Nonsmooth mechanical systems, which are mechanical systems involving dry friction and rigid unilateral contact, are usually described as differential inclusions (DIs), that is, differential equations involving discontinuities. Those DIs may be approximated by ordinary differential equations (ODEs) by simply smoothing the discontinuities. Such approximations, however, can produce unrealistic behaviors because the discontinuous natures of the original DIs are lost. This paper presents a new algebraic procedure to approximate DIs describing nonsmooth mechanical systems by ODEs with preserving the discontinuities. The procedure is based on the fact that the DIs can be approximated by differential algebraic inclusions (DAIs), and thus they can be equivalently rewritten as ODEs. The procedure is illustrated by some examples of nonsmooth mechanical systems with simulation results obtained by the fourth-order Runge-Kutta method.

Mechanical systems involving dry friction and rigid unilateral contact are usually described as differential inclusions (DIs). Conventional approaches for simulating those nonsmooth systems can be broadly categorized into two types: regularization approaches and hard-constraint approaches [

In regularization approaches, also referred to as penalty-based approaches [

In hard-constraint approaches, rigid bodies are considered strictly impenetrable to each other. One major way of this approach is to discretize the equation of motion by backward Euler-like methods. The discretized equation is regarded as an algebraic equation, which is then solved numerically [

This paper introduces a new method to approximately describe nonsmooth mechanical systems by ODEs. This method is derived based on the observations that DIs describing nonsmooth mechanical systems can be approximated by differential algebraic inclusions (DAIs) and that those DAIs are equivalently rewritten as ODEs. In contrast to conventional regularization methods, this method preserves the intrinsic nature of discontinuity in those systems. This method is illustrated by some examples, of which simulation results are obtained through the fourth-order Runge-Kutta (RK4) method.

The rest of this paper is organized as follows. Section

For the discussion throughout this paper, this section introduces three functions:

First, let us define the signum function

The graphs of relevant functions introduced in Section

For

A proof can be given as follows:

Next, let us define the “diode” function

For

A proof can be given as follows:

The graphs of

It must be noted that Theorems

Let us consider the situation where a rigid mass

Some previous friction models can be viewed as approximations of (

where

Other types of regularized friction models are proposed by Kikuuwe et al. [

In hard-constraint approaches, the equations of motion are discretized along time by Euler-like methods. Those discretized equations are usually formulated into complementarity problems, which are then numerically solved. The literature includes some complementarity formulations of dry friction in one-dimensional space [

Let us consider that the one-dimensional system is composed of a rigid mass

One of the trivial methods to approximately realize the contact force

As another example, the nonlinear viscoelastic contact model proposed by Hunt and Crossley [

In hard-constraint approaches for rigid unilateral contact, the equations of motion are usually discretized by Euler-like methods and then solved numerically [

In this section, new ODE approximations are introduced for ((

The new approach for approximating (

Here,

A physical interpretation of ((

In Kikuuwe et al.’s method, (

The observation that motivated the new approach is that ((

As far as the authors are aware, the literature includes no computational methods making use of the equivalence between DAIs of the form of ((

After replacing (

Simulation of the system (

It should be mentioned that ((

The new approach for approximating (

where

A physical interpretation of ((

The new approach presented here is to add another term

One possible interpretation of ((

where

By replacing (

A set of numerical simulation of the ODE (

Simulation of the system (

The methods in Sections

It must be noticed that (

Now we are in a position to present the main contribution of the work. A mechanical system can be generally described by a DI in the following form:

By applying the methods introduced in Sections

First, replace

Next, let

Finally, append

With this procedure, the nonsmooth system (

The presented procedure cannot apply if the function

The presented approach is now illustrated by an example problem. Let us consider a system in which a spherical object with a uniform mass density falls onto a fixed rigid surface, as shown in Figure

Example I: a rolling sphere with collision and slip. In the simulation, the parameters are chosen as

According to the procedure presented in Section

Figure

Simulation results of Example I by using (

In Figure

Next example is the application of the presented method to a system involving many frictional contacts interacting with one another. Let us consider a planar system illustrated in Figure

Example II: multiple frictional-unilateral contacts. In the simulation, the parameters were chosen as

According to the procedure presented in Section

A numerical simulation was performed by using the ODE (

Simulation results of Example II by using (

Also in this simulation, one can observe small penetrations at the time of collisions in Figures

This section shows the application of the proposed method to a system exhibiting periodic motion. Let us consider the system illustrated by Figure

Example III: periodic motion. In the simulation, the parameters were chosen as

According to the procedure presented in Section

Figure

Simulation results of Example III by using (

This paper has introduced a new method to approximate DIs describing nonsmooth mechanical systems involving dry friction and rigid unilateral contact by ODEs. A main difference of the new method from conventional regularization methods is that the resultant ODEs are equivalent to DAIs that are approximations of DIs. As a consequence, the approximated ODEs preserve important features of the original DIs such as static friction and always-repulsive contact force. An algebraic procedure for yielding the ODE approximations has been presented and has been illustrated by using some examples.

Future research should address the theoretical and numerical studies on the influence of the chosen parameters (

One limitation of the presented approach is that it is only for “lumped” contacts. In some situations, the contact force may be distributed across a contact area. It is unclear whether the presented approach is applicable or not to such situations. Anisotropic friction force and elastic contact, such as those seen in vehicle tires, would demand further extension of the presented approach.

This work was supported by Grant-in-Aid for Scientific Research B (no. 24360098) from Japan Society for the Promotion of Science (JSPS).