Expression For Kinetic Energy Research Articles

Выделение гироскопических инерционных сил из центробежных и кориолисовых инерционных сил

The purpose of the study is to identify in the equations of dynamics of mechanical systems generalized gyroscopic inertial forces, consisting of a part of centrifugal and Coriolis inertial forces, indica¬ting the bodies whose movements form these forces. The methods of research refer to mechanics of body systems, systems analysis and robotics. The results of the study allow us to write out general formulas for calculating generalized centrifugal, Coriolis and gyroscopic inertial forces through partial derivatives on generalized coordinates from elements of the matrix of inertial coefficients in the expression of kinetic energy of arbitrary mechanical system. In the process of analyzing these formulas for systems of bodies with an open tree structure, such as orbital stations with manipulators on board or walking vehicles in the single-supported walking phase, the bodies whose movements affect the occurrence of the listed inertial forces are highlighted. For robot manipulation systems we present examples of the equations of dynamics, in which all possible inertial forces are highlighted and the bodies that form them are indicated. Conclusion. Singling out the gyroscopic forces allows you to simplify the formula for calculating the power consumption of drives, because the power of these forces is zero. This fact, in particular, allows to simplify the expression of the minimizing functional in the problem of energy optimal control. The isolated centrifugal forces can be used to realize program motions of some bodies of manipulators with polar, cylindrical, spherical and angular coordinate systems without expenditure of power consumed by their drives.

Analysis and experiments of bi-stable laminate configuration control based on MFC piezoelectric actuation

This paper studies the configuration change control and the snap-through phenomenon of bi-stable laminate with the MFC piezoelectric plate. The kinetic and potential energy expressions of the piezoelectric bi-stable laminate were obtained based on the von-Karman hypothesis. The constitutive equations of the system were given using a six-parameter displacement model, and the potential function of the system was derived using Hamilton’s principle. The configuration changes of the piezoelectric bi-stable laminate under different geometrical conditions were analyzed. The influence of voltage on the bi-stable laminate configuration was studied by applying different driving voltages. The snap-through energy variation of the piezoelectric bi-stable laminate under different geometrical conditions was investigated. In the experimental part, the changes in bi-stable laminate configuration with different geometrical conditions and after attaching the piezoelectric plate were compared. Different driving voltages were applied to measure the difference in piezoelectric bi-stable laminate configuration, and the critical snap-through voltages were recorded. The measured changes in bi-stable laminate configuration and snap-through voltage at different voltages prove the correctness of the theoretical presentation.

Multiple pendulum and nonuniform distribution of average kinetic energy.

Multiple pendulums are investigated numerically and analytically to clarify the nonuniformity of average kinetic energies of particles. The nonuniformity is attributed to the system having holonomic constraints and it is consistent with the generalized principle of the equipartition of energy. With the use of explicit expression for Hamiltonian of a multiple pendulum, approximate expressions for temporal and statistical average of kinetic energies are obtained, where the average energies are expressed in terms of masses of particles. In a typical case, the average kinetic energy is large for particles near the end of the pendulum and small for those near the root. Moreover, the exact analytic expressions for the average kinetic energy of the particles are obtained for a double pendulum.

The Kelvin-Voigt visco-elastic model involving a fractional-order time derivative for modelling torsional oscillations of a complex discrete biodynamical system.

Under external loads trees exhibit complex oscillatory behaviour: their canopies twist and band. The great complexity of this oscillatory behaviour consists to an important degree of torsional oscillations. Using a system of ordinary differential fractional-order equations, free and forced main eigen-modes of fractional-type torsional oscillations of a hybrid discrete biodynamical system of complex structures were done. The biodynamical system considered here corresponds to a tree trunk with branches and is in the form of a visco-elastic cantilever of complex structure. Visco-elasticity corresponds to different ages of a tree. We set up a new model of torsional oscillations of a complex discrete, biodynamical system, using the Kelvin-Voigt visco-elastic model involving a fractional-order time derivative. The analytical expressions describing the characteristic properties of its fractional-type oscillations are determined. Based on mathematical and qualitative analogies, this concept represents a new model of torsional oscillations of a light cantilever that takes into account visco-elastic, dissipative properties of the material. Rigid discs are attached to the cantilever. Expressions for kinetic energy, deformation work and a generalized function of the fractional-type energy dissipation of this biodynamical system are defined. Independent main eigen-modes of the fractional type for free and forced torsional oscillations were determined for a special class of such systems, using formulas for the transformation of independent generalized angle coordinates to the principal main eigen-coordinates of the system. The forms of their approximate analytical solutions are shown. In the general case for inhomogeneous biodynamical systems of fractional type, there are no independent main fractional-type eigen-modes of torsional oscillations. The system behaves as a nonlinear system. A new constitutive relation between coupling of torsion loading to a visco-elastic fractional-type cantilever with fractional-type dissipation of cantilever mechanical energy and angle of torsion deformation is determined using a fractional-order derivative. The main advantages of the proposed model are the possibility to analyse torsional oscillations of more complex structures and the possibility to analyse complex cantilevers with different cross-sectional diameters.

The Motion of a Charged Particle in the Electromagnetic Field of a Polarization-Modulated Wave

This article presents an exact solution of the equation of motion of a charged particle in the electromagnetic field of a high-intensity polarization-modulated wave. Expressions for the average kinetic energy of a particle without regard to its rest energy in the case of circular and linear polarization of a modulated wave are obtained. The motion of a charged particle in the field was analyzed and expressed in terms of dependences of its average kinetic energy on the electromagnetic wave intensity and on various types of modulation depths. The contribution of each type of modulation to the energy characteristics of a charged particle was demonstrated. Solving the equation of motion of a charged particle in the electromagnetic field of a plane wave opens up possibilities for various applications related, in particular, to various developments of multi-frequency lasers and laser modulation technology. This study was proposed due to the growing interest in experiments using high-intensity femtosecond laser radiation and high-temperature plasma

Influence of initial tunneling step on the return energy of high-order harmonic generation

To investigate high-order harmonic generation in a monochromatic laser field, we derive an analytical expression for the return energy of an electron as a function of the time interval between ionization and return. We then expand the expression for kinetic energy to second order with respect to the Keldysh parameter $\ensuremath{\gamma}$. In this expansion, the zero-order term is the return energy in the simple man model and the second-order term corresponds to corrections to this model. The origin of this additional kinetic energy is frequently attributed to the nonzero exit of the initial tunneling step. Here, we show that this commonly used picture is incomplete. We present a framework to fully understand the additional kinetic energy as resulting from additive contributions of zero-order and second-order velocities. Our results show that the nonzero velocity of the initial tunneling step has a quantifiable effect on the cutoff energy measured in high harmonic generation (HHG). This opens the door to experimentally addressing the question of the initial electron velocity at the tunnel exit, with important implications for the correct calibration of the attoclock, as well as our interpretation of the strong field-ionization process more broadly.

The Impact of Non-constant Inertia and Nonlinear Damping on the Torsional Vibration Characteristics of Internal Combustion Engine Including External Forces

Failures of the crankshaft-slider mechanism are the most reasons that affect the durability and operational reliability of the internal combustion engine. An accurate and sophisticated nonlinear dynamic model overcomes the obvious simulation errors of linearized models. The present work studies the effect of the non-conservative forces and nonlinear damping on the torsional vibration of single-cylinder internal combustion engines. Comprehensive dynamic modeling based on a developed expression for the instantaneous kinetic energy of the reciprocating parts and a general model of the overall kinetic energy of the system in terms of the inertia parameters were derived. The effect of variable inertia and nonlinear damping on the damped forced response of slider-crank assembly of the engine was investigated using the numerical integration method. The numerical results show that the phenomenon of secondary rolling excitation torque is well activated and gives arises to variation of frequencies and their corresponding amplitudes. Also, the amplitude of the external excitation torque is strengthened by the secondary excitation inertia torque and introduces multi resonance amplitudes phenomenon and widening the critical range of engine speed which results in producing of dangerous vibrational stress amplitudes. Also, the damped forced results indicate that the presents of damping lead to a vital reduction in the amplitude of torsional displacement and excitation torques. The present work aims to enhance nonlinear dynamic modeling and introduces more reliable design for reciprocating engine crankshaft assembly.

Rotation-Induced Geometrical Stiffening of a Tapered, Pretwisted Blade

A linear geometrical stiffening model is developed for a rotating beam with an asymmetric cross section undergoing coupled stretch–bending–torsion motion. The model is verified for a twisted and tapered Timoshenko beam mounted on a rotating hub with a presetting angle. The floating frame of reference formulation is adopted where the configuration of the deformable beam is described by using different coordinate systems. Three coordinate systems, viz., inertial reference frame attached to the center of the rigid hub, local reference frame attached to an arbitrary point on the elastic axis of the undeformed beam, and body reference frame attached to the point on the elastic axis of the deformed beam, are defined. The global position vector of an arbitrary point on the beam is specified using a coupled set of reference and elastic coordinates. Reference coordinates define the location and orientation of a body reference frame, and elastic coordinates describe the beam deformation about the body reference frame. For a rotating beam problem, reference and elastic coordinates give the angular displacements of the hub and elastic deformations of the beam, respectively. The expressions for the kinetic and strain energy of the deformable beam are derived. The inertial coupling between the reference and elastic coordinates in the kinetic energy expression is identified as a gyroscopic coupling term. Further, the inertial mass matrix, spin softening matrix, and centrifugal stiffening matrix are obtained from kinetic energy expression. The elastic stiffness matrices, which are independent of elastic coordinates, are derived from strain energy expression. According to the Rayleigh–Ritz method, the elastic coordinates are expressed using a series of admissible functions that satisfy the geometric boundary conditions of a fixed-free beam. The ordinary differential equations of motion are then derived using Lagrange’s equations. A set of dimensionless parameters is identified after the nondimensionalization of the equations of motion. Due to the skew-symmetric nature of the gyroscopic coupling matrix, the eigenvalue problem is not amenable to classical modal analysis. Therefore, a complex modal analysis method is used to determine the modal characteristics. The results from the present model are verified with those available in the literature. The results are also compared with those obtained from the three-dimensional finite element method using ANSYS. The effect of these dimensionless parameters on the modal characteristics of a rotating blade is studied.

Exploring the Deviation of the Kinetic Energy Equation in Classical Physics and Theory of Relativity using Taylor Series

Isaac Newton’s system of classical physics continued to be the prevalent physics theory until Albert Einstein’s establishment of the theory of relativity, which is now perceived as modern physics. However, what is the essential difference between these theories? And what are the practical implications of these deviations between classical and modern physics? This essay utilizes Taylor Series to examine the difference in the expressions for kinetic energy in both theories. In conclusion, the equation for kinetic energy in classical physics deviates from that of modern physics in terms of its precision: classical physics can be an efficient tool to employ when calculating the energy for low-velocity objects with less accurate results, whereas modern physics is more generally utilized for high-velocity objects with high demand for accuracy.

The coupling free vibration characteristics of a rotating functionally graded shaft-disk system in thermal field

The numerical model of a functionally graded (FG) shaft-disk rotor system with flexible shaft and flexible various-thickness disk as components is established to investigate the coupling free vibration behavior of this rotor system in thermal field. Considering the thermophysical properties of the material, the properties of functionally graded material vary continuously according to the power law with the thickness of the disk. Based on the theory of rotor dynamics, the kinetic energy expression and strain energy expression of rotating shaft-disk system are obtained. In the framework of Ritz method, the global control differential equation of the rotor system assembled by shaft and FG disk is established by using differential quadrature finite element method. On this basis, a set of numerical examples are presented to perform convergence, correctness and effectiveness verification of the current model. Moreover, the influence of temperature difference, rotating speed, geometric and material parameters on the thermoelastic coupling vibration characteristics of light-weight shaft-disk rotor is discussed in detail, which can help to improve the rotor vibration performance in design work and future research.

Chemistry without the Born-Oppenheimer approximation.

Chemistry without the Born-Oppenheimer approximation.

Structurally coupled characteristics of rotor blade using new rigid-flexible dynamic model based on geometrically exact formulation

In rotor dynamics, blades are normally modelled as a slender beam, in which elastic deformations are coupled with each other. To identify these coupling effects, new rigid-flexible structural model for helicopter rotor system is proposed in this paper. Finite rotations of the whole blade (on flapwise, lagwise, and torsional) are described as three global rigid degrees of freedom. The nonlinear deformation geometrics of the beam is built on geometrically exact beam theory. New expressions for blade strain energy, kinetic energy, and virtual work of various kinds of external forces are derived as functions of finite rotations and elastic deformations. To quantify the coupling characteristics, following the definition of coupling factor in electromagnetics, a new coupling factor between two modal components on each mode is introduced in modal analysis. Simulations show that the new structural model is highly capable of solving static and dynamic problems in rotor system and the maximum deformation that moderate deformation beam theory can predict might be 15% of beam length. After the new coupling factor is applied to study structurally coupled characteristics of rotor blade, it can be concluded that closeness of natural frequencies likely indicates considerable coupling between corresponding DOFs in structure.

Dynamics of a flexible body: a two-field formulation

A two-field formulation of the nonlinear dynamics of an elastic body is presented in which positions/orientations and the resulting velocity field are treated as independent. Combining a nonclassical description of elastic velocity that includes the convection velocity due to elastic deformation with floating reference axes minimizing the relative kinetic energy due to elastic deformation provides a fully uncoupled expression of kinetic energy. A transformation inspired by the classical Legendre transformation concept is introduced to develop the motion equations in canonical form. Finite element discretization is achieved using the same shape function sets for elastic displacements and velocities. Specific attention is brought to the discretization of the gyroscopic forces induced by elastic deformation. A model reduction strategy to construct superelement models suitable for flexible multibody dynamics applications is proposed, which fulfills the essential condition of orthogonality between a rigid body and elastic motions. The problem of expressing kinematic connections at superelement boundaries is briefly addressed. Two academic examples have been developed to illustrate some of the concepts presented.

A unified strong spectral Tchebychev solution for predicting the free vibration characteristics of cylindrical shells with stepped-thickness and internal–external stiffeners

In this paper, a strong spectral Tchebychev (ST) solution is developed to investigate the free vibration behaviors of the cylindrical shell with stepped-thickness (CS-S) and cylindrical shell with internal–external stiffeners (CS-IES) subject to arbitrary boundary constraints. The spring parameter technique is used to impose combined connections and arbitrary boundaries. The potential and kinetic energy expressions of CS-S and CS-IES are derived based on the framework of the first-order shear deformation theory (FSDT). Finally, the unified governing differential equations of CS-S and CS-IES are acquired by the Hamilton’s variational principle. According to the Gauss–Lobatto point discretization process, the Tchebyshev polynomials are used to perform spectral expansion on the axial variables of admissible displacement functions. Circumferential variables are represented by sine and cosine series. In this way, the unified free vibration mathematical model of CS-S and CS-IES is obtained. On the basis of confirming that the model results have reached the state of convergence, the numerical results are compared with those calculated by the reference and the finite element method, which fully validates that the strong spectral Tchebychev model obtained in this paper is feasible for analyzing the free vibration characteristics of CS-S and CS-IES subject to arbitrary boundary constraints. Last but not least, the paper carries out the parametric analysis of how the change of the geometric parameters of the stiffeners will affect the natural frequencies of CS-IES.

Bi-penalty stabilized technique with predictor–corrector time scheme for contact-impact problems of elastic bars

This paper presents a stabilization technique for the finite element modelling of contact-impact problems of elastic bars via a bi-penalty method for enforcing contact constraints while employing an explicit predictor–corrector time integration algorithms. The present proposed method combines three salient features in carrying out explicit transient analysis of contact-impact problems: the addition of a penalty term associated with a kinetic energy expression of gap constraints, in addition to the conventional internal energy penalty term of the gap constraints; an explicit integration method that alleviates spurious oscillations; and, a judicious selection of two penalty parameters such that the stable time steps of the resulting explicit method is least compromised. Numerical experiments have been carried out with three explicit methods: the standard central difference method, the stabilized predictor–corrector method (Wu, 2003 [50]) and a method for mitigating spurious oscillations (Park et al., 2012 [44]) as applied to simulate one-dimensional contact-impact problems of the Signorini problem and the impact of two elastic bars. Results indicate that the proposed method can maintain the contact-free stability limit of the central difference and yield improved accuracy compared with existing bi-penalty methods.

VIBRATION CHARACTERISTICS OF NONUNIFORM BLADES MADE OF FUNCTIONALLY GRADED MATERIAL

The purpose of this study is to examine the vibration characteristics of a rotating blade whose material distribution varies in the spanwise direction. Formulations for functionally graded materials and beam structural models are carried out in detail and the results are displayed in several figures and tables which is a significant source of information for the authors working in this area. Different parameters such as angular speed, radius of the hub, material properties, power law index parameter, boundary conditions and slenderness ratio are considered in the formulation. Finite Element Method where the element matrices are obtained from potential and kinetic energy expressions is applied as the solution procedure. The calculated results are demonstrated in various tables and figures where it is observed that there is a good agreement with literature.

Vibrations Analysis of Cable-Harnessed Plates: Continuum Modeling and Experimental Validation

Abstract Dynamic modeling of spacecraft structures is imperative to their successful design for flight missions. A large number of these structures’ dry mass consists of signal and power cables, dynamics of which is not well-predicted using ad hoc cable lumped mass models. Hence, accurate modeling techniques are required to understand these cable dynamics effects. In the past, efforts toward developing analytical models for cable-harnessed structures have been primarily focused on beam-like host structures. The presented paper is aimed to fill this gap by obtaining analytical solutions through an energy-equivalent homogenization approach for cable-harnessed plate-like structures and to ultimately help with understanding the dynamic effects of signal and power cables on two-dimensional plate-like structures. As a first step, systems of periodic geometries with parallel cable configurations are considered. The strain and kinetic energy expressions for the fundamental repeated elements are found using linear displacement fields and Green–Lagrange strain tensors. The governing partial differential equation (PDE) for the out-of-plane motion of the cable-harnessed system is then found using Hamilton’s principle. Experimental modal testing is performed for the purpose of validations of the frequency response functions (FRFs) obtained from the model for the cable-harnessed plates under clamped-free-free-free boundary conditions. The results clearly show the dynamic effects of the (CFFF) cables which are also well-predicted by the model. Finally, modal assurance criterion (MAC) analysis has been used for further validations of the mode shapes obtained from the model.

Fluid Kinetic Energy Asymptotic Expansion for Two Variable Radii Moving Spherical Bubbles at Small Separation Distance

Two spherical bubbles with changing radii are considered to be moving in ideal fluid along their center-line. The exact expression for the fluid kinetic energy is obtained. The Stokes stream function is expanded in Gegenbauer polynomials in bispherical coordinates. This expansion is used to obtain the exact series for the fluid kinetic energy quadratic form coefficients. The new series are confirmed to be correct by comparison with the known ones. The main advantage of the new kinetic energy form is the possibility to obtain asymptotic expansions at small separation distance between the bubbles. These expansions are obtained and their convergence is analyzed. The results of this work can be used to describe the bubbles approach before the contact and their coalescence in acoustic field.

Effect of axially graded constraining layer on the free vibration properties of three layered sandwich beams with magnetorheological fluid core

The free linear vibration of an adaptive sandwich beam consisting of a frequency and field-dependent magnetorheological fluid core and an axially functionally graded constraining layer is investigated. The Euler-Bernoulli and Timoshenko beam theories are utilized for defining the longitudinal and lateral deformation of the sandwich beam. The Rayleigh-Ritz method is used to derive the frequency-dependent eigenvalue problem through the kinetic and strain energy expressions of the sandwich beam. In order to deal with the frequency dependency of the core, the approached complex eigenmodes method is implemented. The validity of the formulation and solution method is confirmed through comparison with the results available in the literature. Finally, the effects of the magnetic field, axially functionally graded material power-law index, constraining, and core layers thickness on the free vibration behavior of the sandwich beam are studied thoroughly for clamped–clamped, simply supported, and clamped-free boundary conditions. It is shown that any change in each of these parameters would have significant effects on the beam's free vibration properties.

Coupled vibration analysis for equivalent dynamic model of the space antenna truss

A novel equivalent dynamic model is developed for coupled vibration analysis of the space antenna truss to enhance the design capacity of vibration controllers. Based on energy equivalence principle, the space antenna truss with two important features of rigid joints and complicated configuration is equivalent to a spatial anisotropy Timoshenko beam model. According to the kinematic assumptions, strain and kinetic energy expressions of the spatial periodic element can be obtained in accordance with displacement components at its center. Hamilton's principle is carried out to formulate the governing partial differential equations of motion for the equivalent beam model (EBM) which is divided into two sets of PDEs. Each set of PDEs includes three degrees of freedom and describes bending-torsion and bending-extension couplings respectively, due to the asymmetry of the antenna truss. An exact analytical method is developed to solve the two sets of coupled motion equations. The natural characteristics of the EBM are shown to be in excellent agreement with those of the finite element method, which demonstrates that the proposed EBM can provide a satisfactory accuracy for the antenna truss. In addition, since the mode shapes of the EBM are expressed as analytical functions of the spatial coordinate, such an approach may lead the investigations of dynamic property and the design of vibration control law to become convenient for the space antenna truss.