Abstract
This paper investigates the maximum interval of stability and convergence of solution of a forced Mathieu’s equation, using a combination of Frobenius method and Eigenvalue approach. The results indicated that the equilibrium point was found to be unstable and maximum bounds were found on the derivative of the restoring force showing sharp condition for the existence of periodic solution. Furthermore, the solution to Mathieu’s equation converges which extends and improves some results in literature.
Highlights
Consider a harmonically forced Mathieu’s equation defined by d2 dt x + ( w ε cos t ) x =f cos λt (1)For small ε, this equation describes a simple harmonic oscillator whose frequency is a periodic function of time with the boundary condition as; x (= 0) x (2π) x (= 0) x (2π) (2)
This paper investigates the maximum interval of stability and convergence of solution of a forced Mathieu’s equation, using a combination of Frobenius method and Eigenvalue approach
The results indicated that the equilibrium point was found to be unstable and maximum bounds were found on the derivative of the restoring force showing sharp condition for the existence of periodic solution
Summary
For small ε , this equation describes a simple harmonic oscillator whose frequency is a periodic function of time with the boundary condition as;. In [2], Equation (1) was discussed in connection with problem of vibrations in elliptical membrane and developed the leading terms of the series known as Mathieu’s function. Stability is determined by the interval placed on the total derivative of the system form by the given differential equation. Such interval is restricted only on the equilibrium point of the system. Motivated by the above literature and ongoing research in this direction, the objectives of this paper are to investigate the maximum interval of stability and convergence of solution of forced Mathieu’s equation. We further prove that the solution converges in that interval of interest and that if all solutions are bounded, the corresponding point in the w and ε parameter plane is said to be stable
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