## THERMAL SCIENCE

International Scientific Journal

### THERMALLY INDUCED VIBRATION SUPPRESSION IN A THERMOELASTIC BEAM STRUCTURE

**ABSTRACT**

In this paper, the problem of thermally induced vibration suppression in a thermoelastic beam is studied. Physical equivalent of the present problem is that a thermoelastic beam is suddenly entering into daylight zone and vibrations are induced due to heating on the upper surface of the beam or thermoelastic beam in a spacecraft enters to intensive sunlight area just after leaving a shadow of a planet. Thermally induced vibrations are suppressed by means of minimum using of control forces to be applied to dynamic space actuators. Objective functional of the problem is chosen as a modified quadratical functional of the kinetic energy of the thermoelastic beam. Necessary optimality condition to be satisfied by an optimal control force is derived in the form of maximum principle, which converts the optimal vibration suppression problem to solving a system of distributed parameters system linked by initial-boundary-terminal conditions. Solution of the system is achieved via MATLAB© and simulated results reveal that thermally induced vibration suppression by means of dynamic space actuators are very effective and robust.

**KEYWORDS**

PAPER SUBMITTED: 2021-02-12

PAPER REVISED: 2021-02-17

PAPER ACCEPTED: 2021-03-10

PUBLISHED ONLINE: 2021-03-13

**THERMAL SCIENCE** YEAR

**2021**, VOLUME

**25**, ISSUE

**Issue 3**, PAGES [2017 - 2024]

- Yu, Y. Y., Thermally Induced Vibration and Flutter of a Flexible Boom, Journal Spacecraft, 6 (1969), 8, pp. 901-908
- Boley, B. A., Approximate Analysis of Thermally Induced Vibrations of Beams and Plates, Journal Applied Mechanics, 39 (1972), 1, pp. 212-216
- Manolis, G. D., Thermally Induced Vibrations of Beam Structures, Computer Methods and Applied Mechanics and Engineering, 21 (1980), 3, pp. 337-355
- Zener, C., Internal Friction in Solids II. General Theory of Thermoelastic Internal Friction, Physical Review, 53 (1938), 1, pp. 90-99
- Kinra, V. K., Milligan, K. B., A Second Law Analysis of Thermoelastic Dampings, Journal of Applied Mechanics, 61 (1994), 1, pp. 71-76
- Kidawa-Kukla, J., Vibration of a Beam Induced by Harmonic Motion of a Heat Source, Journal of Sound and Vibration, 205 (1997), 2, pp. 213-222
- Kucuk, I., et. al., Optimal Control of a Beam with Kelvin-Voigt Damping Subject to Forced Vibrations Using a Piezoelectric Patch Actuator, Journal of Vibration and Control, 21 (2015), 4, pp. 701-713
- Jayasuriya, S., Choura, S., Active Quenching of a Set of Predetermined Vibratory Modes of a Beam by a Single Fixed Actuator, Int. Journal of Control, 51 (1990), 2, pp. 445-467
- Choura, S., et. al., On the Modelling and Open Loop Control of a Rotating Thin Flexible Beam, ASME J. Dynamic systems, Measurement Control, 113, Mar., pp. 26-33
- Jayasuriya, S., Choura, S., On the Finite Settling Time and Residual Vibration Control of Flexible Structures, Journal Sound Vibration, 148 (1991), 1, pp. 117-136
- Goktepe Korpeoglu, S., Optimal Vibration Control of an Isotropic Beam through Boundary Conditions, Thermal Science, 25 (2021), Special Issue 1, pp. S111-S120
- Kucuk, I., et. al., Optimal Piezoelectric Control of a Plate Subject to Time-Dependent Boundary Moment and Forcing Function for Vibration Damping, Computers and Mathematics with Applications, 69 (2015), 4, pp. 291-303
- Choura, S., Jayasuriya, S., Control of Distributed Parameter Systems by Moving Force Actuators, Journal Guidance, Control, Dynamics, 14 (1991), 6, pp. 1200-1207
- Lagnese, J. E., The Reachability Problem for Thermoelastic Plates, Arch. Rational Mech. Analysis, 112 (1990), Sept., pp. 223-267
- Pamuk, M. T., Numerical Study of Natural Convection in an Enclosure with Discrete Heat Sources on One of Its Vertical Walls, Thermal Science, 25 (2021), 1A, pp. 267-277
- Edberg, D. L., Control of Flexible Structures by Applied Thermal Gradients, AIAA J., 25 (1987), 6, pp. 877-883
- Boley, B. A., Weiner, J. H., Theory of Thermal Stress, John Wiley and Sons, New York, USA, 1960
- Boley, B. A., Thermally Induced Vibrations of Beams, Journal Aeronaut Sci., 23 (1956), pp. 179-181
- Kukla, J. K., Application of the Green Functions to the Problem of the Thermally Induced Vibration of a Beam, Journal of Sound and Vibration, 262 (2003), 4, pp. 865-876
- Pedersen, M., Functional Analysis in Applied Mathematics and Engineering, CRC Press, Boca Raton, Fla., USA, 2018
- Zachmaonoglou, E. C., Thoe, D. W., Intoduction to Partial Differential Equations with Applications, Dover Publ., 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, 3 (2010)