Non-dimensional Equations Of Motion Research Articles

Parametric Instability of an Asymmetric, Rotating Sandwich Beam

The parametric dynamic stability of an asymmetric, rotating sandwich beam and subjected to an axial pulsating load is investigated. A set of Hills equations are obtained from the non-dimensional equation of motion by the application of the general Galerkin method. The zones of parametric instability are obtained using Saito Otomi conditions. The influence of core-loss factor, geometric parameters and rotation parameters on the zones of instability are investigated.

Double poro-magneto-thermoelastic model with microtemperatures and initial stress under memory-dependent heat transfer

The memory response of an initially stressed half-space comprising a double poro-magneto-thermoelastic material with voids and microtemperatures (DPMTMWVM) under the dual-phase-lag heat conduction law is investigated in this article. With the aid of normal mode technique, the non-dimensional equations of motion are solved to derive the theoretical expressions of the displacements, temperature, microtemperatures, void volume fractions, stresses, and heat flux moments. The impacts of magnetic field, kernel functions, and initial stress on the aforementioned distributions are demonstrated through several graphs. Moreover, the memory-dependent Lord-Shulman and dual-phase-lag thermoelasticity theories are compared graphically to exhibit the variations in the field variables. To the best of the authors’ knowledge, the memory response of a double poro-magneto-thermoelastic half-space with voids and microtemperatures is not analyzed so far. Therefore, DPMTMWVM under memory-dependent dual-phase-lag theory is considered in this study.

Parametric stability of Timoshenko taper sandwich beam on Pasternak foundation

The goal of the present study is to analyze the parametric stability of a Timoshenko non-uniform sandwich beam using computational method which is stayed on Pasternak foundation having inconstant spring stiffness parameter. Axial load is applied at terminal points. The governing equation of motion and associated end conditions are defined using extended Hamilton’s principle. The equation of motion and the associated boundary conditions are non-dimensionalized by the application of the main principle of Galerkin method. A set of Hills equations are found from the non-dimensional equation of motion. The regions of parametric instability are analyzed considering the effect of numerous system parameters and geometric parameters and displayed through a series of plots. The computational analysis is done by using MATLAB programming.

Static stability investigation of a tapered asymmetric sandwich beam supported on variable Pasternak foundation

The investigation to analyse a sandwich beam’s static stability with asymmetric configuration, tapered along the thickness, placing on a Pasternak foundation having linearly varying stiffness and influenced by an alive axial load is executed for several boundary conditions employing computational method. Use of Hamilton’s principle results in the equations of motion and related boundary conditions. Hill’s equations are achieved from the non-dimensionalized equations of motion with the use of Galerkin’s method. Then, the effects of various parameters on the static stability for different boundary conditions are obtained and are showcased in a sequence of graphs using the appropriate MATLAB program.

Dynamic Stability Investigation of a Sandwich Beam Tapered along width and thickness with Temperature Gradient

The investigation to analyze a sandwich beam's dynamic stabilitywith asymmetric configuration, tapered along the thickness and width, and influenced by an alive axial load with temperature gradient is executed for several boundary conditions employing computational method. Use of Hamilton’s principle results in the equations of motion and related boundary conditions. Hill’s equations are achieved using non-dimensionalized equations of motion with the Galerkin’s method. Then, the influence ofseveral parameterson the dynamic stability for different boundary conditions are attained by applying Saito-Otomi conditions. The impact of differentparameters on the regions of instabilities observed and is showcased in a sequence of graphs using the appropriate MAT LAB program.

절대절점좌표계에서 구조역학 기반 강성 행렬을 활용한 2차원 보의 무차원 운동방정식

절대절점좌표계는 탄성 다물체 동역학 분야에서 대회전과 대변형을 표현할 수 있는 기법이다. 대변형을 표현할 수 있는 장점이 있는 반면 강성 행렬의 강한 비선형성으로 인해 해석 시간이 증가하는 단점이 있다. 본 연구에서는 절대절점좌표계의 해석 시간을 단축시키기 위하여 무차원 운동방정식을 개발하였다. 2차원에서의 차원 운동방정식은 시간, 길이, 힘 차원을 무차원화하는 변수를 통해 무차원화가 가능하며, 특히 6가지 강성 행렬 모델을 제시하였다. 그 결과 2차원 문제에서 개발된 무차원 운동방정식이 차원 운동방정식에 비해 효율적임을 확인할 수 있었다. 또한 탄성 호스의 직진 거동 예제를 활용하여 무차원 운동방정식의 효율성을 제시하였다.

Static and Dynamic Stability Analysis of an Asymmetric Rotating Tapered Sandwich Beam Subjected to Pulsating Axial Load with Thermal Gradient

The static and dynamic stability of an asymmetric rotating tapered sandwich beam subjected to pulsating axial load in temperature environment is studied under two different boundary conditions. The non-dimensional equations of motion and the boundary conditions are derived by applying Hamilton's energy principle. A coupled Hill's equations with complex coefficients are derived from the non-dimensional equations of motion by the application of the generalized Galerkin method. By the application of the Saito-Otomi conditions, zones of instabilities are obtained and presented graphically. For the calculation of the Young's module for the elastic layers, the effect of temperature has been taken in to consideration by means of a uniform thermal gradient along the longitudinal axes for both the upper and lower elastic layers. The effects of the taper parameter, core loss factor, thermal gradient, rotational speed, hub radius, and core density parameter on the static buckling loads and the regions of instability are investigated.

Non-dimensional simple beam models for the elastic forces in ANCF: 3D equation and simulation

Absolute nodal coordinate formulation can be used to represent large displacement and large deformation in flexible body analysis based on the concept of continuum mechanics. However, the analysis takes a long time because of the increase in the degrees of freedom at a nodal point. In this study, the analysis time was reduced by using non-dimensional variables for the time, length, and force to convert a dimensional equation of motion into a non-dimensional one. Revolute and spherical joint were used to obtain numerical solutions by applying the non-dimensional equation of motion of free-falling pendulum. The non-dimensional equation of motion was verified by comparing the solutions with the numerical solution obtained using a dimensional equation of motion. The motion of a flexible hose was predicted by using the absolute nodal coordinate formulation, and it was shown the efficiency of non-dimensional equation of motion.

Nondimensional analysis of a two-dimensional shear deformable beam in absolute nodal coordinate formulation

Absolute nodal-coordinate formulation is a technique that was developed in 1996 for expressing the large rotation and deformation of a flexible body. It utilizes global slopes without a finite rotation in order to define nodal coordinates. The method has a shortcoming in that the central processing unit time increases because of increases in the degrees of freedom. In particular, when considering the deformation of a cross section, the shortcoming due to the increase in the degrees of freedom becomes clear. Therefore, in the present research, the dimensional equation of motion concerning a two-dimensional shear deformable beam, developed by Omar and Shabana, is converted into a nondimensional equation of motion in order to reduce the central processing unit time. By utilizing an example of a cantilever beam, wherein an exact solution for the static deflection exists, the nondimensional equation of motion was verified. Moreover, by using an example of a free-falling flexible pendulum, the efficiency of the nondimensional equation of motion gained by increasing the number of elements was compared with that of the dimensional equation of motion.

Analytical Prediction of Stability Lobes for Passively Damped Boring Bars

AbstractThe present article proposes the closed-form solution for analytical prediction of stability lobes in internal turning process. The passively damped boring bar is modeled as a cantilevered Euler-Bernoulli beam with constant cross sectional properties in which a Tuned Mass Damper (TMD) is embedded for the purpose of chatter suppression. The non-dimensional equations of motion are derived, assuming that the boring bar dynamics is well-represented by the fundamental mode of vibration. The stability of equivalent two-DOF dynamic model, i.e. boring bar with TMD, is analyzed in frequency domain. The closed- form expressions for critical depth of cut and spindle speed are presented in terms of boring bar and TMD characteristics. The proposed solution considers the effects of boring bar's structural damping and cutting geometry of insert on the stability behavior of passively damped cutting tool. An unconstrained optimization method is utilized to compute the most optimal set of tuning parameters for anti-chatter TMD. In order to improve the boundary of stability in a global sense, maximization of minimum critical depth of cut is selected as the objective of optimization. The superior performance of anti-chatter TMD is compared to H∞and H2TMDs for a wide range of applications. Moreover, the achieved results show a remarkable improvement of stability boundary compared to recent research works.

Unsteady fluid flow in a slightly curved annular pipe: The impact of the annulus on the flow physics

The motivation of the present study is threefold. Mainly, the etiological explanation of the Womersley number based on physical reasoning. Next, the extension of a previous work [Messaris, Hadjinicolaou, and Karahalios, “Unsteady fluid flow in a slightly curved pipe: A comparative study of a matched asymptotic expansions solution with a single analytical solution,” Phys. Fluids 28, 081901 (2016)] to the annular pipe flow. Finally, the discussion of the effect of the additional stresses generated by a catheter in an artery and exerted on the arterial wall during an in vivo catheterization. As it is known, the square of the Womersley number may be interpreted as an oscillatory Reynolds number which equals to the ratio of the inertial to the viscous forces. The adoption of a modified Womersley number in terms of the annular gap width seems therefore more appropriate to the description of the annular flow than an ordinary Womersley number defined in terms of the pipe radius. On this ground, the non-dimensional equations of motion are approximately solved by two analytical methods: a matched asymptotic expansions method and a single. In the first method, which is valid for very large values of the Womersley number, the flow region consists of the main core and the two boundary layers formed at the inner and outer boundaries. In the second, the fluid is considered as one region and the Womersley number can vary from finite values, such that they fit to the blood flow in the aorta and the main arteries, to infinity. The single solution predicts increasing circumferential and decreasing axial stresses with increasing catheter radius at a prescribed value of the Womersley parameter in agreement with analogous results from other theoretical and numerical solutions. It also predicts the formation of pinches on the secondary flow streamlines and a third boundary layer, additional to those formed at the boundary walls. Finally, we show that the insertion of a catheter in an artery may trigger possible disastrous side effects. It may cause unexpected damage to a predisposed but still dormant location of the arterial wall due to high additional radial pressure that induces an excessive distension of the artery.

Efficiency of non-dimensional analysis for absolute nodal coordinate formulation

Absolute nodal coordinate formulation was developed in the mid-1990s. The adoption of the continuum mechanics concept has allowed large displacements and large deformations to be expressed in flexible body analysis. However, the analysis time increases due to the increased number of degrees of freedom at nodal points. Therefore, we aimed to reduce the analysis time by converting a dimensional equation of motion (EOM) to a non-dimensional EOM by using non-dimensional variables of time, length, and force. A non-dimensional mass matrix, a non-dimensional longitudinal stiffness matrix, and a non-dimensional conservative force vector are derived and applied to the non-dimensional EOM. To verify the non-dimensional EOM, a cantilever beam with static deflection, for which an exact solution exists, is considered. As the number of elements is increased, the mean value by the non-dimensional EOM converges to the static deflection. Revolute and spherical joints are used to propose two- and three-dimensional numerical solutions based on the non-dimensional EOM of a free-falling pendulum. These solutions are compared with the numerical solutions from using a dimensional EOM in order to verify the non-dimensional EOM. The analysis results for simple pendulum motion using dimensional and non-dimensional EOMs are in good agreement.

Stability of an asymmetric tapered sandwich beam resting on a variable Pasternak foundation subjected to a pulsating axial load with thermal gradient

The static and parametric stability of an asymmetric tapered sandwich beam resting on a variable Pasternak foundation subjected to a pulsating axial load with thermal gradient under two different boundary conditions is investigated. The complete mathematical modeling of the system has been derived by the application of Hamilton’s principle which helps in getting the admissible path for the system. The equations of motion and boundary conditions obtained from the Hamilton’s equation are non-dimensionalized. A set of Hill’s equation are obtained from the non-dimensional equations of motion by the application of generalized Galerkin’s method. The zones of parametric instability are obtained using Saito–Otomi conditions. The effects of taper, elastic foundation, thermal gradient, core-loss factor, geometric parameter, modulus ratios and shear parameter on static buckling loads and parametric regions of instability are investigated.

Dynamic Stability Analysis of a Circularly Tapered Rotating Beam Subjected to Axial Pulsating Load and Thermal Gradient under Various Boundary Conditions

The dynamic stability of a circularly tapered rotating beam subjected to a pulsating axial external excitation with thermal gradient was studied for all possible combinations of clamped, guided, pinned, fixed, and free boundary conditions. The equations of motion and associated boundary conditions were obtained using the extended Hamilton’s principle. Then these equations of motion and the associated boundary conditions were non-dimensionalised. A set of Hill’s equations were obtained from the non-dimensional equations of motion by the application of the extended Galerkin method. The zones of parametric instability were obtained using Saito-Otomi conditions. The effects of various boundary conditions, thermal gradient, taper, and rotational speed on the regions of parametric instability were investigated and presented through a series of graphs. The results reveal that increasing rotational speed and taper have stabilizing effects, whereas increasing thermal gradient has a destabilizing effect for all boundary conditions of the beam.

Effects of Structural Properties on Natural Frequencies of Axially Moving Beams with an Intermediate Support

In this study, linear vibrations of axially moving beam simply supported between the guides are examined and natural frequencies are calculated numerically. The vibrations of axially tensioned Euler-Bernoulli beam are investigated under clamped-clamped end conditions. Governing differential equations of motion are derived using Hamilton’s Principle for two regions of the beam. The boundaries at the outer ends of the beam are assumed immovable. Non-dimensional equations of motion are derived and the solutions of the linear problem are obtained. Assuming a weakly non-linear system, linear equations are obtained using the Method of Multiple Scales. First seven natural frequencies are calculated numerically based on the flexural rigidity, axial velocity and locations of the intermediate support.

Nonlinear analysis of chatter in turning process using dimensionless groups

A new 3-D nonlinear model of chatter vibration in turning process is presented in this paper. The workpiece is modeled as a rotating clamped-free beam which is excited by cutting forces. π-Buckingham theory is used to extract dimensionless parameters for this problem. Using these parameters, non-dimensional equations of motion are developed. An approximate analytical solution of the nonlinear problem is obtained. A study of the influence of different parameters on the stability results is developed. Using these results, turning velocity intervals for stable and unstable cuts are determined.

Static stability of a viscoelastically supported asymmetric sandwich beam with thermal gradient

The static stability of an asymmetric sandwich beam with viscoelastic core on viscoelastic supports at the ends and subjected to an axial pulsating load and a steady, one-dimensional temperature gradient is investigated by computational method. The equations of motion and associated boundary conditions are obtained using the Hamilton's energy principle. Then, these equations of motion and the associated boundary conditions are non- dimensionalised. A set of Hill's equations is obtained from the non-dimensional equations of motion by the application of the general Galerkin method. The static buckling loads are obtained from the Hill's equations. The effects of shear parameter, geometric parameters, core loss factors, and thermal gradient on the non-dimensional static buckling loads zones have been investigated.

Analysis of Static Instability of an Asymmetric, Rotating Sand-Wich Beam

The static stability of an asymmetric, rotating sandwich beam subjected to an axial pulsating load has been investigated for pinned-pinned and fixed-free boundary conditions. The equations of motion and associated boundary conditions have been obtained by using the Hamilton's energy principle. Then, these equations of motion and the associated boundary conditions have been nondimensionalised. A set of Hill's equations are obtained from the nondimensional equations of motion by the application of the general Galerkin method. The static buckling loads have been obtained from Hill's equations. The influences of geometric parameters and rotation parameters on the nondimensional static buckling loads have been investigated.

Transfer Orbit Design and Control Based on Forbes' Assumption

In order to meet the requirements of space operation, the spacecrafts must maneuver agilely with the artifical propulsion. The orbit under artificial control is termed as Non-Keplerian orbit which does not follow Kepler’s Laws. The transfer orbit design under the continuous thrust is one of the most important topics in this new field. Shape-based method for Non-Keplerian orbit design is developed in this paper. Firstly, the equations of motion are established in polar coordinates system. And then the nondimensional variables are introduced for computation accuracy and speed, which give rise to nondimensional equations of motion. The general equation is derived with which common curves could be utilized in orbit design. In addition, the orbit design method is described based on Forbes’ velocity assumption and the formulation of the radius r with respect to time t, which is a sinusoidal function. The determination of the coefficients causes the orbit design problem to translate into an orbit control problem. The requisite thrust magnitude and direction are available via the simplified nondimensional equations of motion. In the end, an example of the transfer orbit is given. The result demonstrates that the shape-based method is feasible for the transfer orbit design or control problem under the continuous thrust, and the fuel expenditure is practicable.

Numerical study of unsteady interacting compressible jet flows using finite difference method for small Reynolds numbers

Abstract Two dimensional time-dependent flow field for interaction of jets were investigated numerically using finite difference method. The Crank–Nicholson implicit method applied the non-dimensional equations of motion for unsteady flow of the compressible fluids. Successive Over Relaxation (SOR) method was used to solve linear algebraic equations. Four different cases were studied to solve jet flow such as free jet, parallel jets, opposite jets and cross jets for Reynolds numbers 0.5 and 25 in each case. Velocity profiles and velocity vectors were given for different cases.