Harmonic Balance Method to Analyze the Steady-State Response of a Controlled Mass-Damper-Spring Model
Abstract
:1. Introduction
2. System Modelling and Harmonic Balance Method
3. Results and Discussion
4. Conclusions
- A.
- When the controller was OFF:
- The right-bending of the car’s vibration amplitude curve increased with the excitation force amplitude, which made the nonlinearity effect appear due to the spring’s hardening phenomenon.
- The jump phenomena displayed high to low states, and vice versa.
- The damping parameter could suppress the maximum amplitude, in addition to eliminating the existence of saddle-node points and therefore also the jump phenomena.
- The amplitude curve was exposed to a right-bending case (hardening phenomenon), a left-bending case (softening phenomenon), or a linear case depending on the sign of the cubic nonlinearity parameter.
- The excitation force amplitude and the damping factor could play an important role in tightening the range of the jump phenomena on the amplitude curve.
- B.
- When the controller was ON:
- The car’s amplitude took a new V-shaped path, departing from the previous path of higher values.
- The control and feedback gains could control the intersection band between the V-curve and the previous curve, where the bigger the gains were, the wider the V-curve became.
- The apex of the V-curve was at the point at which the system was designed to run.
- Keeping the controller’s damping parameter at lower values took the car’s amplitude to its minimum level.
- The car’s amplitude was saturated at a specific level, keeping the V-curve unchanged even if the excitation force changed.
- The key to keeping the car’s amplitude at almost zero was to guarantee that even if the excitation frequency changed.
- In case of mistuning such that , the car’s amplitude could increase slightly with until a specific value of at which the amplitude could saturate independently of .
- The car’s vibrations were suppressed by about according to the numerical simulations of the studied model.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Symbol | Definition |
Displacement of the car as a function of time | |
Mass of the car | |
Viscosity parameter of the dashpot | |
Linear and cubic stiffness parameters of the spring | |
Amplitude of the external excitation force | |
Angular frequency of the external excitation | |
The control signal as a function of time | |
Damping parameter of the controller | |
Angular natural frequency of the controller | |
Gains of the control signal and the feedback signal | |
Approximate amplitudes of and | |
Approximate phases of and | |
Floquet characteristic exponent |
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Kandil, A.; Hamed, Y.S.; Awrejcewicz, J. Harmonic Balance Method to Analyze the Steady-State Response of a Controlled Mass-Damper-Spring Model. Symmetry 2022, 14, 1247. https://doi.org/10.3390/sym14061247
Kandil A, Hamed YS, Awrejcewicz J. Harmonic Balance Method to Analyze the Steady-State Response of a Controlled Mass-Damper-Spring Model. Symmetry. 2022; 14(6):1247. https://doi.org/10.3390/sym14061247
Chicago/Turabian StyleKandil, Ali, Y. S. Hamed, and Jan Awrejcewicz. 2022. "Harmonic Balance Method to Analyze the Steady-State Response of a Controlled Mass-Damper-Spring Model" Symmetry 14, no. 6: 1247. https://doi.org/10.3390/sym14061247
APA StyleKandil, A., Hamed, Y. S., & Awrejcewicz, J. (2022). Harmonic Balance Method to Analyze the Steady-State Response of a Controlled Mass-Damper-Spring Model. Symmetry, 14(6), 1247. https://doi.org/10.3390/sym14061247