Chain Drive Simulation Using Spatial Multibody Dynamics

Abstract

This paper presents an efficient approach for modeling chain derives using multibody dynamics formulation based on the spatial algebra. The recursive nonlinear dynamic equations of motion are formulated using spatial Cartesian coordinates and joint variables to form an augmented set of differential-algebraic equations. The spatial algebra is used to express the kinematic and dynamic equations leading to consistent and compact set of equations. The connectivity graph is used to derive the system connectivity matrix based on the system topological relations. The connectivity matrix is used to eliminate the Cartesian quantities and to project the forces and inertia into the joint subspace. This approach will result in a minimum set of equation and can avoid iteratively solving the system of differential and algebraic equations to satisfy the constraint equations. In order to accurately capture the full dynamics of the chain links, each link in the chain is modeled as rigid body with full 6 degrees of freedom. To avoid singularities in closed loop configurations, the chain drive is considered a kinematically decoupled subsystem and the interaction between the links and other system components is modeled using force elements. The out-of-plane misalignment between the sprockets can be easily modeled using a compliant force element to model the joints between each two adjacent links. The nonlinear three dimensional contact forces between the chain links and the sprockets are modeled using elastic spring-damper element and accounts for the sliding friction. The proposed approach can be used to model complex drive chain, bicycle chain as well as conveyance systems. Results show that realistic behavior of the chain as well as out-of-plane vibration can be easily captured using the presented approach. The proposed approach for chain drive subsystem could be easily appended to any other multibody simulation system.

Mixed convection flow about a solid sphere with a constant surface heat flux embedded in a porous medium filled with a nanofluid

“…Equations (15) and (16) subject to the boundary conditions (17) have been solved numerically using an efficient implicit finite-difference scheme known as the Keller-box method along with the Newton's linearization technique as described by Cebeci and Bradshaw [22]. The numerical solutions obtained are from the lower stagnation point of the sphere ( 0 x | ) and proceed round the sphere up to the separation point.…”

“…A good collection of current and future application of nanofluids can be found in the book by Das et al [15] and the work by Kaufui and Omar [16].…”

A walking roller chain

“…They were 40 times faster than implicit methods and 5 times faster than standard explicit methods, controlled for numerical error. 5 We used the eight-order RungeÀKuttaÀFehlberg 7(8) method that has a seventh-order error estimate, the Butcher tableau of which can be found in [14]. Our simulation also benefited from the odeint C++ library, available within the Boost C++ libraries package found in reference [15].…”

“…Previous studies of roller chains have predominantly investigated the mechanics of a chain within a chain drive system, its power transmission efficiency, longevity, durability and many other topics of practical significance [1][2][3][4]. Recently, successful attempts at modelling and simulating roller chains using multibody dynamics 2 have been developed [5][6][7], although still limited to chain drive motion. Currently there is no analysis of the motion of a free roller chain, rapidly spinning or otherwise, available in the literature.…”

Decoupling Kinematic Loops for Real-Time Multibody Dynamic Simulations

“…In most earth-moving equipment, the translational motion is controlled by a hydraulic actuator, as shown in Figure 1. The kinematic topology of the FBL and FBSL lead to a Corresponding author: momar@taibahu.edu.sa, momar23@gmail.com kinematic closed-loops [1,2,3]. Real-time simulation for such systems poses a challenge for software developers and for analysts [4,5,6,7].…”