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Numerical Computation of Separation Events

Authored by Kevin Murphy

Impacting systems possess stiffness discontinuities (nonlinearities) and integrating the system through these abrupt transitions in stiffness is numerically challenging. Henon’s method [1] for identifying numerically the instant of impact - and the associated values of the state variables - is well established. However, in cases where distinct components move together, there is not a well-developed technique for identifying when they separate because this separation event is not expressed in terms of the generalized coordinates. This work presents a numerical technique for doing just that. If one couples this method with Henon’s method, the time response of a system undergoing repeated impacts and separations may be obtained with a high degree of precision. Several specific examples are presented to demonstrate the utility of the method.

Impact oscillators give rise to nonlinear behavior through the abrupt, impulsive interaction between the distinct, interacting bodies. Modeling this behavior accurately is a challenge because the instant of impact is not known a-priori. Moreover, capturing the instant of impact is crucial because nonlinear systems can display an extreme sensitivity to initial conditions, i.e., an impact event can be viewed as an initial condition for a new trajectory. For cases where the motion is found numerically via a time-stepping algorithm, Henon [1] developed a simple numerical technique for ascertaining this instant of impact. This method has been used effectively in a number of impact problems and produces results that agree with other techniques and, more importantly, with independent experimental observations. It has also been incorporated into commercial software packages.

One common model for an impact oscillator uses the coefficient of restitution (CoR) [2]. Here, the normal component of the rebound velocity is given as a percent of the inbound (pre-collision) velocity. It is assumed that, provided e≠0, the interacting bodies come into contact and then immediately separate. When numerically timestepping a solution, Henon’s method is invaluable because it clearly indicates the instant when the impact event occurs, i.e., when the CoR model should be applied.

Another class problem that comes up from time to time is that of separation. One can imagine two (or more) contacting bodies moving together; in this case, the system can be modeled as a single degree of freedom problem with one combined mass. As the motion evolves, these bodies may separate. This requires a sudden change in the model from one degree of freedom to multiple degrees of freedom. Therefore, like the impact problem, determining the moment of separation is critical in order to generate the response accurately. However, unlike the impact problem, the separation event is more subtle. It is not dictated by a condition placed on one of the state variables. Instead, the system must be unpacked a bit before a Henon-like approach may be taken (Figure 1).

In this paper a numerical technique is developed to determine this instant of separation for these systems. This method takes its inspiration from Henon’s method. Consequently, Henon’s method is briefly reviewed. Next, the separation problem is described and a numerical technique for the instant of separation is developed. A number of examples are provided, which demonstrate the utility of the method.

The Numerical Technique

The numerical solution of nonlinear ordinary differential equations involves time-stepping. This process begins by writing the set of equations in first order form: ẋ = f(x(t), t). x is the unknown state vector and t is the independent variable. Solutions are obtained by starting at some initial condition and integrating the state variables through some discrete time step △t from to t0 to t1. This integration is accomplished by some numerical scheme, such as Runge-Kutta, Geer’s method, etc. [3]. Using x(t1) as a new initial condition, another time integration step △t may be taken to t2. This gives x(t2). This process continues and gives the timediscretized trajectory of the system.

As will be described, both of the impact and separation problems are nonlinear in the sense that there is a model change at the impact/separation event. So, it is critical to ascertain the instant of these events or else the computed trajectories will not faithfully reflect the system response. As motivation, the impact problem is outlined and a brief description of the Henon technique [1] is given in the next section. This is followed, in the subsequent section, by a description of the separation problem and the new numerical technique for handling that problem.

Impact events

For the moment, consider a single mass approaching a rigid wall along its normal direction; this is shown schematically in Figure (1a). The equations of motion for the ball may be integrated until the mass impacts the wall. At the moment of impact, the velocity is instantaneously changed. The post-impact velocity υ’ (which serves as the new initial condition for the next step in the integration) is found by reversing the direction of the pre-impact velocity v and multiplying it by the coefficient of restitution: υ’= -eυ. However, to apply this condition, it is assumed that the integration routine will land exactly at the impact event (the wall). In general, this does not happen.

To demonstrate this, again consider Figure (1a). The equations of motion for the mass have been integrated through i time steps to ti. This leaves the mass very near the wall, △x away from an impact. The integration routine steps to ti+1; the updated position of the mass is determined by the governing differential equations. This turns out to be on the other side of the impact condition, which is physically meaningless since the mass can’t penetrate the wall. Traditional routines would return to ti and proceed to take smaller time steps, in the hope of landing “close enough” to the impact condition. But this process of refining the time step is computationally laborious. Moreover, it won’t effectively find the moment of impact because the same issue will continue to crop up, just at smaller spatial and temporal scales (Figure 2).

Henon [1] came up with a practical and easy-to-implement technique to circumvent this problem. Here is an overview. Consider Figure (1b). The problem begins at the same instant ti and position xi as before. Because the distance to the wall is well known (but the time needed to get there is not), the problem is recast. Specifically, the equations are rewritten with x as the independent variable and ẋ and t as the dependent variables. Then, rather than taking a time step, the system undertakes a single displacement step of △x. This, by definition, lands the mass at the impact condition. The associated velocity and time are determined by the recast differential equations. At this instant, the velocity is changed to the post-impact value: υ’ = -eυ. Then the equations are returned to their original form (with t as the independent variable) and the time stepping algorithm is continued until the next impact event.

To integrate the system through an impact event, Eqs. (2) are time integrated until the position violates the impact condition (i.e., the two bodies pass through one another, similar to Figure (1a)). Taking a time step back, one then switches to Eq. (3) and takes a prescribed relative displacement step (r1) to the impact condition. The other dependent variables (including time, t) are obtained through the integration process. Once at the impact condition, the velocities are changed using the CoR model and time stepping may resume using Eq. (2).

Separation events

Various conditions may cause two (or more) bodies to move together. Initial conditions could be such that they all begin together. Or two objects may approach one another, and their velocities coalesce at impact, causing them to move together afterwards. Regardless of the cause, consider two contacting bodies that are moving together. This scenario opens the door to the possibility that they may separate. Normal time integration of the governing equations will typically step through the separation event - just as time integration usually steps through an impact event (see Figure (1a)). Now the parallel question is asked: can the equations be recast in such a way as to end up at the moment of separation? As a specific example, recall the two degree of freedom model of Figure (2a) with u1 = u2 (or r1 = 0). There is an interaction force Fi between the bodies, as shown in Figure (2b). Applying Newton’s law to each mass and ignoring the external forces (for the moment) gives:

This work presents a new technique for calculating numerically the moment of separation of two moving, contacting bodies. This approach builds off of the technique developed by Henon [1] for determining numerically the moment of impact for two moving bodies. Specifically, the governing equations are integrated numerically until the impact condition is violated (i.e., the last time step went beyond the impact condition, allowing the bodies to partially pass through one another). The system then goes back one-time step, where the distance to impact is known. The governing equations are recast so with the relative position of the two bodies being the independent variable.

Time t is a dependent variable to be determined by the integration routine. A single displacement step is taken exactly to the condition of first contact. The associated time is then computed from the recast equations, along with the other state variables. In a similar way, the onset of separation of two contacting bodies may be identified. The system may be integrated forward in time until a separation event takes place. Going back one-time step, the governing equations may be recast so that the independent variable is the interaction force between the two bodies. Then a single force step may be taken to the exact moment of separation (F → 0). The process of recasting the equations is demonstrated with a simple two mass problem. To highlight the implementation, a number of impact/separation cases are considered both numerically and analytically. In each scenario, the instant of contact/separation are precisely identified, and the ensuing motion found. In general, the method developed here is straightforward and effective. And it may be easily extended for systems containing more degrees of freedom.

 

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