Rigid Body Dynamics Equations Research Articles

Going with the Flow

Given a sequence of poses of a body we study the motion resulting when the body is immersed in a (possibly) moving, incompressible medium. With the poses given, say, by an animator, the governing second-order ordinary differential equations are those of a rigid body with time-dependent inertia acted upon by various forces. Some of these forces, like lift and drag, depend on the motion of the body in the surrounding medium. Additionally, the inertia must encode the effect of the medium through its added mass. We derive the corresponding dynamics equations which generalize the standard rigid body dynamics equations. All forces are based on local computations using only physical parameters such as mass density. Notably, we approximate the effect of the medium on the body through local computations avoiding any global simulation of the medium. Consequently, the system of equations we must integrate in time is only 6 dimensional (rotation and translation). Our proposed algorithm displays linear complexity and captures intricate natural phenomena that depend on body-fluid interactions.

Experimental and numerical investigation on the unsteady interaction in longitudinal stage separation for parallel-staged two-body configuration

Hypersonic stage separation is a significant process, probably involving complex aerodynamic interaction, which determines the survival of two-stage-to-orbit (TSTO) vehicles. The longitudinal stage separation (LSS) scheme is proposed to improve the safety of the parallel-staged TSTO vehicle, where a small interstage gap may result in weak interference. Therefore, an experimental and numerical study of LSS for the parallel-staged TSTO vehicle at Mach 7 with different angles of attack (AoA) is carried out. The dynamic interaction, including variations in the shock structure, wall pressure distribution, and unsteady aerodynamics, is investigated by testing and numerical simulation. The LSS experiments for the TSTO vehicle were performed using a high-speed pneumatic ejection launch system in the JF-12 shock tunnel, and the method was developed using high-speed visualization and image processing techniques to capture the separating trajectory. The numerical simulations were carried out using the overset grid method and solving the Navier–Stokes equations coupled with the rigid body dynamics equations to obtain the laminar flows over the TSTO vehicle during LSS. The qualitative and quantitative comparison of the test and numerical results showed good agreement in terms of aerodynamic performance, flowfield pattern, wall pressure, and separation trajectory. They show that the small interstage gap of the LSS leads to weak type I and VI shock–shock interactions, with short-duration weak shock reflection at a higher AoA. Furthermore, no shock reflection or interstage gap is observed at lower AoA. Moreover, no stage recontact is observed, and the safety and feasibility of LSS for parallel-staged TSTO vehicles are demonstrated.

VTOL UAV digital twin for take-off, hovering and landing in different wind conditions

With UAVs becoming increasingly popular in the industry, vertical take-off and landing (VTOL) convertiplanes are emerging as a compromise between the advantages of planes and multicopters. Due to their large wing surface area, VTOL convertiplanes are subject to a strong wind dependence on critical phases such as take-off, landing, and hovering. Developing a new and improved unmanned aerial vehicle (UAV) is often expensive and associated with failures and accidents. This paper proposes the dynamic characterization of a commercial VTOL convertiplane UAV in copter mode and provides a novel method to estimate the aerodynamic forces and moments for any possible wind speed and direction. Starting from Euler’s equations of rigid body dynamics, we have derived the mathematical formulation to precisely consider aerodynamic forces and moments caused by any wind speed and direction. This unique approach will allow for VTOL convertiplane UAVs to be trained and tested digitally in take-off, hovering, and landing maneuvers without the cost and hassle of physical testing, and the dependence on existing wind conditions. A digital twin of a VTOL convertiplane UAV in copter mode has been modeled and tested in the Gazebo robotics simulator. Take-off, hovering and landing maneuvers have been compared with and without the wind physics model. Finally, the simulator has been tested against real flight conditions (reproducing the mean wind speed and direction only), showing a natural and realistic behavior.

Influence of Nonlinear Dynamics Behavior of the Roller Follower on the Contact Stress of Polydyne Cam Profile

The effect of the cam profile on the nonlinear dynamics phenomenon of the follower is studied at three involutes’ profiles for the cam. The value of the Lyapunov exponent parameter is calculated at different internal distances of the follower guide from inside and at different cam speeds. The effect of the Lyapunov exponent value on the contact stress is studied based on the clearance between the follower and its guides. The contact between the cam and the square grooving key and between the cam and the follower has been taken into consideration at different locations. The finite element method is used to calculate the contact stress numerically using the SolidWorks program. The nonlinear response of the follower is calculated analytically using the Newton–Euler equations of rigid body dynamics of translation and rotation motions while the follower position is tracked experimentally using a high-speed 3-D camera device. The contact stress is checked and verified using photo-elastic apparatus. The data of the contact stress against time are used in the Wolf algorithm code to extract the value of the largest Lyapunov exponent. A phase-plane diagram is drawn at different cam speeds and different internal distances of the follower guide from the inside. The periodic and non-periodic motions of the contact stress are examined at different contact locations for the follower with the cam profile. The higher the value of contact stress, the higher the value of the largest Lyapunov exponent parameter and the higher the value of bending deflection.

Numerical Analysis of the Take-Off Performance of a Seaplane in Calm Water

Nowadays, with the escalating tensions in maritime dispute and the development of marine economy, there has been renewed interest in seaplanes for their special capacity of taking off and landing on water. Prediction of take-off performance, involving aerodynamic analysis and hydrodynamic analysis, is a main challenge in seaplane design, while the prediction methods have been little improved since the 1960s. This paper aims to investigate the attitude and resistance characteristics of a seaplane at different speeds during the take-off by numerically modeling the air-water flow field using RANS equations with VOF method. The trim and heave motion of seaplane in response to aerodynamic forces, hydrodynamic forces, hydrostatic forces, and propeller thrust was realized by solving rigid body dynamics equations and adopting dynamic overset mesh technique. The variations in heave, trim angle, and resistance characteristics during the takeoff were investigated, and their inherent relationships with the aerodynamic, hydrodynamic, and hydrostatic performance were revealed. Particular investigation on the hydrodynamic resistance indicates that the stagnation line located at the convex bow would contribute a considerable increase of pressure resistance at the first hump, and the trim angel of a seaplane should be operated in an optimum trim range, typical between 4–6 deg, to minimize the hydrodynamic resistance at the second hump. Additionally, the dynamic motion convergence study proves that the utilization of damping terms was an effective way to accelerate the convergence of the dynamic motion ending with a quasi-static state.

Longitudinal aerodynamic modeling and verification for air-launch-to-orbit system during stage separation

The stage separation is one of key issues for air-launch-to-orbit system. In this process, the elastic deformation of carrier causes periodic pulsation in interference aerodynamics and may lead to modal coupling and dynamics instability of spacecraft. The dynamic responses and stability of the two vehicles should be considered carefully. The CFD-based dynamic simulation method has high accuracy but requires large computation cost. In this study, a new reduced-order modeling method is developed, which is for unsteady aerodynamics of the air-launch-to-orbit system with considering the elastic deformations of carrier. Based on the physical mechanism of interference aerodynamics factors, the model input parameters are decoupled and the modeling difficulty is reduced. By coupling the unsteady aerodynamic model, structural dynamics equations and rigid body dynamic equations, the longitudinal dynamic responses of air-launch-to-orbit system during stage separation are simulated. Through comparison, those results have good agreement with traditional CFD-based method, and the simulation efficiency of new method is improved greatly. The new modeling method is applicable to the air-launch-to-orbit system and can provide guidance for similar engineering issues.

GENERALIZATION OF KIRCHHOFF KINETIC ANALOGY TO THIN ELASTIC SHELLS

The Kirchhoff kinetic analogy is generalized from thin elastic rods to thin elastic shells. The generalization makes thin shell deformations physically correspond and mathematically equivalent to rigid body motions. Hence theories and methods of rigid body dynamics can be applied to investigate deformations of thin elastic shell, and also provide a novel discretization for continuous thin elastic shells. An orthogonal spatial axis system is established along the coordinate lines of the middle surface under the straight normal assumption. The moving of the axis system along the coordinate lines in unit velocity forms its angular velocity, which is the curvature-twist vector with two independent variables. The curvature-twist vector along two coordinate lines expresses the deformation and the configuration of a thin elastic shell. It is demonstrated that curvature-twist vectors are compatible, and curvature-twist vectors and tangential vectors of middle surface are compatible. Nonholonomic constraints and differential equations of middle surfaces are established in the Euler angles and the Lam$\acute{e}$ coefficient form. The strain, the stress and the internal forces are formulated in the curvature-twist vectors and the Lam$\acute{e}$ coefficients. The equilibrium partial differential equations are presented with distributed internal forces intensity of thin elastic shells. The forms of the equations are similar to the Euler equations of rigid body dynamics and Kirchhoff equations of thin elastic rods. The fact means that the Kirchhoff kinetic analogy of thin elastic rods is generalized to thin elastic shells. The analogy relations between thin elastic shells and dynamics of rigid body or thin elastic rods are concluded. Finally, an example is given to show the application of this method. The proposed analogy leads to novel views and approaches to model and to analyze deformation of thin elastic shells. It is possible to generalize further the analogy for dynamics of thin elastic shells.

A multi-time scale solution strategy for an elastomeric aircraft

This paper develops a multi-time scale kinetics solution strategy for the flight dynamic simulation of elastomeric aircraft. The combination of computational fluid dynamics (CFD) solver and reduced-order model is utilized to calculate the aerodynamics of elastic aircraft. CFD is applied to the aerodynamic solution of rigid motion, and reduced-order model is to the elastic deformation. The governing equations of rigid-body dynamics and structure dynamics are decoupled in time-scale, and their time step sizes can be set differently. This method overcomes the problem of large difference in characteristic times between rigid-body dynamics and structure dynamics, and improves the calculation efficiency of CFD based multi-disciplinary coupling solution. For current work, the multi-time scale kinetics solution method is applied to the simulation of store separation problem considering elastic deformation of carrier aircraft. The autoregressive with exogenous input model is employed to establish the generalized aerodynamic model of carrier. By comparison, the time responses of store obtained by the multi-time scale kinetics solution method are in good agreement with those by traditional CFD based multi-disciplinary coupling simulation, which validates high precision of the new method. Furthermore, the calculation amount of the new coupled calculation method is reduced by about an order of magnitude compared with that of the traditional method. The multi-time scale kinetics solution method can provide an effective solution strategy for similar engineering problems.

SPH method with applications of oscillating wave surge converter

Hydrodynamics characteristics of land hinged Oscillating Wave Surge Converter (OWSC) are investigated by smoothed particle hydrodynamics (SPH). δ-SPH model is used to remove the spurious high-frequency oscillations. Boundary force model is used to avoid the penetration of fluid particles across the wall. The classic equations of rigid body dynamics are used to control the motions of OWSC. Standing waves, regular waves, the interaction between wave and rigid plate and bottom hinged OWSC are simulated to validate the numerical methods. Comparing SPH results with the analytical solutions or reference results, relative good agreements are observed. Then, SPH method is employed to simulate interactions between waves and land hinged OWSC. The results show that the active power of land hinged OWSC strongly depends on both the PTO damping coefficients and the wave periods.

Measured and simulated analysis of a model rocket

A comparison between two types of sensors and two types of simulation software are investigated here for a student built rocket. Many students use an open source software package called OpenRocket which uses empirical aerodynamics based on the shape of the rocket. This software is compared to the standard set of rigid body dynamic equations using coefficients for the aerodynamics based on windtunnel and computational fluid dynamics tests. During experimentation two sensors are used and price and resolution is compared. The first sensor is a turn-key sensor called the TeleMega which has many features such as telemetry and on board data logging. In an effort to reduce costs, the Arduino Mega platform has been augmented with a custom made shield capable of measuring Global Positioning System (GPS), angular velocity, and attitude information with on board data logging as well. Although this sensor has limited functionality, the cost is substantially reduced. It is shown that all sensors and simulation software have their strengths and weaknesses with appropriate usage for each.

Finite-Element-Based Modeling for Flight Dynamics and Aeroelasticity of Flexible Aircraft

Fully coupled equations of rigid-body and structural dynamics of flexible aircraft are deduced from the weak formulation of Cauchy’s equation for an unconstrained elastic continuum. An appropriate choice of the body coordinate system enables to describe inertial coupling using a reduced set of coefficients, which are estimated for complex configurations via finite element method discretization. A fully coupled linearized formulation around aeroelastic trim conditions is presented and implemented using data from a generic finite element method solver. Small-disturbance, fully unsteady aerodynamics is modeled in the frequency domain via doublet lattice method and recast in time-domain state-space form by means of a rational function approximation. A state-space representation of the linearized system is ultimately obtained, which simultaneously includes rigid-body, elastic, and aerodynamic state variables. Numerical results for two representative configurations are presented and discussed to point out the influence of inertial and aerodynamic coupling on integrated flight dynamic/aeroelastic stability and response.

Kirchhoff’s equations for the rigid body motion revisited

The purpose of the note is to relate Kirchhoff’s equations with Euler’s equations of rigid body dynamics and to establish the conservation laws arising from given external forces. This is done in vector form, for an inhomogeneous rigid body, by considering the kinetic energy and showing how the terms occurring in the Euler’s equations may be given the form of partial derivatives of the kinetic energy. Next it is shown how the vector form, via derivatives of the kinetic energy, provides the conservation laws and the corresponding first integrals along with their immediate mechanical meaning. An outline is given of different approaches in the literature to the derivation of Kirchhoff’s equations.

On the uncertainty quantification of the unsteady aerodynamics of 2D free falling plates

The aim of this paper is to conduct a statistical analysis of the effects of the fillet radii on the dynamics of the falling plate using the nonintrusive spectral projection (NISP) method. The free fall of two-dimensional cards immersed in a fluid was studied using a deterministic and stochastic numerical approach. The motion is characterized by the fluid-body interaction described by coupling the Navier–Stokes and rigid body dynamic equations. The model’s predictions have been validated using both experimental and numerical data available in the literature. In the stochastic simulations, the fillet radius of the plate was considered a random variable characterized by a uniform probability density function introducing, in this way, some uncertainties in the plate’s trajectory. To take into account the uncertainties, we employed the NISP method based on polynomial chaos expansion. The analysis was focused on finding the ensemble mean trajectory and error bar for a confidence interval of 95 % for both tumbling and fluttering regimes.

Computational Aeromechanics of a Manuevering Unmanned Aerial Vehicle with Variable-Incidence Wings

The aerodynamics and flight mechanics of the dynamic maneuver of a low-speed propeller-powered puller-type unmanned aerial vehicle are modeled by solving the unsteady, incompressible Navier–Stokes equations and the six-degree-of-freedom rigid-body flight dynamics equations together. The flight maneuver of this unmanned aerial vehicle is accomplished by swiveling the outboard wings relative to the inboard wing fixed to the fuselage to change the angle of attack toward stall angle to slow down the unmanned aerial vehicle for symmetric as well as asymmetric flight conditions. To set the stage for a computational aeromechanics modeling, the performance characteristics of a propeller are assessed numerically first. Time-accurate unsteady, unmanned aerial vehicle aerodynamic characteristics are estimated on the basis of high-fidelity computational aerodynamics, and the flight trajectory of the unmanned aerial vehicle is computed as part of the overall computed solution, considering the variation of aerodynamic characteristics during the flight maneuver by coupling the aerodynamics and flight dynamics.

Numerical Integration of Coupled Equations for High-Speed Electromechanical Devices

In this paper, a new approach for the numerical solution of coupled electromechanical problems is exploited with the aim of simulate electromagnetic devices with high-speed moving parts. The considered problem formulation rests on the low-frequency integral formulation of the Maxwells equations and the Newton–Euler rigid-body dynamic equations. The numerical integration of the coupled problem is discussed and developed with a two-step method. The method has been validated by comparison with experimental data (when available) and with results obtained by other numerical formulations. An example of application to a 2 degrees of freedom device based on permanent magnets is finally reported.

Simulation Analysis of Designing a New Technique KER Warhead

The major technique of an air-defense warhead is to improve the damage after effect. Therefore, a new Kinetic Energy Rod (KER) warhead named profiled rod warhead is proposed in this paper. The detonation process is simulated by ANSYS/LS-DYNA, and the deployment velocity and initial flight attitude of rod are achieved. Rigid body dynamics equations of rod, which accounts for the influence of air resistance, are set up to predict the flight trajectory of long-distance. The results show that profiled rods are conducted to determine a higher penetrability compared with traditional cylindrical rods, and the profiled rods may provide a better penetration angle that still maintains a significant penetrability against projectiles when the rods move off long-distance range.

CFD Transonic Trajectory Predictions of Three-Store Ripple Release Using Chimera Method

The complicated unsteady flows with moving boundary were simulated numerically by coupling solving unsteady compressible Navier-Stokes equations and 6DOF rigid-body dynamics equations. The Chimera grid technology was used to handle the relative motion. The three-store ripple release of the wing-store configuration was simulated using this method. The computational results are in good agreement with data from other literature, showing that the method used has a strong applicability to complex multi-body separation problem.

Modeling of electromechanical devices by GPU‐accelerated integral formulation

SUMMARYThis paper discusses the practical implementation on graphics processing unit (GPU) of a computer code for the analysis of electromechanical devices. The software is based on a low frequency integral formulation of the Maxwell equations coupled with the rigid body dynamic equations. The formulation here considered is based on the development of an equivalent network whose parameters are functions of the relative positions and of the velocities of the subparts of the device. Positions and velocities on their turn depend on the force between them that are functions of the electromagnetic quantities. The choice of the numerical methods to manage such a complex coupled problem is an open issue in the electromagnetic community. The use of multicore CPUs can reduce the computation times, but a true breakthrough is achieved by running these codes on GPUs. The practical implementation of an existing code is used as a case study for discussing a number of issues that may arise in the implementation of other GPU‐accelerated electromagnetic codes. The implementation here reported has been designed for a multi‐GPUs environment where the efficient cooperation between GPUs is a further aspect to be considered. The overall performance of the accelerated code has been evaluated by considering the dynamic analysis of a passive magnetic bearing. Copyright © 2012 John Wiley & Sons, Ltd.

THE CORIOLIS EFFECT – WHAT’S GOING ON?

The subject of dynamics is fundamental to engineering programs in general, and can be challenging to teach. In particular, the topic of Coriolis acceleration and the “Coriolis effect” may present the greatest challenge of all. Although mathematical formulations and physical explanations are available in many textbooks and technical articles, undergraduate students still have trouble making meaningful and long-lasting connections between the mathematical expressions and the physical manifestations of the phenomenon. In our opinion, a significant part of the problem can be attributed to the traditional formulation of kinematic equations using rotating frames of reference. Normally, the Coriolis acceleration is defined, derived, and explained using at least one noninertial frame. At the University of Saskatchewan, our approach is to express the equations of rigid body dynamics solely with respect to a single inertial frame of reference, and to thereby solve all the problems that would otherwise be solved using rotating frames. We also use this approach for deriving and explaining the Coriolis effect. Although our approach is, in a sense, equivalent to the traditional method, we have found that students are better able to understand and apply the principles of dynamics when working within a single inertial frame. In this paper, our pedagogical approach will be presented and then applied to the question of “what’s going on” with the Coriolis effect

Analysis of Rifle Recoil Based on Different Methods of Support

In order to analysis and study the factors influencing of measuring the rifle recoil, consider the different methods of support, based on rigid body dynamics equations and collision theory, three kinds of rifle dynamics model, which included the holder, standing and the prone position had been established. The rifle recoil curve of the different conditions had been calculated, analysis of the effects of different methods of support on the measurement of the rifle recoil.From the theoretically clarify the recoil is not a inherent properties of rifle.