Modeling and Stability Analysis for the Vibrating Motion of Three Degrees-of-Freedom Dynamical System Near Resonance

The focus of this article is on the investigation of a dynamical system consisting of a linear damped transverse tuned-absorber connected with a non-linear damped-spring-pendulum, in which its hanged point moves in an elliptic path. The regulating system of motion is derived using Lagrange’s equations, which is then solved analytically up to the third approximation employing the approach of multiple scales (AMS). The emerging cases of resonance are categorized according to the solvability requirements wherein the modulation equations (ME) have been found. The stability areas and the instability ones are examined utilizing the Routh–Hurwitz criteria (RHC) and analyzed in line with the solutions at the steady state. The obtained results, resonance responses, and stability regions are addressed and graphically depicted to explore the positive influence of the various inputs of the physical parameters on the rheological behavior of the inspected system. The significance of the present work stems from its numerous applications in theoretical physics and engineering.

Analyzing the Stability for the Motion of an Unstretched Double Pendulum Near Resonance

This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics.

An Analytical Spectral Model for Infragravity Waves over Topography in Intermediate and Shallow Water under Non-breaking Conditions

The theoretical model for group-forced infragravity (IG) waves in shallow water is not well established for non-breaking conditions. In the present study, analytical solutions of the group-forced IG waves at (, hx =bottom slope, Δk =group wavenumber, h =depth) in intermediate water and at in shallow water are derived separately. In case of off-resonance (, where is the resonant departure parameter, cg = group speed) in intermediate water, additional IG waves in quadrature with the wave group forcing (hereinafter as the non-equilibrium response or component) are induced at relative to the equilibrium bound IG wave solution of Longuet-Higgins and Stewart (1962) in phase with the wave group. The present theory indicates that the non-equilibrium response is mainly attributed to the spatial variation of the equilibrium bound IG wave amplitude instead of group-forcing. In case of near-resonance () in shallow water, however, both the equilibrium and non-equilibrium components are at the leading order. Based on the nearly-resonant solution, the shallow water limit of the local shoaling rate of bound IG waves over a plane sloping beach is derived to be ~ h−1 for the first time. The theoretical predictions compare favorably with the laboratory experiment by Van Noorloos (2003) and the present numerical model results using SWASH. Based on the proposed solution, the group-forced IG waves over a symmetric shoal are investigated. In case of off-resonance, the solution predicts a roughly symmetric reversible spatial evolution of the IG wave amplitude, while in cases of near- to full- resonance the IG wave is significantly amplified over the shoal with asymmetric irreversible spatial evolution.

Experimental evaluation of the effect of soil water fluctuations in the dynamic behavior of machines

In this study the dynamic response of a machine-foundation-soil system was investigated experimentally and theoretically. The objective of this work is to analyze the effects of the water table fluctuations in the soil on the response of the foundation and machine subjected to dynamic loads at frequencies ranging from 30 to 50 Hz. A physical model test was developed to simulate a machine-foundation-soil system, with measurements of the machine vibrations and the shear wave velocity of the soil. It is found that the water level produced significant changes in the shear wave velocity of the soil and, thus, in the natural frequencies of the system. For a fully saturated soil the vibration levels increased due to a working condition near resonance. The results showed a good agreement between the experimental vibration measurements and the predictions based on the theory used in foundation design, when considering the appropriate soil parameters. It is concluded that proper estimation of soil parameters is of high importance in the design process of machine foundations.

Sources of Harmonic Force and Speed in Mechatronic Automatic Systems

To study resonance and near-resonance phenomena, a symbolic (complex) method was used, which makes it possible to significantly increase productivity, simplify and formalize mathematical transformations. Parallel and sequential connections of elements of a mechanical system with a source of harmonic force or a source of harmonic speed as a source of external mechanical harmonic action are considered. The analytical descriptions of resonance in theoretical mechanics courses correspond to parallel connection. There are devices, in a satisfactory approximation, capable of performing the functions of sources of force and sources of speed. The source of harmonic speed can be a crank-yoke drive and a flywheel with a large moment of inertia. The source of the harmonic force can be the rod of the pneumatic cylinder, the cavity of which communicates with the cavity of another pneumatic cylinder, the diameter of which is immeasurably higher than that of the first, and the piston performs harmonic oscillations. The mechanical harmonic influences described in the courses of theoretical mechanics correspond to the source of the force. Four modes are described — resonances and antiresonances of forces and velocities. The use of the symbolic (complex) method has significantly simplified the study of resonance and near-resonance phenomena, in particular, it has made it possible to deeply unify and formalize the consideration of various mechanical systems. The cumbersome and time-consuming operations associated with the preparation and solution of differential equations have been replaced by simple algebraic transformations. Resonance and antiresonance of forces, resonance and antiresonance of velocities are determined.

On the Existence of Time Delay for Rotating Beam with Proportional–Derivative Controller

A rotating beam at varying speed mathematical model is studied. Multiple time scales method is applied to the nonlinear system of differential equations and investigated the system behavior approximate solution in the instance of resonance case. We studied the system in case of applying the delayed control on the displacement and the velocity with Proportional–derivative (PD) controller. The consistency of the steady state solution in the near-resonance case is reviewed and analyzed using the Routh-Huriwitz approach. The factors on the steady state solution of the various parameters are recognized and discussed. Simulation effects are obtained using MATLAB software package. Different response curves are involved to show and compare controller effects at various system parameters.