Direct floquet method for stability limits determination—I

Weyh, Bernhardt; Kostyra, H.

doi:10.1016/0094-114x(91)90077-h

Cited by 7 publications

(2 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since the matrices A(τ) change periodically with τ, Equation 20is still a periodic variable coefficient dynamic system. According to the Floquet-Lyapunov theory, the stabil-ity of the periodic variable coefficient system can be studied according to the eigenvalue λ of its transition matrix P [37]: If the modulus of all eigenvalues of P are less than 1, the system is asymptotically stable. If P has an eigenvalue whose modulus is greater than 1, the system is unstable.…”

Section: Posture Stability Analysis 41 the Floquet-lyapunov Theorymentioning

confidence: 99%

Posture Dynamic Modeling and Stability Analysis of a Magnetic Driven Dual-Spin Spherical Capsule Robot

et al. 2021

View full text Add to dashboard Cite

In order to realize the intervention operation in the unstructured and ample environments such as stomach and colon, a dual-spin spherical capsule robot (DSCR) driven by pure magnetic torque generated by the universal rotating magnetic field (URMF) is proposed. The coupled magnetic torque, the viscoelastic friction torque, and the gravity torque were analyzed. Furthermore, the posture dynamic model describing the electric-magnetic-mechanical-liquid coupling dynamic behavior of the DSCR in the gastrointestinal (GI) tract was established. This model is a second-order periodic variable coefficient dynamics equation, which should be regarded as an extension of the Lagrange case for the dual-spin body system under the fixed-point motion, since the external torques were applied. Based on the Floquet–Lyapunov theory, the stability domain of the DSCR for the asymptotically stable motion and periodic motion were obtained by investigating the influence of the angular velocity of the URMF, the magnetic induction intensity, and the centroid deviation. Research results show that the DSCR can realize three kinds of motion, which are asymptotically stable motion, periodic motion, and chaotic motion, according to the distribution of the system characteristic multipliers. Moreover, the posture stability of the DSCR can be improved by increasing the angular velocity of the URMF and reducing the magnetic induction intensity.

show abstract

Section: Posture Stability Analysis 41 the Floquet-lyapunov Theorymentioning

confidence: 99%

Posture Dynamic Modeling and Stability Analysis of a Magnetic Driven Dual-Spin Spherical Capsule Robot

et al. 2021

View full text Add to dashboard Cite

show abstract

“…It can be numerically determined in different ways. For example, a high order Runge±Kutta approach with 2n integrations of equation (47) over the time interval 0Y T has been used in [28]. In this work, we have tested two methods [29] to evaluate B.…”

Section: Numerical Evaluation Of the Monodromy Matrixmentioning

confidence: 99%

Fast Fourier nonlinear vibration analysis

Cardona

Lerusse

Géradin

1998

Computational Mechanics

View full text Add to dashboard Cite

We present an implementation of the multiharmonic balance method (MHB) where intensive use of the Fast Fourier Transform algorithm (FFT) is made at all stages of calculations. The MHB method is not modi®ed in essence, but computations are organized to obtain a very attractive method that can be applied systematically on general nonlinear vibration problems. The resulting nonlinear algebraic problem is solved by a particular implementation of a continuation method. Nonlinear vibration results are analyzed a posteriori by a Floquet method to determine their stability. The technique is applied on a series of problems of different nature, demonstrating the robustness and¯exibility of the approach. IntroductionThe multi-harmonic balance method (MHB) has been widely used to solve nonlinear vibration problems under periodic excitation. It ®nds applications in several ®elds of mechanical engineering, e.g. machine dynamics, vehicle dynamics, helicopter rotor blade analysis, structural dynamics.The method is well-known from literature. Urabe [1] investigated the convergence conditions of the method and presented numerical applications [2] using the classical Fourier transform. Lau et al. [3±5] and more recently [6] developed an incremental form of the method. Pierre et al. [7, 8] and Ferri [9] followed the same approach to solve nonlinear vibration problems involving dry-friction. Ling and Wu [10] introduced the use of the Fast Fourier Transform (FFT) algorithm in Urabe's formulation. The same path was followed by other authors [11±16]. The use of FFT permits an overwhelming gain of CPU time when computing the Fourier transform. Most authors use the FFT to switch displacements, velocities, accelerations and forces between the time and frequency domains. However, even if the original proposal by Urabe included the analytical expression of the Jacobian of the nonlinear algebraic problem, most of them avoid computing this matrix. Ling and Wu [10] used a Broyden method; Cameron and Grif®n [11] used either a Picard iteration or a Jacobian matrix evaluated by ®nite differences; Lewandowski [14] gave particular expressions for the contributions of nonlinear terms to the Jacobian for the case of geometric nonlinearities in the form of trigonometric expansions.In reference [15] we presented an approach which differs from most others in that an analytical expression for the Jacobian matrix of the nonlinear algebraic problem is developed for general applications. This fact allows to solve the nonlinear algebraic problem with utmost ef®-ciency, reaching quadratic convergence rate. We have shown how the Jacobian matrix can be computed from the Fourier transform of the time domain stiffness, damping and mass matrices of the system under analysis.The MHB leads to a nonlinear algebraic problem in which the solution should be searched in the Fourier transformed displacements versus excitation period space. The solution is found in the form of a nonlinear dynamic equilibrium path in this space for varying values of the excitation period. St...

show abstract