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Research Papers

Sources of Error in a Simulation of Rigid Parts on a Vibrating Rigid Plate

[+] Author and Article Information
Stephen Berard1

Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180-3590sberard@cs.rpi.edu

Binh Nguyen, J. C. Trinkle

Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180-3590

Kurt Anderson

Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590

1

Corresponding author.

J. Comput. Nonlinear Dynam 5(4), 041003 (Jun 29, 2010) (14 pages) doi:10.1115/1.4001820 History: Received October 07, 2008; Revised December 01, 2009; Published June 29, 2010; Online June 29, 2010

We present a simulation study of an important rigid-body contact problem. The system in question is composed of a rigid plate and a single rigid body (or particle). The plate follows a prescribed periodic motion of small amplitude and high frequency, such that the net force applied to the part appears to be from a time-independent, position-dependent velocity field in the plane of the plate. Theoretical results obtained by Vose were found to be in good agreement with simulation results obtained with the Stewart–Trinkle time-stepping method. In addition, simulations were found to agree with the qualitative experimental results of Vose After such verification of the simulation method, additional numerical studies were done that would have been impossible to carry out analytically. Specifically, we were able to demonstrate the convergence of the method with decreasing step size (as predicted theoretically by Stewart). Further analytical and numerical studies will be carried out in the future to develop and select robust simulation methods that best satisfy the speed and accuracy requirements of different applications. With the accuracy of our time-stepper verified for this system, we were able to study the inverse problem of designing new plate motions to generate a desired part motion. This is done through an optimization framework, where a simulation of the part interacting with the plate (including the full dynamics of the system) is performed, and based on the results of the simulation the motion of the plate is modified. The learned (by simulation) plate motion was experimentally run on the device, and without any tuning (of the simulation parameters or device parameters) our learned plate motion produced the desired part motion.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

The plate is kinematically controlled by the vector function g(t). There are three forces acting on the part: the force due to gravity λapp, the nonpenetration constraint force Wnλn, and the frictional force Wfλf.

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Figure 2

Friction cone approximated by an eight-sided pyramid defined by friction direction vectors dj

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Figure 3

A part on a plate rotating about an axis below the plate. The fixed world frame is centered on the axis of rotation.

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Figure 4

The assumptions of contact maintenance and no sticking are satisfied

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Figure 5

Comparison between the numerically computed asymptotic velocity (simulated) to the value determined in the work of Vose (theoretical)

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Figure 6

With the axis of rotation closer to the plate there are periods of sticking during the cycle and the asymptotic velocity found during simulation does not match (as expected) the theoretical value determined in the work of Vose

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Figure 7

A 6 s simulation of all the fields described in Fig. 5 of Ref. 22 using a step size of 0.0001 s. The green circle indicates the starting location of the particle.

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Figure 8

Direct comparison of the particle’s position between our simulation results and those of Vose for the centrifuge motion

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Figure 9

Direct comparison of the particle’s velocity between our simulation results and those of Vose for the centrifuge motion

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Figure 10

Trajectory and error of a particle with the plate subject to the centrifuge motion. The green circle indicates the starting location of the particle. The error is shown as a function of step size, assuming that a step size of 0.00005 s is ground truth.

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Figure 11

The log of the error versus log of the step size illustrates that the time-stepper appears to be somewhere between O(h) and O(h2), as expected. The slope of the line between h=0.0001 s and h=0.0005 s is 1.41. The slope of the line between h=0.0005 s and h=0.001 s is 1.19.

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Figure 12

Trajectory and error caused by the polygonal approximation of the friction cone. The green circle indicates the starting location of the particle. The error at time t is the Euclidean distance of the particle between the LCP and NCP formulations. All simulations used a step size of 0.0001 s.

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Figure 13

The log of the error versus log of the number of friction directions

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Figure 14

Trajectory and error caused by the polygonal approximation of the friction code. The error at time t is the Euclidean distance of the particle between the LCP and NCP formulations. The initial position of the particle was (4,0) and all simulations used a step size of 0.0001 s.

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Figure 15

The log of the error versus log of the number of friction directions

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Figure 16

Timing of a single LCP function call for a block sliding on a plane as a function of number of friction directions. The solution time for an equivalent NCP formulation with quadratic friction law is also plotted. At approximately eight friction directions; the solution time of the LCP is larger than the NCP.

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Figure 17

Results of the circle trajectory described in Ref. 23 with the particle starting at various radii. The simulation used a constant step size of 0.0001 s and ran for 41.89 s. When the particle starts at (4, 0), there are periods of sticking (17 b) during a cycle of the plate’s motion, and the part does not travel with the desired speed.

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Figure 18

Results of the learned circle trajectory with a constant step size of 0.0001 s and a penalty for particle plate separation

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Figure 19

Results of the learned circle velocity field without a contact maintenance requirement using a constant step size of 0.0001 s. The optimization problem took advantage of contact separation in the solution found.

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Figure 20

Results of the learned saddle trajectory for various initial positions and a step size of 0.0001 s

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Figure 21

The x and y angles of rotation for one period of the centrifuge plate motion, when starting with the identity unit quaternion. The rotation about the x-axis is not symmetric about zero.

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Figure 22

The x and y angles of rotation for one period of the centrifuge plate motion, when starting with the computed initial unit quaternion

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