Newton-Euler method

Orin, D.E., McGhee, R.B., Vukobratovic, M., and Hartoch, G., Kinematic and kinetic analysis of open-chain Unkages utUizing Newton-Euler methods. Math. Biosci. 43 107-130,1979. 

Different methods are available to derive the dynamical equations of motion for a motor task. In the Newton-Euler method (Pandy and Berme, 1988), free-body diagrams are eonstructed to show the external forces and torques acting on each body segment. The relationships between forces and... 

The Newton-Euler method is well suited to a recursive formulation of the kinematic and dynamic equations of motion (Pandy and Berme, 1988) however, its main disadvantage is that all of the intersegmental forces must be eliminated before the governing equations of motion can be formed. In an dtemative formulation of the dynamical equations of motion, Kane s method (Kane and Levinson, 1985), which is also referred to as Lagrange s form of D Alembert s principle, makes explicit use of the fact that constraint forces do not contribute directly to the governing equations of motion. It has been shown that Kane s formulation of the dynamical equations of motion is computationally more efficient than its counterpart, the Newton-Euler method (Kane and Levinson, 1983). 

A. M. Mathiowetz, A. Jain, N. Karasawa, W. A. Goddard III. Protein simulations using techniques suitable for very large systems the cell multipole method for nonbond interactions and the Newton-Euler inverse mass operator method for internal coordinate dynamics. CN 8921. Proteins 20 221, 1994. 

In the case of rigid molecules, the force equations for the molecular center-of-mass motions are supplemented with an additional set of equations describing the torques on the molecules, the Newton-Euler equations. In this case more complex finite difference methods are used to generate the trajectories [32]. When simulating liquid water, a time step of about 10 s is used, whether a rigid molecule or flexible molecule model is under investigation. 

Because Xk+i appears on both sides of this equation, additional steps are required to solve for x +i before the approximation can be used to calculate it. (This can be done via iteration, such as through the Newton-Raphson method.) Hence, the backward Euler method is also referred to as an implicit method. The trapezoidal algorithm averages the information from the forward and backward Euler algorithms such that the iteration equation to be used is... 

Lee and Dudukovic [18] described an NEQ model for homogeneous RD in tray columns. The Maxwell-Stefan equations are used to describe interphase transport, with the AIChE correlations used for the binary (Maxwell-Stefan) mass-transfer coefficients. Newton s method and homotopy continuation are used to solve the model equations. Close agreement between the predictions of EQ and NEQ models were found only when the tray efficiency could correctly be predicted for the EQ model. In a subsequent paper Lee and Dudukovic [19] presented a dynamic NEQ model of RD in tray columns. The DAE equations were solved by use of an implicit Euler method combined with homotopy continuation. Murphree efficiencies calculated from the results of an NEQ simulation of the production of ethyl acetate were not constant with time. 

The common factor in the implicit Euler, the trapezoidal (Crank-Nicolson), and the Adams-Moulton methods is simply their recursive nature, which are nonlinear algebraic equations with respect to y +j and hence must be solved numerically this is done in practice by using some variant of the Newton-Raphson method or the successive substitution technique (Appendix A). 

Another numerical method to supplement the MC method should be the numerical integration of the equations of motion. This kind of calculation for simple molecular systems is called molecular dynamics (MD) mediod where Newton or Newton-Euler equation of motion is solved numerically and some dynamic properties of flie molecule involved can be obtained. 

The kinematics table, shown in Table 7.1, introduces a method (referred to within this chapter as the table method) for efficiently managing the mathematics involved in analyzing multibody and multiple coordinate system problems, and can be used in either the Lagrangian or the Newton-Euler approach. A schematic diagram, which defines an inertial or body-fixed coordinate system, must accompany every kinematics table. The purpose of the schematic is to identify the point on the body at which the absolute velocity and acceleration is to be determined. The corresponding schematic for Table 7.1 is shown in Fig. 7.7. 

For a multibody system, each body would require a kinematics table and a corresponding schematic. The following examples illustrate the steps required for solving problems by the table method. Note that one example includes the expressions for acceleration to demonstrate the use of the table method with the Newton-Euler approach, while all other examples consider only the velocity. 

Mazur et al. [103, 104] demonstrated the conformational dynamics of biomacromolecules. However, their method scaled exponentially with size and relied on an expensive expression for the inter-atomic potentials in internal coordinates. Subsequently, our group pioneered the development of internal coordinate constrained MD methods, based on ideas initially developed by the robotics community [102, 105-107], reaching 0(n) serial implementations, using the Newton-Euler Inverse Mass Operator or NEIMO [108-110] and Comodyn [111] based on a variant of the Articulated Body Inertia algorithm [112], as well as a parallel implementation of 0(log n) in 0(n) processors using the Modified Constraint Force Algorithm... 

Vaidehi N, Jain A, Goddard WA (1996) Constant temperature constrained molecular dynamics The Newton-Euler inverse mass operator method. J Phys Chem 100(25) 10508-10517... 

NPH-MD = the layer spacing remains a dynamical quantity and the method of Parrinello and Rahman was used to solve the resulting equation of motion, which replaces the usual Newton-Euler equations GCMC-MD = MD simulation with MC attempts to insert or delete a water molecule at specific frequency (e.g., each third) of MD time step. 

The Euler method based on proportional loading was applied in combination with the Newton-... 

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