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Chapter S4: Special Functions and Their Properties
Published in Andrei D. Polyanin, Valentin F. Zaitsev, Handbook of Ordinary Differential Equations, 2017
Andrei D. Polyanin, Valentin F. Zaitsev
The Mathieu functions cen(x,q) $ ce_{n} (x, q) $ and sen(x,q) $ se_{n} (x, q) $ are periodic solutions of the Mathieu equation
Mathieu Stability of Compliant Structures
Published in Srinivasan Chandrasekaran, Advanced Steel Design of Structures, 2019
Structural systems that are designed as form dominant are also compelled to remain as positive buoyant while encountering the lateral loads. Offshore floating platforms, ships and large vessels, semi-submersibles and offshore production platforms that are moored to the seabed fall in this category. While they are flexible in the horizontal plane, they remain very stiff in the vertical plane, imposing motion constraints to the topside. Mathieu equation describes the stability of compliant structures to their parametric oscillations whose solution is dependent on Mathieu parameters and represented as Mathieu stability chart (Chandrasekaran & Kiran, 2017). Mathieu stability conditions should be satisfied to ensure the safe operations of these structures. Position restrained by high pretension tethers, floating and compliant offshore platforms exhibit larger stiffness in the vertical plane. These platforms are generally moored to the seabed using mooring lines or tethers, which ensure position restraint of the platform under the lateral loads caused by ocean waves. They experience coupled response between the displacements in a horizontal and vertical plane under the environmental loads, resulting in the dynamic tether tension variations in the mooring lines (Chandrasekaran, 2015b). Under the displaced position, the horizontal component of tether force enables recentering of the deck while the vertical component imposes heave restraint. Thus, a major contribution to their stability under operational loads is achieved by tethers. Classic examples of such structural systems are tension leg platforms, triceratops, buoyant-leg storage, regasification platforms and semi-submersibles. Their dynamic tension variations impose a challenge to its stability, which can be described using the Mathieu equation. Detailed studies on spar platforms showed Mathieu-type instability in systems where the natural pitch period is about twice as that of the heave (Haslum & Faltinsen, 1999; Koo et al., 2004). Dynamic behavior under such unstable conditions showed chaotic behavior, which is critical to ensure safe functionality of the platform (Rho et al., 2005).
NMR of orientationally ordered short-chain hydrocarbons
Published in Liquid Crystals, 2018
E. Elliott Burnell, Cornelis A. De Lange
Experimental values for the quantities that determine the height and shape of the rotational barrier are known, and are kcal/mol and kcal/mol [38]. This leads to a one-dimensional Schrödinger equation where represents the potential energy. Truncating the potential after the term, this differential equation leads to the so-called Mathieu equation whose eigenvalues and eigenfunctions have been fully tabulated [39–41]. If so desired, the term can be included by perturbation theory. The full quantum-mechanical treatment of the torsional problem can thus be performed, albeit with considerable effort. However, since in practise the differences between a quantum-mechanical and semi-classical treatment are small, we have employed the latter simpler approach. Hence, the torsional motion is taken into account by classical averaging over the internal rotational motion and by using appropriate Boltzmann factors. The classical averaging over the torsional motion is incorporated into the calculated dipolar couplings [2].