Comparing double-step and penalty-based semirecursive formulations for hydraulically actuated multibody systems in a monolithic approach

Abstract

The simulation of mechanical systems often requires modeling of systems of other physical nature, such as hydraulics. In such systems, the numerical stiffness introduced by the hydraulics can become a significant aspect to consider in the modeling, as it can negatively effect to the computational efficiency. The hydraulic system can be described by using the lumped fluid theory. In this approach, a pressure can be integrated from a differential equation in which effective bulk modulus is divided by a volume size. This representation can lead to numerical stiffness as a consequence of which time integration of a hydraulically driven system becomes cumbersome. In this regard, the used multibody formulation plays an important role, as there are many different procedures for the constraint enforcement and different sets of coordinates to choose from. This paper introduces the double-step semirecursive approach and compares it with a penalty-based semirecursive approach in case of coupled multibody and hydraulic dynamics within the monolithic framework. To this end, hydraulically actuated four-bar and quick-return mechanisms are analyzed as case studies. The two approaches are compared in terms of the work cycle, energy balance, constraint violation, and numerical efficiency of the mechanisms. It is concluded that the penalty-based semirecursive approach has a number of advantages compared with the double-step semirecursive approach, which is in accordance with the literature.

Efficiency comparison of various friction models of a hydraulic cylinder in the framework of multibody system dynamics

“…In Eq. ( 31 (32). The total friction force, F μ , generated from the bending of the bristles and lubricant film dynamics can be written as…”

“…2.2, with the friction force models of a hydraulic cylinder, introduced in Sect. 3, in a monolithic approach [32,49]. The force vector, Q , in Eq.…”

“…where λ is obtained as shown in Eq. (10), whereas the approximated tangent matrix, ∂f(x) ∂x , can be numerically obtained, as in [32,49], by using the forward differentiation rule as…”

“…(A.4) Equations (A.2)-(A.4) are nonlinear system of equations that can be represented asf (x k+1 ) = 0, where x = z T , p T T for the Bengisu-Akay and Brown-McPhee approaches, x = z T , p T , q T T for the LuGre approach, and x = z T , p T , q T , h T T for the modified LuGre approach. Such nonlinear systems of equations can be solved iteratively using the Newton-Raphson method as [32,49]…”

“…In this approach, the equations of motion describe a force equilibrium for the mechanical system under consideration. This approach allows to describe systems of other physical nature, such as hydraulic actuators [32,33,43]. Hydraulic systems are highly nonlinear systems, which are caused, in part, by the friction model of the hydraulic cylinders.…”