A General and Efficient Method for Dynamic Analysis of Mechanical Systems Using Velocity Transformations

This paper presents a matrix formulation for the dynamic analysis of spatial mechanisms with common types of kinematic joints. The formulation is derived in two steps. Initially an equivalent constrained system of particles that replaces the rigid bodies is constructed and used to define the configuration of the mechanical system. This results in a simple and straightforward procedure for generating the equations of motion in terms of the rectangular Cartesian coordinates of the particles without introducing any rotational coordinates. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. An example is used to demonstrate the generality and efficiency of the proposed method.

This research extends the generalized recursive formulas for the rigid body dynamics to the flexible body dynamics using the backward difference formula (BDF) and the relative generalized coordinate. When a new force or joint module is added to a general purpose program in the relative coordinate formulations, the modules for the rigid bodies are not reusable for the flexible bodies. Since the flexible body dynamics handles more degrees of freedom than the rigid body dynamics does, implementation of the flexible dynamics module is generally complicated and prone to coding mistakes. In order to overcome the implementation difficulties, a virtual rigid body is introduced at every joint and force reference frames. A virtual flexible body joint is introduced between two body reference frames of the virtual and original bodies. Since the multibody system dynamics are formulated by highly nonlinear algebraic and differential equations and there are many different types of joints, a tremendous amount of computer implementation is required to develop a general purpose dynamic analysis program using the relative coordinate formulation. The implementation burden is relieved by the generalized recursive formulas. The notationally compact velocity transformation method is used to derive the equations of motion in the joint space. The terms in the equations of motion which are related to the transformation matrix are classified into several categories each of which recursive formula is developed. Whenever one category of the terms is encountered, the corresponding recursive formula is invoked. Since computation time in a relative coordinate formulation is approximately proportional to the number of the relative coordinates, computational overhead due to the additional virtual bodies and joints is minor. Meanwhile, implementation convenience is dramatically improved.

1. There are two very different ways of executing linear regression analysis. One is Model I, when the x-values are fixed by the experimenter. The other is Model II, in which the x-values are free to vary and are subject to error. 2. I have received numerous complaints from biomedical scientists that they have great difficulty in executing Model II linear regression analysis. This may explain the results of a Google Scholar search, which showed that the authors of articles in journals of physiology, pharmacology and biochemistry rarely use Model II regression analysis. 3. I repeat my previous arguments in favour of using least products linear regression analysis for Model II regressions. I review three methods for executing ordinary least products (OLP) and weighted least products (WLP) regression analysis: (i) scientific calculator and/or computer spreadsheet; (ii) specific purpose computer programs; and (iii) general purpose computer programs. 4. Using a scientific calculator and/or computer spreadsheet, it is easy to obtain correct values for OLP slope and intercept, but the corresponding 95% confidence intervals (CI) are inaccurate. 5. Using specific purpose computer programs, the freeware computer program smatr gives the correct OLP regression coefficients and obtains 95% CI by bootstrapping. In addition, smatr can be used to compare the slopes of OLP lines. 6. When using general purpose computer programs, I recommend the commercial programs systat and Statistica for those who regularly undertake linear regression analysis and I give step-by-step instructions in the Supplementary Information as to how to use loss functions.

<div class="htmlview paragraph">In this paper methods for the nonlinear dynamic analysis of multibody vehicle systems are presented. A total Lagrangian finite element formulation for deformable bodies of the vehicle systems that undergo large translational and rotational displacements is developed. A set of nonlinear dynamic differential equations are developed in a closed form for each finite element in terms of the reference as well as the nodal elstic coordinates and their time derivatives. These equations are expressed in terms of a set of element invariants that depend on the assumed displacement field. The invariants of the deformable body that undergoes large reference translational and rotational displacements can be obtained by assembling the invariants of its elements using a standard finite element procedure. The nonlinear dynamic equations of the deformable body can then be obtained by assembling the nonlinear element equations. Both the consistent and lumped mass finite element formulations are discussed. The dynamic formulations presented in this paper are exemplified using multibody vehicle systems that consist of interconnected rigid and deformable bodies. Numerical results are obtained using the general purpose computer program <b>DAMS</b> (Dynamic Analysis of Multibody Systems).</div>

Different finite‐element‐based strategies used to represent the components' flexibility in multibody systems lead to various sets of co‐ordinates. For systems in which the bodies only experience small elastic deformations it is common to use mode component synthesis to reduce the number of generalized elastic co‐ordinates and, consequently, the equations of motion are written in terms of modal co‐ordinates. However, when the system components experience non‐linear deformations the use of reduction methods is not possible, in general, and the finite element nodal co‐ordinates are the generalized co‐ordinates used. Furthermore, depending on the type of finite elements used to represent each flexible body, the nodal co‐ordinates may include all node rotations and translations or only some of each. Regardless of the type of generalized co‐ordinates adopted it is required that kinematic joints are defined. The complete set of joints available in a general‐purpose multibody code must include, for each particular type of joint, restrictions involving only rigid bodies, or only flexible bodies, or flexible and rigid bodies. Therefore, the effort put into the development and implementation of any joint is at least three times as much as the initial work done in the implementation of joints with rigid bodies only. The concept of virtual bodies provides a general framework to develop general kinematic joints for flexible multibody systems with minimal effort, regardless of the flexible co‐ordinates used. Initially, only a rigid constraint between the flexible and a massless rigid body is developed. Then, any kinematic joint that involves a flexible body is set with the massless rigid body instead, using the regular joint library of the multibody code. The major drawback is that for each kinematic joint involving a flexible body it is required to use six more co‐ordinates per virtual body and six more kinematic constraints. It is shown in this work that for small elastic deformations, for which the mode component synthesis is applied, the use of sparse matrix solvers can compensate for the computational overhead of involving more co‐ordinates and kinematic constraints in the system, due to the use of virtual bodies. For non‐linear deformations, where the generalized co‐ordinates are the global positions of the finite‐element nodes, the use of the virtual body concept does not require an increase in the number of system co‐ordinates or kinematic constraints. By introducing the rigid joint between the flexible body nodal co‐ordinates and the virtual body, with the use of Lagrange multipliers, and then solving the equations explicitly for these multipliers the resulting equations of motion for the subsystem have the same degrees of freedom as the original flexible body alone. The difference is that degrees of freedom associated to the virtual body are used as co‐ordinates of the subsystem instead of the nodal co‐ordinates of the nodes of the flexible body attached to the virtual body. Copyright ©2003 John Wiley & Sons, Ltd.

In the present study, the equations of motion for planar mechanical systems that consist of interconnected rigid bodies with common types of kinematic joint are derived on the basis of particle dynamics and a recursive approach. The system of rigid bodies is replaced by a dynamically equivalent constrained system of particles. Then, for the resulting equivalent system of particles, the concepts of linear and angular momenta are used to generate the equations of motion without either introducing any rotational coordinates or distributing the external forces and force couples over the particles. For an open-loop system, the equations of motion are generated recursively along the open chains. For a closed-loop system, the system is transformed to open chains by cutting suitable kinematic joints with the addition of cut-joint kinematic constraints. Examples of multibranch closed-loop systems are chosen to demonstrate the generality and simplicity of the proposed method.

Analytical dynamics of mechanisms—a computer oriented overview

In this paper, issues related to the dynamic modelling of constrained deformable bodies that undergo large rigid body displacements are discussed. Particular attention is focused on finite element formulations. It is shown that the use of nodal co‐ordinates and shape functions to describe the finite rotation of some of the commonly used finite elements leads to a linearization of the kinematics and dynamic relationships. The structure of the non‐linear dynamic equations that govern the motion of deformable bodies that undergo large displacements is examined. Comments on the finite element formulation of the invariants of motion, the definition of the generalized forces and moments in flexible body dynamics and the computational strategy used for the automatic generation of the equations of motion are made. The computer formulation of the joint constraints between deformable bodies as well as the numerical algorithms currently used in many of the general purpose computer programs that are based on the augmented formulation are discussed. A decoupled joint‐elastic acceleration recursive formulation is also presented. This formulation leads to a small system of acceleration equations whose dimensions are independent of the number of the elastic degrees of freedom of the system. In this paper, the coupling between the displacements of the deformable bodies is classified as kinematic, inertia and elastic. In view of this classification, comments on the validity of using the updated finite element Lagrangian formulation and the 4 × 4 transformation matrix in the dynamic analysis of flexible multibody systems are made. The coupling between the finite rotation and the wave motion in constrained deformable bodies is also discussed.

Some invariant relations in the problem of the motion of a body on a smooth horizontal plane

: Our work for the AFOSR has three primary components: Development of new quantum theory for the accurate description of molecular structure and spectra. Implementation of this new theory into general purpose computer programs (ACES II and pralel ACES III) to make it possible for many investigators, besides ourselves, to readily apply these new methods to problems of their interest. Application of these new methods to challenging molecules chosen to test the new methods, and to other systems of interest to AFOSR. These include high energy, density molecules, where the new methods allow an assessment of their energy content, stability, possible synthetic paths, activation barriers, and provide spectroscopic fingerprints for identification. We also make applications of interesting clusters, atmospheric systems, potential interstellar molecules, and to gas phase molecular reactions of the sort that can occur in flames.

The nonlinear formulation using the principle of virtual work-energy for free vibration of a large-sag extensible catenary riser in two dimensions is presented in this paper. A support at one end is hinged and the other is a free-sliding roller in the horizontal direction. The catenary riser has a large-sag configuration in the static equilibrium state and is assumed to displace with large amplitude to the motion state. The total virtual work of the catenary riser system involves the virtual strain energy due to bending, the virtual strain energy due to axial deformation, the virtual work done by the effective weight, and the inertia forces. The nonlinear equations of motion for two-dimensional free vibration in the Cartesian coordinate system is developed based on the difference between the Euler's equations in the static state and the displaced state. The linear and nonlinear stiffness matrices of the catenary riser are obtained and the eigenvalue problem is solved using the Galerkin finite element procedure. The natural frequencies and mode shapes are obtained. The results are validated with regard to the reference research addressing the accuracy and efficiency of the proposed nonlinear formulation. The numerical results for free vibration and the effect of the nonlinear behavior for catenary riser are presented.

Analytical Mechanics. Foundations of Engineering Mechanics Series

This paper proposes new concepts of configuration and attitude control of underwater towed vehicle to perform operation in both vertical and horizontal axis with stable attitude.The principal configuration composed of fuselage, main wing, horizontal tail, vertical canard, upper and lower vertical tails designed taking into account stability and manoeuvrability in vertical and horizontal planes. The manoeuvrability was estimated by use of lamped mass method for equilibrium equations of motion of towed cable and vehicle. The dynamic stability was analysed by use of eigenvalue analysis based on linear equations of motion.CCV (Control Configured Vehicle) technology was adopted to perform the attitude control. Three types of decoupled mode were considered, i) pitch angle and vertical velocity, ii) pitch angular velocity and vertical velocity, iii) roll angle, yaw angle and side slip velocity. Fuzzy algorithm was applied to CCV attitude control.The effect of motion of towing mother ship in waves was simulated numerically based on non-linear equations of motion of cable and vehicle to examine the effectiveness of attitude control in vertical axis. Not only drifts of velocity components and pitch angle, but also oscillation of pitch angle are remarkably suppressed.The effect of tidal current was simulated numerically to examine the attitude control in horizontal axis. Oscillations of side slip velocity, roll angle and yaw angle are almost eliminated.

Recent Advances in Mobile Robotics 128 gradient field Unfortunately, path planning approach based on the function family of potential field cannot obtain the optimal objective function which, in turn, cannot guarantee the desired optimal trajectories. Additionally, some of them, for example, potential panel method and harmonic potential function, can plan a path for an autonomous vehicle, but the computing burdens is huge and real time performance hardly satisfies the practical requirement. Inspired by biological intelligence, many approaches, such as ant colony optimization (ACO) They mostly rely on the stochastic searching, known as non-deterministic algorithm. These methods will eventually find an optimal solution, but no estimate on the time of convergence can be given. Thus, it may take long even infinite time to find the best solution. Furthermore, all of them have a great number of parameters to tune and that is never an easy job particularly when users are short of prior knowledge. Some comparisons Another kind of method is mathematic programming. Based on the relative velocity coordinates (RVCs), linear programming (LP) and mixed integer linear programming (MILP) can be employed for path planning problem. For example, Wang et al. converted the 2D path planning of an unmanned underwater vehicle to constrained optimization or semiinfinite constrained optimization problem Zu et al. discussed the path planning in 2D and an LP method was proposed for the problem of dynamic target pursuit and obstacle avoidance This method tried to plan the variables of the linear acceleration and the angular acceleration of a ground vehicle. Schouwenaars et al. proposed a MILP formulation with receding horizon strategy where a minimum velocity and a limited turn rate of aircraft are constrained However, these results still focused on the two dimensions. In this chapter, we consider the same problem of TPOA but in three dimensions. To our problem, the uncertain environment has one target or some of them, denoted by G, and many obstacles O that are all velocity-changeable with their moving actions. The aim of the vehicle A is to find an optimal path for pursuing the target while, at the same time, avoiding collision threaten from obstacles. The position and velocity of movers, including the target and obstacles, are assumed known or estimated at current time. To be more specific, it is assumed that the noise contained in the data can be eliminated with certain filters. However, this paper has no intend to discuss the filtering algorithm for achieving the feasible data of on-board sensors. The trajectory from a start to a destination location typically needs to be computed gradually over time in the fashion of receding horizon The target and obstacles, probably static or dynamic, are modeled by spheres having a certain radius for their impact area, and the vehicle is modeled by a mass point. In order to optimize the vehicle's acceleration online, we construct a linear programming model in Cartesian orthogonal coordinates based on relative velocity space This chapter is organized as follows. First, the relative velocity coordinates are introduced in both two dimensions and three dimensions and then the TPOA principles are introduced, including obstacle-avoidance and target-pursuit principles. The formulation of the standard linear programming is mathematically introduced in Section 3.1, and the path planning model is described in detail in Section 3.2. Simulations of path planning in 2D and 3D are www.intechopen.com Path Planning of Mobile Robot in Relative Velocity Coordinates 129 A, G, O Vehicle, target, obstacle Planning period

To produce images of tissue elasticity, the vibro-elastography technique involves applying a steady-state multi-frequency vibration to tissue, estimating displacements from ultrasound echo data, and using the estimated displacements in an inverse elasticity problem with the shear modulus spatial distribution as the unknown. In order to fully solve the inverse problem, all three displacement components are required. However, using ultrasound, the axial component of the displacement is measured much more accurately than the other directions. Therefore, simplifying assumptions must be used in this case. Usually, the equations of motion are transformed into a Helmholtz equation by assuming tissue incompressibility and local homogeneity. The local homogeneity assumption causes significant imaging artifacts in areas of varying elasticity. In this paper, we remove the local homogeneity assumption. In particular we introduce a new finite element based direct inversion technique in which only the coupling terms in the equation of motion are ignored, so it can be used with only one component of the displacement. Both Cartesian and cylindrical coordinate systems are considered. The use of multi-frequency excitation also allows us to obtain multiple measurements and reduce artifacts in areas where the displacement of one frequency is close to zero. The proposed method was tested in simulations and experiments against a conventional approach in which the local homogeneity is used. The results show significant improvements in elasticity imaging with the new method compared to previous methods that assumes local homogeneity. For example in simulations, the contrast to noise ratio (CNR) for the region with spherical inclusion increases from an average value of 1.5–17 after using the proposed method instead of the local inversion with homogeneity assumption, and similarly in the prostate phantom experiment, the CNR improved from an average value of 1.6 to about 20.