Temporal Equation Of Motion Research Articles

Circuit quantum electrodynamic model of dissipative-dispersive Josephson traveling-wave parametric amplifiers

We present a quantum mechanical model for a four-wave mixing Josephson traveling-wave parametric amplifier including substrate losses and associated thermal fluctuations. Under the assumption of a strong undepleted classical pump tone, we derive an analytic solution for the bosonic annihilation operator of the weak signal photon field using temporal equations of motion in a reference timeframe, including chromatic dispersion. From this result, we can predict the asymmetric gain spectrum of a Josephson traveling-wave parametric amplifier due to non-zero substrate losses. We also predict the equivalent added input noise including quantum fluctuations as well as thermal noise contributions. Our results are in excellent agreement with recently published experimental data.

Nonlinear Vibrations and Frequency Response Analysis of a Cantilever Beam Under Periodically Varying Magnetic Field

In this paper, nonlinear vibration of a cantilever beam with tip mass subjected to periodically varying axial load and magnetic field has been studied. The temporal equation of motion of the system containing linear and nonlinear parametric excitation terms along with nonlinear damping, geometric and inertial types of nonlinear terms has been derived and solved using method of multiple scales. The stability and bifurcation analysis for three different resonance conditions were investigated. The numerical results demonstrate that while in simple resonance case with increase in magnetic field strength, the system becomes unstable, in principal parametric or simultaneous resonance cases, the vibration can be reduced significantly by increasing the magnetic field strength. The present work will be very useful for feed forward vibration control of magnetoelastic beams which are used nowadays in many industrial applications.

Nonlinear response of a vertically moving viscoelastic beam subjected to a fluctuating contact load

In the present work, the nonlinear response of a vertically moving viscoelastic beam subjected to a periodically varying contact load is investigated. The generalized Galerkin’s method is used to discretize the nonlinear partial differential equation of motion into the temporal equation of motion. The temporal equation of motion contains many nonlinear terms such as cubic geometric and inertial nonlinear terms, nonlinear damping term, and nonlinear parametric excitation terms in addition to forced excitation and parametric excitation terms. The first-order approximate solutions are obtained by using the method of multiple scales, and the stability and bifurcations of the obtained steady-state responses are studied. Extensive numerical simulations are presented to illustrate the influences of various types of system parameters for different resonance conditions. A significant amount of vibration reduction is obtained with the increase in the material loss factor. The results obtained by numerically solving the temporal equation of motion are found to be in good agreement with the results determined by the method of multiple scales. The obtained results are useful for reduction in the vibration of the viscoelastic flexible beam with prismatic joint or single-link viscoelastic Cartesian manipulator with payload subjected to a sinusoidally varying contact load.

Analysis of dynamic instability for arbitrarily laminated skew plates

The dynamic instability and nonlinear response of rectangular and skew laminated plates subjected to periodic in-plane load are studied. Based on von Karman plate theory, the large amplitude dynamic equations of thin laminated plates are derived by applying the approach of generalized double Fourier series. On the assumed mode shape, the governing equations are reduced to the Mathieu equation using Galerkin's method. The incremental harmonic balance (IHB) method is applied to solve the nonlinear temporal equation of motion, and the region of dynamic instability is determined in this work. Calculations are carried out for isotropic, angle-ply and arbitrarily laminated plates under two cases of boundary conditions. The principal region of dynamic instability associated with the effect of the stacking sequence of lamination and the skew angle of plate are also investigated and discussed. The results obtained indicated that the instability behavior of the system is determined by the several parameters, such as the boundary condition, number of the layers, stacking sequence, in-plane load, aspect ratio, amplitude and the skew angle of plate.

On the Steady State Response of a Cantilever Beam Partially Immersed in a Fluid and Carrying an Intermediate Mass

This paper presents a study on the nonlinear steady state response of a slender beam partially immersed in a fluid and carrying an intermediate mass. The model is developed based on the large deformation theory with the constraint of inextensible beam, which is valid for most engineering structures. The Lagrangian dynamics in conjunction with the assumed mode method is utilized in deriving the non-linear unimodal temporal equation of motion. The distributed and concentrated sinusoidal loads are accounted for in a consistent manner using the assumed mode method. The non-linear equation of motion is, analytically, solved using the single term harmonic balance (SHB) and the two terms harmonic balance (2HB) methods. The stability of the system, under various loading conditions, is investigated. The results are presented, discussed and some conclusions on the partially immersed beam nonlinear dynamics are extracted.

LATEST DEVELOPMENTS ON THE DYNAMIC STABILITY AND NONLINEAR PARAMETRIC RESPONSE OF GEOMETRICALLY IMPERFECT RECTANGULAR PLATES

The authors present a rational analysis of the effect of initial geometric imperfections on the dynamic stability and nonlinear parametric response of general rectangular plates, the plate theory used in the analysis may described as the dynamic analog of the von Kármán’s large deflection theory and is derived in terms of the stress function, the lateral displacement and the initial geometric imperfection. The governing equations are satisfied using the orthogonality properties of the assumed functions. The temporal response of the system is analyzed using a first-order asymptotic method and various types of resonances are investigated. The temporal equations of motion describing the nonlinear dynamic behaviour of the imperfect plates are also solved using a direct integration method and the results are compared with those obtained by the asymptotic method.