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Numerical Techniques for Non-Linear Dynamic Contact Problems
Numerical Techniques for Non-Linear Dynamic Contact Problems
In the summer of 2021, I was a RIYA Scholar at the Department of Mechanical and Aerospace Engineering at The Ohio State University. I worked on this project in The Gear Lab with Dr. Talbot and Dr. Ryali as my mentors. The RIYA program is an undergraduate research opportunity organized by Prof. Raj Singh, Faculty Emeritus, The Ohio State University.
Keywords: Optimization, Frequency domain analysis, numerical simulation, Physics informed neural networks, Dynamical systems
Problem Setup
Problem Setup
Meshing gears do not have the perfect involute profile, intentionally or unintentionally. The clearances arise for lubrication, to avoid interference, due to manufacturing errors or due to wear. Two meshing spur gears can be shown to be equivalent to a spring mass damper system with a clearance (Kahrman & Singh 1990).
The objective of this project was to use this simplified model and explore numerical techniques to solve for the response of this system. Difficulties arise due to the presence of the clearance nonlinearity in the system.
The first numerical technique studied were numerical time integration methods such as Forward & Backward Euler 1st Order, Adams-Bashforth 4th Order, Adams-Moulton 3rd Order, and finally Runge-Kutta 4th Order.
Nonlinear phenomenon in the frequency response of the system when subject to external excitations were observed.
The jump phenomenon and the existence of multiple solutions for the same operating conditions are shown in the following two plots.
The two plots represent the steady state solution of the position of the block, broken down into its mean component and amplitude of oscillation.
In each plot, the external forcing function is varied to get each of those curves. In this case they correspond to different mean values for the forcing function.
Optimization Based Solver
Optimization Based Solver
I proposed another method to solve for the steady state solution of the nonlinear system. By projecting the solution into a Fourier basis of limited terms, the problem boils down to finding the optimal coefficients for a given cost function. The cost function is given as the "degree of violation" of the ODE of the system.
The solution obtained from the optimization based solver were compared against analytical solutions (“Non-Linear Dynamics of a Spur Gear Pair” by A. Kahrman & R. Singh (1990)).
To obtain some of the solutions corresponding to double sided impact of the block on both springs, the initial guess given to the solver had to be very different from the one that the Runge-Kutta method would provide.
The main benefit of this method is that it arrives at the steady state solution much faster than the Runge-Kutta method and the degree of violation of the ODE is much lesser than the Runge-Kutta method's.
One of the main drawbacks is that in such nonlinear dynamical systems, a chaotic response is possible where the solution is not periodic. A fundamental assumption of this method is that the period of the solution is known. Even if it is not known exactly, it is known to be a multiple of a certain minimum period for this system.
Physics Informed Neural Networks (PINNs)
Physics Informed Neural Networks (PINNs)
Another approach I used to tackle this problem was to adapt the SciANN library to essentially train a neural network to represent the steady state solution of the system with a given forcing function.
Throwing a neural network and asking it to represent the solution is possible only because of the Universal Approximation Theorem. This approach (the way I tried to use it at least) also suffers from not being able to capture chaotic responses of the system. Also as elucidated by the zoom-in by the green arrow, the solution is not periodic in time, because I did not encode this necessity in my loss function.
All figures are mine unless stated otherwise.
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