Abstract
Several theoretical studies deal with the stability transition curves of the Mathieu equation. A few others present numerical and asymptotic methods to describe the stability of coupled Mathieu equations. However, sometimes the averaging and perturbation techniques deal with cumbersome computations, and the numerical methods spend considerable resources and computation time. This contribution extends the definition of linear Hamiltonian systems to periodic Hamiltonian systems with a particular dissipation. This leads naturally to a generalization of symplectic matrices, to μ-symplectic matrices. This definition enables an efficient way for calculating the stability transition curves of coupled Mathieu equations.
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Notes
- 1.
The matrix product \(\left ( \frac {d}{dt}N^{T}JN\right ) N^{T}JN=N^{T}JN\left ( \frac {d}{dt}N^{T}JN\right ) \) is commutative.
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Acknowledgement
The first and third author are thankful for the financial support within the Austrian IWB project LaZu-CLLD_IWB_TIROL_OSTT_005 (Campus Technik Lienz).
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Ramírez Barrios, M., Collado, J., Dohnal, F. (2020). Stability of Coupled and Damped Mathieu Equations Utilizing Symplectic Properties. In: Lacarbonara, W., Balachandran, B., Ma, J., Tenreiro Machado, J., Stepan, G. (eds) Nonlinear Dynamics of Structures, Systems and Devices. Springer, Cham. https://doi.org/10.1007/978-3-030-34713-0_14
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DOI: https://doi.org/10.1007/978-3-030-34713-0_14
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