exhibit passive motion, on a plane inclined at slight angle from horizontal the plane. We analytically investigate changes in the stability of equilibrium points of the free links connected to the passive joints using high-frequency horizontal excitation of the first link. We derive autonomous averaged equations from the dimensionless equations of motion using the method of multiple scales. We clarify that the two free links can be swung up through pitchfork bifurcations and stabilized at some configurations by producing nontrivial and stable equilibrium points due to the high-frequency excitation. Furthermore, it is experimentally verified that increasing the excitation frequency multiplies stable and nontrivial equilibrium points.

Absorption and Evaporation Mass Transfer Simulation of a Droplet by Finite Volume Method and SIMPLEC Algorithm

In this paper, a numerical model is developed to simulate single droplet heat and mass transfer in a two-pieces solution with a saturated solvent vapor environment. Finite volume method and transient SIMPLEC algorithm in spherical coordinates system used for simulation. For simulation of the mass transfer, dimensionless equations of motion, heat transfer and mass transfer (based on mass ratio) are solved simultaneously. All the thermodynamic and transitional solution properties have been considered as a function of temperature and concentration. Verification of method is done by compare these numerical results with analytical and numerical analysis of other studies. Evaporation, absorption and condensation contours in a water droplet at uniform temperature at superheated water vapor and Distribution of Droplet's Surface Temperature, Constant Temperature and concentration lines as the modeling results are presented. The results are shown that 0.5% increase of concentration of droplet cause increase 8 degree C of mean temperature of droplet.

Free flapewise vibration analysis of rotating double-tapered Timoshenko beams

Free vibration analysis of a rotating double-tapered Timoshenko beam undergoing flapwise transverse vibration is presented. Using an assumed mode method, the governing equations of motion are derived from the kinetic and potential energy expressions which are derived from a set of hybrid deformation variables. These equations of motion are then transformed into dimensionless forms using a set of dimensionless parameters, such as the hub radius ratio, the dimensionless angular speed ratio, the slenderness ratio, and the height and width taper ratios, etc. The natural frequencies and mode shapes are then determined from these dimensionless equations of motion. The effects of the dimensionless parameters on the natural frequencies and modal characteristics of a rotating double-tapered Timoshenko beam are numerically studied through numerical examples. The tuned angular speed of the rotating double-tapered Timoshenko beam is then investigated.

Multi-pulse orbits dynamics of composite laminated piezoelectric rectangular plate

The multi-pulse orbits and chaotic dynamics of a simply supported laminated composite piezoelectric rectangular plate under combined parametric excitation and transverse excitation are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motion for the laminated composite piezoelectric rectangular plate are derived from von Karman-type equation and third-order shear deformation plate theory of Reddy. The two-degree-of-freedom dimensionless equations of motion are obtained by using the Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plate. The four-dimensional averaged equation in the case of primary parametric resonance and 1:3 internal resonances is obtained by using the method of multiple scales. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on the normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plate. The analysis of the global dynamics indicates that there exist multi-pulse jumping orbits in the perturbed phase space of the averaged equation. Based on the averaged equation obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the laminated composite piezoelectric rectangular plate are also found by numerical simulation. The results obtained above mean the existence of the chaos in the Smale horseshoe sense for the simply supported laminated composite piezoelectric rectangular plate.

Vibro-acoustic analysis of Mindlin rectangular plates resting on an elastic foundation

This study investigates the acoustic radiation of rectangular Mindlin plates in different combinations of classical boundary condition. A set of exact close-form sound pressure equations is given for the first time, using the so-called Mindlin plate theory (a first-order shear deformation theory), for plates having two opposite edges that are simply supported. The other two edges may be given any possible combination of free, simply-supported and clamped boundary condition. It is assumed that mechanical in-plane loading occurs on the plate structure. In order to study the transverse vibration of moderately thick rectangular plates, the dimensionless equations of motion are derived, based on the Mindlin plate theory. Structural-acoustic coupling is implemented for vibrating plate models. The radiation field of a vibrating plate with a specified distribution of velocity on the surface can be computed using the Rayleigh integral approach. The acoustic pressure distribution of the radiator is analytically obtained in its far field. Additionally, the influence of six possible combinations of boundary condition, foundation parameter, loading case, aspect ratio and thickness ratio, on the sound pressure is examined and discussed in detail.

Element-free Galerkin method for free vibration of rectangular plates with interior elastic point supports and elastically restrained edges

The element-free Galerkin method is proposed to solve free vibration of rectangular plates with finite interior elastic point supports and elastically restrained edges. Based on the extended Hamilton’s principle for the elastic dynamics system, the dimensionless equations of motion of rectangular plates with finite interior elastic point supports and the edge elastically restrained are established using the element-free Galerkin method. Through numerical calculation, curves of the natural frequency of thin plates with three edges simply supported and one edge elastically restrained, and three edges clamped and the other edge elastically restrained versus the spring constant, locations of elastic point support and the elastic stiffness of edge elastically restrained are obtained. Effects of elastic point supports and edge elastically restrained on the free vibration characteristics of the thin plates are analyzed.

A mathematical analysis of hollow fiber spinning: Bore and dope velocity profiles in the air gap

Hollow fiber membranes are the most applicable form of membranes in laboratory and industrial scale with their large surface to volume ratios. Hollow fiber membranes usually are fabricated by dry–wet solution spinning, where a polymer solution is co-extruded through an annular region with a bore fluid and the nascent hollow fiber passes through an air gap and then enters a liquid coagulation bath. The spinneret dimension, dope and bore fluid flow rates, air gap length, bore fluid and dope compositions and their physical properties, coagulant composition and condition, shear stress within a spinneret, the ratio of dope to bore fluid volumetric flow rate, and the take-up-to-initial velocity ratio (draw ratio) are the primary factors that determine the final hollow fiber morphology and separation properties. All of the aforementioned parameters are not independent and some of them are functions of others. The objective of this paper is to elucidate the interdependence between some of the above parameters. For this purpose, dope and bore fluid axial velocity in the air gap was assumed as a function of the axial distance from the spinneret outlet, z, and radial velocity as a function of the radial distance from the center of the hollow fiber, r, and z. Dimensionless equations of motion and continuity were then simplified and solved simultaneously based on preceding assumptions. It was found from the analysis that the dimensionless axial velocity of dope was a function of the Reynolds number, Capillary number and Stokes number indicating that the axial velocity acceleration in the air gap is determined by viscous, capillary, and gravity force gradients. It was also concluded that the optimum value of the ratio of bore liquid flow rate to dope flow rate is equal to 0.8 of the ratio of the cross-sectional area of the bore fluid to that of the dope at the spinneret outlet.

Exact acoustical analysis of vibrating rectangular plates with two opposite edges simply supported via Mindlin plate theory

This study investigates acoustic radiation of rectangular Mindlin plates in different combinations of classical boundary conditions. A set of exact close-form sound pressure equations are given for the first time using the so-called Mindlin plate theory (a first-order shear deformation theory) for the plates having two opposite edges that are simply supported. The other two edges may be given any possible combination of free, simply-supported and clamped boundary conditions. It is assumed that no fluid loading occurs on the plate structure. In order to study the transverse vibration of moderately thick rectangular plates, the dimensionless equations of motion are derived based on the Mindlin plate theory. Structural–acoustic coupling is implemented for vibrating plate models. The radiation field of a vibrating plate with a specified distribution of velocity on the surface can be computed using the Rayleigh integral approach. The acoustic pressure distribution of the radiator is analytically obtained in its far field. To reveal the excellent accuracy of our exact acoustical solution, a comparison is first made with the existing data. Additionally, a few 3-D plots of the directivity pattern and their corresponding contour plots are illustrated for moderately thick rectangular plates with different boundary conditions. Finally, the influence of six possible combinations of boundary conditions, aspect ratios and thickness to length ratios on the sound pressure, frequency and critical distance parameters are examined and discussed in detail.