# American Institute of Mathematical Sciences

January  2019, 24(1): 83-107. doi: 10.3934/dcdsb.2018162

## Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping

 1 School of Mechanical, Industrial, & Manufacturing Engineering, Oregon State University, Corvallis, OR 97331-6011, USA 2 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA

* Corresponding author

† H. Shoori and J. Singler were supported in part by National Science Foundation grant DMS-1217122; B. Batten was supported in part by the Department of Energy under Award Number DE-FG36-08GO18179.

Received  February 2017 Revised  January 2018 Published  January 2019 Early access  June 2018

We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

Citation: Belinda A. Batten, Hesam Shoori, John R. Singler, Madhuka H. Weerasinghe. Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 83-107. doi: 10.3934/dcdsb.2018162
##### References:

show all references

H. Shoori and J. Singler were supported in part by National Science Foundation grant DMS-1217122; B. Batten was supported in part by the Department of Energy under Award Number DE-FG36-08GO18179.

##### References:
The cable mass system
Eigenvalues of the linear system and energy decay for the nonlinear system with $\gamma = \alpha_{l} = 0.1$, $k_{0} = k_{l} = 1$, and $\alpha_{0} = \alpha = 0$
Eigenvalues of the linear system and energy decay for the nonlinear system with $\gamma = 0$ and $\alpha = \alpha_{0} = \alpha_{l} = k_{0} = k_{l} = 0.01$
Example 1, Input 2: Output of the ROM and FOM for $\alpha_{0} = \alpha = 0$, $\alpha_{l} = k_{0} = k_{l} = 0.1$, and $\gamma = 0.001$
Example 5, Input 4: Output of the nonlinear ROM and FOM for $\alpha = \alpha_{0} = \alpha_{l} = 0$, $\gamma = 0.001$, and $k_{0} = k_{l} = 0.1$
Example 2, Input 1: Output of the ROM and FOM for $\gamma = 0$, $\alpha = \alpha_{0} = \alpha_{l} = 0.1$, and $k_{0} = k_{l} = 0.001$
Example 1, Input 4: Output of the ROM and FOM for $\alpha = \alpha_0 = 0$, $\gamma = \alpha_{l} = 0.1$, and $k_{0} = k_{l} = 0.001$
Example 5, Input 4: Output of the ROM and FOM for $\gamma = 0.1$, $\alpha = \alpha_{0} = \alpha_{l} = 0$, and $k_{0} = k_{l} = 0.001$
Example 3: Output of the nonlinear ROM and FOM for $\alpha_{0} = \alpha_{l} = 0$ and $\alpha = \gamma = k_{0} = k_{l} = 0.001$
Fixed simulation parameters
 $l$ $m_0$ $m_l$ $k_3$ $\beta$ 1 1 1.5 1 1
 $l$ $m_0$ $m_l$ $k_3$ $\beta$ 1 1 1.5 1 1
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