## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Optimal vibration control of an isotropic beam through boundary conditions

**ABSTRACT**

An isotropic structure modelled as a Timoshenko beam is considered for the optimal vibration control problem. The beam model to be controlled is described by a distributed parameter system with the selection of Timoshenko's shear correction factor. Control of the vibrations is achieved through a function placed on the boundary conditions. The performance index which seeks to be minimized indicates that the goal is to minimize the magnitude of performance measure without consuming control effort in large quantities. It is shown how to derive the optimal control function using Pontryagin's principle that turns the control problem into solving optimality system of partial differential equations with terminal values. Wellposedness of the optimal solution on the control set is presented and controllability of the problem is analyzed. Numerical simulations are given in terms of computer codes produced in MATLAB© in the forms of graphical and tables in order to show the applicability and effectiveness of the control acting on the boundary conditions.

**KEYWORDS**

PAPER SUBMITTED: 2020-05-13

PAPER REVISED: 2020-10-11

PAPER ACCEPTED: 2020-10-23

PUBLISHED ONLINE: 2021-01-24

- Timoshenko, S. P., On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars, Philosophical Magazine, (1921), 41, pp. 744-746.
- Timoshenko, S. P., On the Transverse Vibration of Bars of Uniform Cross-section, Philosophical Magazine, (1922), 43, pp. 125-131.
- Cowper, G. R., The Shear Coefficient in Timoshenko's Beam Theory, Journal of Applied Mechanics,33, (1966), 2, pp. 335-340.
- Kaneko, T., On Timoshenko's Correction for Shear in Vibrating Beams, Journal of Physics D: Applied Physics, 8, (1975), 16, pp. 1927-1936.
- Hutchinson, J. R., Shear Coefficients for Timoshenko Beam Theory, Journal of Applied Mechanics, 68, (2001), 1, pp. 87-92.
- Leissa, A.W., So, J., Comparisons of Vibration Frequencies for Rods and Beams from One-Dimensional and Three-Dimensional Analyses, The Journal of the Acoustical Society of America, (1995), 98, pp. 2122-2135.
- Kennedy, G. J. et. al., A Timoshenko Beam Theory with Pressure Corrections for Layered Orthotropic Beams, International Journal of Solids and Structures, (2011), 48, pp. 2373-2382.
- Kirk, D. E., Optimal Control Theory: An Introduction, Dover Publications, New York, USA, 2004.
- Gudi, T., Sau, R. C., Finite Element Analysis of the Constrained Dirichlet Boundary Control Problem Governed by the Diffusion Problem, ESAIM: Control, Optimisation and Calculus of Variations, (2019).
- Fattorini, H. O., Murphy, T., Optimal Control Problems for Nonlinear Parabolic Boundary Control Systems: The Dirichlet Boundary Condition, Differential and Integral Equations, 7, (1994), 5-6, pp. 1367-1388.
- Fattorini, H. O., Murphy, T., Optimal Problems for Nonlinear Parabolic Boundary Control Systems, SIAM Journal on Control and Optimization, 32, (1994), 6, pp. 1577-1596.
- Nowakowski, A., A Neumann Boundary Control for Multidimensional Parabolic "Minmax" Control Problems, In Advances in Dynamic Games and Their Applications, Birkhäuser Boston, Boston, 2009, pp. 1-13.
- Yildirim, K. et. al., Dynamics Response Control of A Mindlin-type Beam, International Journal of Structural Stability and Dynamics, 17, (2017), 3, pp. 1750039 (1-14).
- Goktepe Korpeoglu, S.et. al., Optimal Boundary Control for A Second Strain Gradient Theory-Based Beam Model, Sigma: Journal of Engineering & Natural Sciences/Mühendislik ve Fen Bilimleri Dergisi, 37 (2019), 4, pp. 1281-1292.
- Yildirim, K., Kucuk, I., Active Piezoelectric Vibration Control for A Timoshenko Beam, Journal of the Franklin Institute, 353, (2016), 1, pp. 95-107.
- Zachmanoglou, E. C., Thoe, D. W., Introduction to Partial Differential Equations with Applications, Dover Publications, New York, USA, 1986.
- Guliyev, H. F., Jabbarova, K. S., The Exact Controllability Problem for the Second Order Linear Hyperbolic Equation, Differential Equations and Control Processes, (2010), 3, pp. 10-19.
- Pedersen, M., Functional Analysis in Applied Mathematics and Engineering, CRC press, USA, 1999.