Equivalent Electronic Circuit of a System of Oscillators Connected with Periodically Variable Stiffness
Abstract
:1. Introduction
2. System Description
3. Mathematical Model
3.1. Dimensional Equations of the System
- 1
- ; . is the resistance force of the bearings and the term is the smooth approximation of the function sign().
- 2
- is the force due to the magnetic spring.= ; . The idea of the above formula is considered by the two assumptions: (i) the repulsive force between two magnets is defined by the simplest expression of the inverse square law, where the dipole expression has been considered. (ii) when = , the first term of the expression becomes the unity, when = , the second term becomes unity, and the expression becomes negative value.
- 3
- The stiffness coupling of the considered system, = . It varies periodically having the linear frequency, , where is the angular frequency of oscillation.
3.2. Non-Dimensional Equations of the System
4. Numerical Results
4.1. Behavior of the Dry-Friction and Resistance Force Terms
4.2. Behavior of Magnetic Spring Force Term
4.3. Time-Series and Phase-Space Plots of the Considered System
4.4. Bifurcation Diagrams and the Corresponding Maximal Lyapunov Exponent
4.5. Co-Existence of Two Attractors
5. Equivalent Circuit of the Coupled Mechanical Oscillators
- 1.
- = + = ; . is the non-dimensional resistance force of the bearings. is the linear damping force, and is the force due to the dry-friction. The values of , , , and are , , , , and respectively.
- 2.
- = ; ., is the non-dimensional force due to the magnetic spring. The values of , , , and are , , , and 1 respectively. To reduce the complexity of the equations, we have chosen the most straightforward expressions of the non-dimensional force due to the magnetic spring, which are in the Equations (7) and (8).
- 3.
- The non-dimensional stiffness couplings of the considered system, = and = . The parameter values are , . The non-dimensional angular frequency, , will have to be varied in order to obtain the bifurcation diagram.
5.1. Equivalent Circuit Diagram of the Dry-Friction Term
5.2. Equivalent Circuit Diagram of the Magnetic Spring Force Term
5.3. Equivalent Circuit Diagram of
6. Results Obtained from Simulating the Circuit
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Seth, S.; Kudra, G.; Witkowski, K.; Awrejcewicz, J. Equivalent Electronic Circuit of a System of Oscillators Connected with Periodically Variable Stiffness. Appl. Sci. 2022, 12, 2024. https://doi.org/10.3390/app12042024
Seth S, Kudra G, Witkowski K, Awrejcewicz J. Equivalent Electronic Circuit of a System of Oscillators Connected with Periodically Variable Stiffness. Applied Sciences. 2022; 12(4):2024. https://doi.org/10.3390/app12042024
Chicago/Turabian StyleSeth, Soumyajit, Grzegorz Kudra, Krzysztof Witkowski, and Jan Awrejcewicz. 2022. "Equivalent Electronic Circuit of a System of Oscillators Connected with Periodically Variable Stiffness" Applied Sciences 12, no. 4: 2024. https://doi.org/10.3390/app12042024
APA StyleSeth, S., Kudra, G., Witkowski, K., & Awrejcewicz, J. (2022). Equivalent Electronic Circuit of a System of Oscillators Connected with Periodically Variable Stiffness. Applied Sciences, 12(4), 2024. https://doi.org/10.3390/app12042024