Derive Equations Of Motion Research Articles

A Detailed Analysis of the Dynamic Behavior of a MEMS Vibrating Internal Ring Gyroscope.

This paper presents the development of an analytical model of an internal vibrating ring gyroscope in a Microelectromechanical System (MEMS). The internal ring structure consists of eight semicircular beams that are attached to the externally placed anchors. This research work analyzes the vibrating ring gyroscope's in-plane displacement behavior and the resulting elliptical vibrational modes. The elliptical vibrational modes appear as pairs with the same resonance frequency due to the symmetric structure of the design. The analysis commences by conceptualizing the ring as a geometric structure with a circular shape possessing specific dimensions such as thickness, height, and radius. We construct a linear model that characterizes the vibrational dynamics of the internal vibrating ring. The analysis develops a comprehensive mathematical formulation for the radial and tangential displacements in local polar coordinates by considering the inextensional displacement of the ring structure. By utilizing the derived motion equations, we highlight the underlying relationships driving the vibrational characteristics of the MEMS' vibrating ring gyroscope. These dynamic vibrational relationships are essential in enabling the vibrating ring gyroscope's future utilization in accurate navigation and motion sensing technologies.

Matematyczne modele hydromechaniki przepływu wielofazowego o zmiennej masie

The paper discusses the mathematical model of hydromechanics of multiphase flows with varying mass. A multiphase flow is considered a continuum consisting of a set of a large number of different groups of particles. The derivation of motion equations and similarity criteria are given taking into account both the externally attached (or detached) mass and phase transitions within the medium. The equations of mass, momentum and energy transfer for individual phases and the medium as a whole are derived based on fundamental conservation laws. It was demonstrated that in the absence of sources (or flow-offs) of mass, momentum and energy, the known equations of single- and multi-phase flow hydromechanics follow as a special case from the obtained systems of motion equations and similarity criteria. The obtained motion equations are valid for the description of an ingredient of mixture and the medium as a whole, regardless of their physical and mechanical properties. Thermodynamic and rheological state equations, as well as expressions for heat flow, interfacial mass forces phase transitions, and heat exchange between phases can be used to close them. The implemented models make it possible to simulate both the stationary distribution of parameters along the wellbore during production and non-stationary processes that occur, for example, when the pump shaft speed changes during oil production. The developed approaches were implemented in the DataFlow software tool for analysis of the hydrodynamics of multiphase hydrocarbon flows, taking into account heat exchange with the rocks surrounding the well, and phase transitions in the fluid. Using the software package, test calculations were carried out to demonstrate the performance of the proposed and implemented models.

Nonlinear Structural Vibration of Multi-Megawatt Vertical Axis Wind Turbine Blades-Part 1: Derivation of Motion Equations

Replacing fossil fuels with renewable energy is one of the important means to improve energy scarcity and environmental protection, and is of great significance for achieving sustainable development of human society. Vertical axis wind turbine (VAWT) is one of the key equipment for applying green energy — wind energy. During the operation of VAWT, its blades not only have to withstand periodic aerodynamic loads, but also suffer from centrifugal force effects and the Coriolis force effect caused by coupling with multi-dimensional deformation. The complex stress situation makes the blades susceptible to damage, resulting in structural resonance and instability of aeroelastic stability. This paper applies the Euler beam model and Hamilton principle to establish a nonlinear structural vibration motion equation for large-span vertical axis wind turbine blades with fully coupled four-dimensional deformation (i.e. lateral bending deformation, chord bending deformation, axial torsion deformation, and axial tensile deformation), and retains the nonlinear term of the motion equation to the third-order infinitesimal. Separating the motion equation into equilibrium equation and dynamic equation, it can be seen that the displacement of equilibrium state caused by the centrifugal force of the rigid body exists in the form of a coefficient in the dynamic equation, which will have an impact on the dynamic characteristics. This modeling work can provide a research foundation for exploring the dynamic characteristics and aeroelastic stability of VAWT blades in the future.

A new gravitational action for the trace anomaly

The question of building a local diff-invariant effective gravitational action for the trace anomaly is reconsidered. General Relativity (GR) combined with the existing action for the trace anomaly is an inconsistent low energy effective field theory. This issue is addressed by extending GR into a certain scalar-tensor theory, which preserves the GR trace anomaly equation, up to higher order corrections. The extension introduces a new mass scale – assumed to be below the Planck scale – that governs four high dimensional terms in a local diff-invariant trace anomaly action. Such terms can be kept, while an infinite number of Planck-suppressed invariants are neglected. The resulting theory maintains two derivative equations of motion. In a certain approximation it reduces to the conformal Gallileon, which could have physical consequences.

Chiral gravitational waves in Palatini-Chern-Simons gravity

We study the parity-breaking higher-curvature gravity theory of Chern-Simons (CS), using the Palatini formulation in which the metric and connection are taken to be independent fields. We first show that Palatini-CS gravity leads to first-order derivative equations of motion and thus avoid the typical instabilities of CS gravity in the metric formalism. As an initial application, we analyze the cosmological propagation of gravitational waves (GWs) in Palatini-CS gravity. We show that, due to parity breaking, the polarizations of GWs suffer two effects during propagation: amplitude birefringence (which changes the polarization ellipticity) and velocity birefringence (which rotates the polarization plane). While amplitude birefringence is known to be present in CS gravity in the metric formalism, velocity birefringence is not present in metric CS gravity for high frequency waves, but now appears in Palatini-CS due to the fact that left-handed and right-handed GW polarizations have a different dispersion relation. In the approximation of small deviations from general relativity (GR), we do find however that velocity birefringence appears at least quadratically in the CS coupling parameter $\ensuremath{\alpha}$, while amplitude birefringence appears linearly in $\ensuremath{\alpha}$. This means that amplitude birefringence will be the most relevant effect in Palatini-CS and hence this model will behave similarly to metric CS. We confirm this by applying current constraints on amplitude and velocity birefringence to Palatini-CS, and showing that those from amplitude birefringence give the tightest bounds.

Hardware in the loop simulation and implementation of a dragonfly-like MAV using clap and fling mechanism

The main aim of this paper is to model and simulate flight dynamics and to design a controller for a dragonfly-like micro aerial vehicle (MAV). This MAV has flapping wings using a clap and fling mechanism with a rigid abdomen. Kane’s method is applied to derive the longitudinal motion equations of the MAV as a multi-body system. Then, the aerodynamic lift and drag forces of the MAV are obtained. These equations are substituted in the derived motion equations. Furthermore, a new structure for a dragonfly-like flapping wing MAV is presented in which abdomen motion is considered in the longitudinal mode. In this work, the use of the abdomen is also formulated. A change in the motor’s speed generates differential thrust that creates a pitch moment as a control input. State space linearized equations of motion are presented by using appropriate assumptions in the aerodynamics, and dynamic nonlinear equations. To validate the linearized model, the response of the open-loop linear system is compared with the nonlinear one. In addition, the positive role of the abdomen in the open loop is discussed and analyzed. Finally, an LQR controller is designed for the pitch mode which is robust against disturbances. The validity, effectiveness, and performance of the derived equations and designed autopilot are shown by implemented hardware in the loop testbed using a fixed abdomen. Results demonstrate a good agreement between theoretical and experimental responses with accurate hovering at the desired angle.

Causality in strain gradient elasticity: An internal variables approach

The absence of higher order time derivatives in equations of motion causes the infinite velocity of wave propagation in strain gradient elasticity models. This issue needs to be avoided from both a theoretical and practical viewpoint. It is shown that dual internal variables approach solves the causality problem. Furthermore, the primary internal variable can be interpreted as the Laplacian of the strain gradient resulting in an Aifantis-type strain gradient model.

Nonlinear dynamic analysis of the extended telescopic joints manipulator with flexible links

In this paper, the open-chain arm containing flexible links connected through telescopic joints is investigated so as to replace robotic chains with revolute joints. In this manipulator, telescopic joints are considered by a simultaneous motion combination of the prismatic and revolute joints. In comparison with revolute–prismatic joints in which prismatic joint hubs are utilized, telescopic joints lack the hub, and the linear motion is generated via a gear. Accordingly, the flexible link is not placed inside the hub. In fact, the link is merely divided into two sections each at a time step. The front section which is considered in the motion equations is placed at the back of the hinge’s point. This part solely creates vibration, which probably affects the motion of other sections. This type of manipulator’s joints due to the structure benefits consists of the optimum weight, simple mechanism, extended workspace acceptable for robotic tools, industrial machinery, and domestic systems. To derive motion equations of N-link flexible manipulator with telescopic joints system, the Euler–Lagrange formulation is employed. The obtained equations are in the time-varying form, and the manipulator link length changes during motion. The derived equations are simulated for a two-link system with different elasticity values, and the results are then analyzed to verify the outcomes. The bifurcation analysis has also been used to prevent unstable vibrations and chaotic responses. It should be noticed that this type of manipulator can be used in a fast and precise mechatronics' system for utilization in space exploration, space station, and spacecraft as well.

Horndeski theories and beyond from higher dimensions

The Einstein–Hilbert action with a cosmological constant is the most general local four-dimensional action leading to second-order derivative equations of motion that are symmetric and divergence free. In higher dimensions, additional terms can appear. We investigate a generalised metric decomposition involving a scalar degree of freedom to express the higher-dimensional action as an effective four-dimensional scalar–tensor theory. From the higher-dimensional Ricci scalar alone and a subclass of our metric ansatz, we recover the subset of Horndeski theories with luminal speed of gravitational waves. More generally, beyond-Horndeski terms appear. When including a Gauss–Bonnet scalar in the higher-dimensional action, we generate contributions to all cubic-order second-derivative terms present in the degenerate higher-order scalar–tensor theory (DHOST) as well as higher-derivative terms beyond that. We discuss this technique as a way to generate healthy four-dimensional gravity theories with an extra scalar degree of freedom and outline further generalisations of our method.

Reconfigurable Mobile Robot with Adjustable Width and Length: Conceptual Design, Motion Equations and Simulation

In this article, the dynamic equations of a reconfigurable non-holonomic mobile robot used for space explorations and rescue operations are presented. Dimensions of mobile robots usually remain unchanged while they are programmed to determine a new path to bypass obstacles. The aim of developing this robot is to upgrade the mechanical structure so it can adapt its structure to pass the obstacles without path deviation. Furthermore, by means of mentioned reconfigurations, less motor power is needed, leading to optimized energy consumption. To this end, longitudinal and transverse adjustments are defined for the robot. In view of the motion restrictions existing in conventional wheels, omnidirectional wheels are used in robot structure and their features are considered when deriving motion eqs. Accordingly, the robot is able to move in the direction of wheels’ axis. To evaluate the designed mechanism, the system is simulated in ADAMS and the results are compared with the derived motion equations. The findings prove that not only is this system implementable, but the results are also consistent with ADAMS with an acceptable error. Additionally, the calculated energy consumption in a robot that uses transverse adjustment to cross obstacles decreases by 12% compared to a robot that climbs the obstacle and 10% decrease occurs compared to path planning method. As the system dynamics are obtained in a general form, the derived equations can be applied in various applications and different configurations.

Minimum-time Trajectory Planning for Residual Vibration Suppression of Flexible Manipulator Carrying a Payload

For the purpose of eliminating residual vibration of a two-link flexible manipulator as well as minimizing time consumption, a scheme of multi-objective trajectory planning in joint space is proposed in this paper. Piecewise polynomial is applied to generate a smooth joint trajectory. Lagrangian approach and the assumed modes method is used to derive motion equations of FMs. Finally, Multi-objective particle swarm optimization (MOPSO) algorithm is implemented to solve time of each polynomial. The effectiveness of proposed algorithm is demonstrated by simulation results.

Analytical Investigation of the Vibrational and Dynamic Response of Nano-Composite Cylindrical Shell Under Thermal Shock and Mild Heat Field by DQM Method

In this paper, the vibrations and dynamic response of an orthotropic thin-walled composite cylindrical shell with epoxy graphite layers reinforced with carbon nanotubes under heat shock and heat field loading are investigated. the carbon nanotubes were uniformly distributed along the thickness of the composite layer. The problem is that at first there is a temperature change due to the thermal field in the cylinder and the cylinder is coincident with the thermal field, then the surface temperature of the cylinder rises abruptly. Partial derivative equations of motion are coupled to heat equations. The differential quadrature method (DQM) is used to solve the equations. In this study, the effects of length, temperature, thickness and radius parameters on the natural frequencies and mid-layer displacement are investigated. The results show that increasing the outside temperature reduces the natural frequency and increases the displacement of the system. Radial displacement results were also compared with previous studies and were found to be in good agreement with previous literature. Increasing the percentage of carbon nanotubes also increased the natural frequency of the system and decreased the mobility of the middle layer.

Free and forced vibration analyses of simply supported Z-reinforced sandwich structures with cavities through a theoretical approach

This paper develops a theoretical approach to study free and forced vibrations of simply supported Z-reinforced sandwich structures with cavities in vacuum and water. A multi-level homogenization model is firstly proposed to transform the core layer of the sandwich structure to an orthotropic material. Then the first-order shear deformation theory (FSDT) is adopted to describe displacement fields and Hamilton’s principle is used to derive motion equations of the sandwich structure. In water domain, the acoustic pressure and the continuity on the fluid-structure coupling face are governed by Helmholtz equation and Euler’s equation respectively. By the aid of Green’s function, the transverse vibration formula is rebuilt through regarding the submerged sandwich structure as a thin plate. New motion equations are thus achieved. Under simply supported boundaries, natural frequencies and forced vibration responses are obtained via the Navier method. By comparing theoretical results with those from finite element (FE) and boundary element (BE) analyses, the high accuracy of the present approach is verified and a clear application frequency range is found. Besides, effects of diameters of reinforcements and cavities on free vibration are discussed. Influences of the external force location and fluid loading on forced vibration are also studied.

The Dynamic and Vibration Response of Composite Cylindrical Shell Under Thermal Shock and Mild Heat Field

In this article, the vibration and dynamic response of an orthotropic composite cylindrical shell under thermal shock loading and thermal field have been investigated. The problem is that the shell is initially located at a first temperature, and some tension caused by a mild heat field is created, then the surface temperature of the cylinder suddenly increases. The partial derivative equations of motion are in the form of couplings with the heat equations. First, the equations of motion are derived by the Hamilton principle, here first-order shear theory and considering strain-shift relations of Sanders are used. Then, the equation system including the equations of motion and energy equations by the Runge–Kutta fourth-order methodare solved. In this study, the effects of length, temperature, thickness and radius parameters on natural frequencies and intermediate layer displacement are investigated. The results show that the increase in external temperature decreases the natural frequency and increases the displacement of the system. Also, the results of radial transitions were evaluated with previous studies and it was found that it is in good agreement with the results of previous papers.

Damping Potential, Generalized Potential, and D’Alembert’s Principle

An alternative form of Lagrange’s equations is used to introduce a new equation for deriving a monogenic force from a velocity-dependent potential. Based on this and corresponding to a linear damping force, a bilinear potential function is introduced, which obviates the need for a dissipation function to represent a damping force. This idea is then generalized by introducing a generalized potential function which allows D’Alembert’s principle, Lagrange’s equations, and Hamilton’s principle to be represented solely in terms of potential functions. The introduced potential functions and equations are used to derive motion equations of a floating body.

Stability analysis of an axially moving nanocomposite circular cylindrical shell with time-dependent velocity in thermal environments

This article investigates the instability of an axially moving simply supported nanocomposite circular cylindrical shell reinforced by single-walled carbon nanotubes (SWCNTs) with time variant velocity in thermal environments considering the effect of viscous structure damping. The nanocomposite material properties are obtained from the extended mixture rule for uniformly distributed CNT, furthermore three different types of functionally graded distribution CNT in the matrix are represented. In addition, the Hamilton principle via the simplification assumption of the Donnell shallow-shell theory is employed to derive motion equations. At first, related displacements are obtained in terms of displacement in radius direction which is expressed in a combination of driven and companion modes from motion equations in axial and circumferential directions. The dimensionless motion equation in radius direction is discretized by the Galerkin method. The critical speed values of different types of CNT distribution reinforcement, various types of volume-fraction of CNT reinforcement and different temperatures are established using the multiple scale method. Moreover, the first order solution of the perturbation theory makes it possible to predict the unstable regions of principle parametric resonances and combination resonances. It occurs under the effect of temperature rising condition, changing the kind of SWCNTs as well as increasing the value of viscous structure damping, volume-fraction of CNT distribution, constant part of velocity function and fluctuation domain. Results show a system with the symmetric distribution of CNT reinforcement, while the outer and inner are CNT-rich, represents higher stability in axially moving nanocomposite cylindrical shells with constant and time-dependent velocity. Communicated by Eleonora Tubaldi

Emergence of Pseudo-Phononic Gaps in Periodically Architected Pendulums

Rejection of unmitigated vibrational disturbances represents an ongoing dilemma in complex linkage systems. In this work, we present an inherent and self-reliant vibration isolation mechanism in architected periodic chains of serially pivoted pendulums. Absorption of external excitations is achieved by virtue of Bragg-type band gaps which stem from the emergent chain dynamics. Owing to its coupled dynamics, the self-repeating cell of a Phononic Crystal (PC) is not trivial and cannot be readily identified from the periodic arrangement by inspection. As such, this work entails the extraction of a "pseudo" unit cell of an equivalent PC lattice from the derived motion equations of a finite chain, which departs from traditional wave dispersion methods. The model presented herein comprises a chain of linked pendulums with periodic variations of inertial/geometrical properties, carrying a payload at one end. We ultimately show evidence of forbidden wave propagation in prescribed frequency regimes, reminiscent of band gaps in PC lattices. The presented framework can be invaluable in applications that require vibration reduction in the delivery of payloads including gantry cranes, robotic arms and space tethers.

Mathematical Model of the RRR Anthropomorphic Mechanism for 2D Biomechanical Analysis of a Deep Squat and Related Forms of Movement

The aim of this study was to create a mathematical model of the RRR anthropomorphic mechanism for a 2D biomechanical analysis of a deep squat and related forms of movement. The segment stick model is designed to diagnose the movement with sagittal plan symmetry. Based on the input data from kinematic and dynamometric analysis, and from the anthropometric data of the monitored person, it is possible to estimate the resulting momentum of the forces acting on the main joints of the lower body. The technology may be applied in analysing deep squats, studying the dynamics of vertical reflection as well as in the biomechanical analysis of related forms of movement (e.g. standing-up, squatting with a dumbbell, skiing in downhill posture, etc.). The derived motion equations may be used to analyse the dynamics of the movement of anthropomorphic or mechatronic systems with the same geometry.

Geometrically nonlinear transient analysis of laminated composite super-elliptic shell structures with generalized differential quadrature method

In this study, geometrically nonlinear dynamic behavior of laminated composite super-elliptic shells is investigated using generalized differential quadrature method. Super-elliptic shell can represent cylindrical, elliptical or quasi-rectangular shell by adjusting parameters in super-ellipse formulation (also known as Lamé curve formulation). Geometric nonlinearity is taken into account using Green–Lagrange nonlinear strain–displacement relations that are derived using differential geometry and theory of surfaces. Transverse shear effect is considered through the first-order shear deformation theory. Equation of motion is obtained using virtual work principle. Spatial derivatives in equation of motion is expressed with generalized differential quadrature method and time integration is carried out using Newmark average acceleration method. Several super-elliptic shell problems under uniform distributed load are solved with the proposed method. Effects of layer orientations, boundary conditions, ovality and ellipticity on dynamic behavior are investigated. Transient responses are compared with finite element solutions.

The self-excited vibrations of an axially retracting cantilever beam using the Galerkin method with fitted polynomial basis functions

In the present study, the self-excited vibrations of an axially retracting beam under gravity is investigated. A GFB method is presented to discretize the system energy functions, in which the orthogonality relationships between the basis functions are needless. In this way, the motion equations can be derived in a much simpler way of matrix. The presented method and the derived motion equations are validated by comparing the computed results with which from previous literature. Based on the computed dynamic responses, the effect of gravity on beam dynamic behavior is investigated. An interesting phenomenon of self-excited vibration is observed as the beam retracts with a certain axial speed. Finally, the effects of retracting speed, structural damping and static tip deflection on self-excited vibration are investigated.