Lagrangian Equations Of Motion Research Articles

Dynamic Model and PD Control with Forces Compensation of Dual-Stage Gough-Stewart Platform

This paper investigates a dual-stage Gough-Stewart platform. The lower platform is responsible for simulating the oscillations of moving vehicles such as cars, ships, and airplanes. The upper platform is connected to devices that require either balance stabilization or motion stabilization according to specific requirements. The dynamic model of the robot system is derived in a general form based on the Lagrange equations of motion with Lagrange multipliers. Using these equations in a compact form, a PD controller with forces compensation in task space is designed for the robot system. Oscillation generation and balance stabilization are computed and simulated using the kinematic and dynamic parameters of two Bosch Rexroth robots. The computation and simulation results demonstrate the dynamic model's accuracy and the controller's effectiveness.

Particle response to oscillatory flows at finite Reynolds numbers

The response of spherical particles to oscillatory fluid flow forcing at finite Reynolds numbers exhibits significant deviations from classical analytical predictions due to nonlinear convective contributions. This study employs finite element simulations to explore the long-term (stationary) behavior of such particles across a wide range of conditions, including various external and particle Reynolds numbers, Strouhal numbers, and fluid-to-particle density ratios. Key contributions of this work include determining the range of validity of Tchen's equation of motion for infinitesimal and finite Reynolds numbers and correlating particle response for a wide range of density ratios and flow conditions at high frequency oscillations. This work introduces a modified form of the history drag term in a newly proposed Lagrangian equation of motion. The new equation incorporates a parameter-dependent fractional-order derivative tailored to accommodate nonlinearities due to convective effects. These novel correlations not only extend the operational range of existing model equations but also provide accurate estimates of particle response under a range of external flow conditions, as validated by comparison with numerical solutions of the Navier–Stokes flow around the particles.

Polarization effects in higher-order guiding-centre Lagrangian dynamics

The extended guiding-centre Lagrangian equations of motion are derived by the Lie-transform perturbation method under the assumption of time-dependent and inhomogeneous electric and magnetic fields that satisfy the standard guiding-centre space–time orderings. Polarization effects are introduced into the Lagrangian dynamics by the inclusion of the polarization drift velocity in the guiding-centre velocity and the appearance of finite-Larmor-radius corrections in the guiding-centre Hamiltonian and guiding-centre Poisson bracket.

Lagrangian formulation for free 6D infinite spin field

We construct a Lagrangian that describes the dynamics of a six-dimensional free infinite (continuous) spin field in 6D Minkowski space. The Lagrangian is formulated in the framework of the BRST approach to higher spin field theory and is based on a system of constraints defining an irreducible representation of the corresponding Poincaré group. The field realization of generators in the 6D Poincaré algebra and the second-, fourth-, and sixth-order Casimir operators are obtained in explicit form using additional spinor coordinates. Specific aspects of such a realization in six dimensions are discussed. We derive the conditions that determine the irreducible representation 6D infinite spin field and reformulate them as operators in the Fock space forming a first-class algebra in terms of commutators. These operators are used to construct the BRST charge and the corresponding Lagrangian. We prove that the conditions of the irreducible representation are reproduced as the consequence of the Lagrangian equations of motion, which finally provides the correctness of the results obtained.

Physical based motion reconstruction from videos using musculoskeletal model

AbstractWe propose a novel method that combines human pose estimation and physical simulation of character animation. Our approach allows characters to learn from the actor's skills captured in videos and subsequently reconstruct the motions with high fidelity in a physically simulated environment. Firstly, we model the character based on the human musculoskeletal system and build a complete dynamics model of the proposed system using the Lagrange equations of motion. Next, we employ the pose estimation method to process the input video and generate human reference motion. Finally, we design a hierarchical control framework consisting of a trajectory tracking layer and a muscle control layer. The trajectory tracking layer aims to minimize the difference between the reference motion pose and the actual output pose, while the muscle control layer aims to minimize the difference between the target torque and the actual output muscle force. The two layers interact by passing parameters through a proportional differential controller until the desired learning objective is achieved. A series of complex experimental results demonstrate that our proposed method can learn to produce comparable high‐quality motions with high similarity from videos of different complexity levels and remains stable in the presence of muscle contracture weakness perturbations.

A critical assessment of the immersed boundary method for modeling flow around fixed and moving bodies

This paper reports the verification and validation of the immersed boundary methods implemented in the open-source toolbox FOAM-Extend version 4.0 and 4.1. Five test cases with increasing complexity are studied and the simulation results are compared with the experimental and numerical data found in the literature. A solver is developed based on the Immersed Boundary Surface Method (IBS) which is an advanced cut-cell finite volume immersed boundary technique. By integrating the Lagrangian equations of motion with the IBS, this solver is capable of simulating fluid–particle interactions. The benchmark studies indicated that the IBS yields excellent results for stationary immersed bodies. Even though the flow field is captured satisfactorily for moving bodies, spurious force oscillations are present in these cases. Simulating different size particles in a channel flow shows that the spurious force oscillations can negatively affect the prediction of particle dynamic motion, although the severity depends on the value of the particle Stokes number.

Lagrangian Equation of Coupled Spring-Pendulum System

A coupled spring-pendulum system in a conservative field was studied where the equation of motion of the system using Lagrangian and Hamiltonian equation were obtained. The equation of motion represented by a second-order differential equation from the three generalized coordinate were used. The potential energy equal to zero when the system is in its equilibrium position. The generalized coordinate that being used were the angle of the first pendulum θ_1, the angle of the second pendulum θ_2, and the increase in the length of the spring x. The resulting equation of motion can be used to determine the dynamics behavior of the system at any time. Students' understanding is expected to be more complete by providing a procedure to derive the Lagrangian equation of motion.

Hidden Degrees of Freedom in Implicit Vortex Filaments

This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.

Operational classical mechanics: holonomic systems

We construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, we rediscover several results from analytical mechanics from an entirely new perspective. We start by expressing the position and velocity of point particles as the eigenvalues of self-adjoint operators acting on a suitable Hilbert space. The concept of holonomic constraint is shown to be equivalent to a restriction to a linear subspace of the free Hilbert space. The principal results we obtain are: (1) the Lagrange equations of motion are derived without the use of D’Alembert or Hamilton principles, (2) the constraining forces are obtained without the use of Lagrange multipliers, (3) the passage from a position–velocity to a position–momentum description of the movement is done without the use of a Legendre transformation, (4) the Koopman–von Neumann theory is obtained as a result of our ab initio operational approach, (5) previous work on the Schwinger action principle for classical systems is generalized to include holonomic constraints.

Nonlinear flutter of 2D variable stiffness curvilinear fibers composite laminates by a higher-order shear flexible beam theory with Poisson’s effect

In this work, the nonlinear supersonic panel flutter characteristics of two-dimensional variable stiffness curvilinear fibres based laminated composite panels are studied using a higher-order shear flexible theory represented by sine function coupled with first-order approximation leading to quasi-aerodynamic theory. The structural formation takes care of geometric nonlinearity with von Karman’s assumptions. The beam constitutive equation is modified for the laminated beam with general lay-up by accounting for Poisson’s effect. The nonlinear dynamic equilibrium equations developed by Lagrangian equations of motion are solved using finite element approach in conjunction with the direct iterative solution procedure. For limit cycle oscillation, critical dynamic pressure is predicted iteratively through eigenvalue analysis, thereby identifying the first coalescence of vibrational modes. Also, the flutter behavior of two-dimensional panel under static differential pressure is investigated considering nonlinear static equilibrium position of panel obtained by Newton-Raphson’s iterative approach and then followed by modes coalescence approach. These solution procedures are tested against the results in literature. A thorough numerical investigation is done to show the effect of the curvilinear fiber path orientation, limited cycle amplitude, static differential pressure, panel thickness, panel end condition flexibilities and thermal environment on the nonlinear supersonic panel flutter of two-dimensional variable stiffness laminated panels.

Spin-induced internal resonance in circular cylindrical shells

Internal resonances in circular cylindrical shells, induced by the spinning motion, are explored in this research. The mathematical model of spinning cylindrical shells is established by employing Donnell’s nonlinear shell theory and the Lagrange equations of motion. Then, the nonlinear forced vibration responses of the shells are solved by using the pseudo-arclength continuation technique. Some comparisons are conducted to verify the accuracy of the present model and method The results show that internal resonances can be caused by the spinning motion of the shells, which cover a wide range of spinning speeds. Different types of internal resonances can occur, such as 1:1, 1:1:1, and 1:2 internal resonances. Energy exchanges between different modes with internal resonance relationships. Both softening and hardening nonlinearities exist in the frequency response of the spinning cylindrical shells. The excitation amplitude has a threshold to excite the internal resonances. It is found that the internal resonances can be avoided by tuning the spinning speed of the shells.

Power requirements for bat-inspired flapping flight with heavy, highly articulated and cambered wings.

Bats fly with highly articulated and heavy wings. To understand their power requirements, we develop a three-dimensional reduced-order model, and apply it to flights of Cynopterus brachyotis, the lesser dog-faced fruit bat. Using previously measured wing kinematics, the model computes aerodynamic forces using blade element momentum theory, and incorporates inertial forces of the flapping wing using the measured mass distribution of the membrane wing and body. The two are combined into a Lagrangian equation of motion, and we performed Monte Carlo simulations to address uncertainties in measurement errors and modelling assumptions. We find that the camber of the armwing decreases with flight speed whereas the handwing camber is more independent of speed. Wing camber disproportionately impacts energetics, mainly during the downstroke, and increases the power requirement from 8% to 22% over flight speed U = 3.2-7.4 m s-1. We separate total power into aerodynamic and inertial components, and aerodynamic power into parasitic, profile and induced power, and find strong agreement with previous theoretical and experimental studies. We find that inertia of wings help to balance aerodynamic forces, alleviating the muscle power required for weight support and thrust generation. Furthermore, the model suggests aerodynamic forces assist in lifting the heavy wing during upstroke.

Design of rotary inverted pendulum swinging-up and stabilizing

The mechanical design of inverted pendulum systems may be very diverse. This study makes use of mathematics to explain the expanded unstable Rotary planer inverted pendulum arrangement (Rotary -inverted system). Pin connections are used in planer-inverted pendulums to attach the pendulum to the rotating- Rotary actuation base. This design of a Rotary -inverted pendulum is explored in the development of underactuated robotic systems because it best simulates the balance of a broomstick in the hand by treating the elbow and shoulder as revolute joints. It is necessary to utilize the Lagrangian equation of motion when creating the dynamical equation for the Rotary -inverted pendulum. MATLAB Simulink is used to simulate a nonlinear computational model of the system so as to test the accuracy of the mathematical model.

Low-Order Modeling of Flapping Flight with Highly Articulated, Cambered, Heavy Wings

A three-dimensional low-order modeling technique is described for modeling both wing inertial and aerodynamic forces of animals and robots flapping heavy and morphing wings. The model is applied to prescribed kinematics from a previously measured flight of a lesser-nosed, dog-faced fruit bat (Cynopterus brachyotis) in straight and climbing flight. Quasi-steady blade element momentum theory is used to model the aerodynamic forces of the highly articulated wing, and a Lagrangian equation of motion is adopted to describe the overall wing–body dynamics that includes the wing inertia. These two distinct and independent models yield a good prediction of the thrust/drag balance, and are in excellent agreement of lift production. Unique to this approach, the quantitative effect of wing camber was examined within the framework, which contributes as much as 27% of the cycle-averaged lift, with a smaller price (10%) paid to drag.

Figure-eight orbits in three post-Newtonian formulations of triple black holes

There are three types of dynamical equations for a post-Newtonian Lagrangian formulation of three black holes. They are approximate post-Newtonian Lagrangian equations of motion, coherent post-Newtonian Lagrangian equations of motion, and post-Newtonian Hamiltonian canonical equations. The energy integral is conserved approximately in the first model, but exactly in the second and third models. The three formulations are not exactly equivalent, while they are approximately related in the orbital dynamical behavior. Figure-eight orbits in the three problems may have different initial conditions. Imai, Chiba, and Asada adjusted the initial velocities satisfying the figure-eight orbits for given initial positions in the first model. The initial velocities are still suitable for the figure-eight orbits of the latter two models when the initial separations of the first body are enough large. However, they are not generally applicable to relatively small initial separations in the three systems. We can obtain the desired initial velocities by scanning the initial velocities of the third body in the three systems. The figure-eight orbits in these models exhibit sensitive dependence on the scanned initial velocities for smaller initial separations. Although the differences in the scanned initial velocities among the three problems are minor for the small initial separations, the scanned initial velocities for any one of the three systems do not yield the figure-eight orbits for the other two systems.

Solution of Lagrange's equation of motion form the first principle for volumetric modulated arc therapy delivery

To calculate the Lagrange’s equation of motion from first principle for volumetric modulated arc therapy delivery. The delivery of volumetric modulated arc therapy (VMAT) is based on a rotational co-ordinate system. This delivery technique differs from any other delivery technique with static beams. VMAT delivery is a complex many-body problem, to represent the VMAT uniquely, it is necessary to find the equation of motion in a non-inertial (rotating) fame. In the non-inertial frame of reference, Newton's equations of motion do not hold, hence Lagrange's equation of motion (Lagrange's equation of motion of second kind) needs to be solved. If A is the aperture created by Multileaf collimator (MLC), MU is delivered Monitor Unit of radiation dose and θ is the gantry angle, then equation of motion for volumetric arc therapy delivery is represented by following three equations 1. $$\frac{d\theta }{dt}=\dot{\theta }=$$ constant between two control points, 2. $$m\ddot{x}-m x \dot{\theta }2= Fx$$ ; where Fx is the external force along $$\overrightarrow{x}$$ direction, causing the MLC movement along the x-direction only and m is the mass of the MLC, and $${MU}_{delivered}\left(t\right)={\int }_{t=0}^{t=t}MU{(t)}_{planned}dt$$ . This piece of derivation presents the calculation of the equation of motion for VMAT delivery in terms of MLC movement, gantry angle, and delivered dose by deriving Lagrangian formulation from the first principle.

Passive-based adaptive control with the full-order observer for induction motor without speed sensor

The conventional linear control methods are difficult to meet the control requirements of high-performance speed regulation of asynchronous motor due to the nonlinear and multi-variable problems of induction motor. A passive-based control method of induction motor with the full-order state observer is proposed with the Euler–Lagrange equation of motion of the induction motor. Based on the relationship between passivity and stability of induction motor, the state feedback is used for torque and speed tracking. The full-order state observer is adopted with rotor current and rotor flux as state variables, and the adaptive speed controller is designed to realize the passive-based control. The experimental results show that the errors between the estimated value based on the proposed full-order observer and the actual value of rotor current, speed and flux are small; the speed with the proposed adaptive control can reach the expected value quickly. The proposed control method can effectively meet the high-performance requirements of induction motor.

Finite element analysis of coupled phase-field and thermoelasticity equations at large strains for martensitic phase transformations based on implicit and explicit time discretization schemes

In this paper, a staggered, nonlinear finite element procedure is developed to solve the large-strain based coupled system of time dependent Ginzburg-Landau (GL) and thermoelasticity equations for phase transformations at the nanoscale. Geometrical nonlinearities are included based on the total Lagrangian description where the total deformation gradient is defined as the multiplicative decomposition of elastic, transformational and thermal deformation gradient tensors. The principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion whose discretized form is solved using the iterative methods which gives the thermal and transformational deformation gradient tensors. Next, the elastic deformation gradient tensor and the first Piola-Kirchhoff stress are calculated and substituted in the GL equation which gives the phase order parameter. The weighted residual method is used to derive the corresponding finite element form which allows for the phase-dependent surface energy BCs. Explicit and different implicit methods belonging to the generalized trapezoidal family are used for time discretization of the GL equation. Various examples of phase transformations are simulated and discussed. The effect of different time integration schemes are also investigated. The current study allows for a proper modeling of various phase field problems at large and small strains.

Explicit nonlinear finite element approach to the Lagrangian-based coupled phase field and elasticity equations for nanoscale thermal- and stress-induced martensitic transformations

In this paper, a nonlinear finite element procedure is developed to solve the coupled system of phase field and elasticity equations at large strains for martensitic phase transformations at the nanoscale. The transformation is defined based on an order parameter which varies between 0 for austenite to one for martensite. The phase field equation relates the rate of change of the order parameter to the Helmholtz free energy. Both the Helmholtz free energy and the transformational strain tensor depend on the order parameter, and the coupling between the phase field and elasticity equations occurs due to the presence of elastic energy in the Helmholtz free energy and the transformational strain in the total strain. The deformation gradient tensor is introduced as a multiplicative decomposition of elastic and transformational gradient tensors. A staggered strategy is used to solve the coupled system of equations so that first, the principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion. It is then discretized and solved using the Newton–Raphson method which gives the displacements and consequently, the deformation gradient tensor. Next, since the order parameter is obtained from the phase field equation from the previous time step, the elastic deformation gradient tensor and the first Piola–Kirchhoff stress can be calculated and substituted in the phase field equation to find the order parameter for the current time step. The weighted residuals method is used to derive the finite element form of the phase field equation and the explicit method is used for its time discretization. The finite element procedure is well verified by comparing the obtained results with those from other simulations. Examples of thermal- and stress-induced phase transformations are presented, including austenite–martensite interface propagation and growth of preexisting martensitic regions. The developed algorithm and code can be advanced to solve kinetic problems coupled with mechanics at large strains for phase transformations and similar phenomena at the nanoscale.

Use of smart intelligent sensor & actuator mechanical materials (PVDF / PZT) in developing MIMO mathematical model of the smart structure and its use to control the active vibrations using discrete sliding mode control theory with output samples

A multi-input, multi-output based modelling of intelligent flexible mechanical aluminium structures using smart materials with Euler-Bernoulli beam theory & state space techniques is being presented with multi-sensor data fusion concepts. An aluminium cantilever beam of rectangular cross section with known specifications is considered for the modelling purposes. To start with, from the fundamental principles of E-B theory, a Lagrange equation of motion is obtained both for the regular beam element as well as for the PZT & PVDF sensor actuator pair. Following which, a dynamic equation of motion of the smart intelligent structure developed using the smart materials consisting of PZT& PVDF is obtained. The dynamic equation is transformed into a state space model with 2 i/o’s using the concepts of state space theory in control engineering. The model is developed considering the first 2 modes of vibration, which are the most dominant modes in the vibration theory. The developed state space model is used for active vibration control. An external force of 1 N is applied at the end of the smart flexible multivariable multimodel cantilever beam after which the beam is subjected to vibrations. A Discrete Sliding Mode Control (DSMC) based controller on the o/p samples is designed and put in loop with the plant, after which the plant vibrations decays to zero in no time. The mathematical model of the DSMC controller is used to control the vibrations of the plant using the concept of destructive interference. The mathematical model of the DSMC controller can be used to control the vibrations of the plant using the concept of destructive interference.