Resonance in the Cart-Pendulum System—An Asymptotic Approach

The major objective of this research is to study the planar dynamical motion of 2DOF of an auto-parametric pendulum attached with a damped system. Using Lagrange’s equations in terms of generalized coordinates, the fundamental equations of motion (EOM) are derived. The method of multiple scales (MMS) is applied to obtain the approximate solutions of these equations up to the second order of approximation. Resonance cases are classified, in which the primary external and internal resonance are investigated simultaneously to establish both the solvability conditions and the modulation equations. In the context of the stability conditions of these solutions, the equilibrium points are obtained and graphically displayed to derive the probable steady-state solutions near the resonances. The temporal histories of the attained results, the amplitude, and the phases of the dynamical system are depicted in graphs to describe the motion of the system at any instance. The stability and instability zones of the system are explored, and it is discovered that the system’s performance is stable for a significant number of its variables.

Some possible Lagrangians for the quadratically damped system

In this paper, some possible Lagrangians for the quadratically damped systems are investigated. The corresponding classical Hamiltonians are investigated with Hamilton equations. The quantum Hamiltonians are also constructed so that they may be Hermitian. As an example, the quantum mechanics for the harmonic oscillator with a small quadratic damping is discussed.

Resonant Motions of Dynamic Offshore Structures in Large Waves

In marine engineering, the dynamics of fixed offshore structures (for oil and gas production or for wind turbines) are normally found by modelling of the motion by a classical mass-spring damped system. On slender offshore structures, the loading due to waves is normally calculated by applying a force which consists of two parts: a linear “inertia/mass force” and a non-linear “drag force” that is proportional to the square of the velocity of the particles in the wave, multiplied by the direction of the wave particle motion. This is the so-called Morison load model. The loading function can be expanded in a Fourier series, and the drag force contribution exhibits higher order harmonic loading terms, potentially in resonance with the natural frequencies of the system. Currents are implemented as constant velocity terms in the loading function. The paper highlights the motion of structures due to non-linear resonant motion in an offshore environment with high wave intensity. It is shown that “burst”/“ringing” type motions could be triggered by the drag force during resonance situations.

On the escape of a resonantly excited couple of particles from a potential well

The escape dynamics of a damped system of two coupled particles in a truncated potential well under biharmonic excitation are investigated. It is assumed that excitation frequencies are tuned to the modal natural frequency of the relative motion and to the modal frequency of the centre of mass on the bottom of the potential well. Although the escape is essentially a non-stationary process, the critical force strongly depends on the stationary amplitude of the relative vibrations within the pair of masses. The characteristic escape curve for the critical force moves up on the frequency-escape threshold plane with increasing relative vibrations, which can be interpreted as a stabilizing effect due to the high-frequency excitation. To obtain the results, new modelling techniques are suggested, including the reduction in the effect of the high-frequency excitation using a probability density function-based convolution approach and an energy-based approach for the description of the evolution of the slow variables. To validate the method, the coupled pair of particles is investigated with various model potentials.

Ein semiaktiver Ansatz zur Verbesserung des dynamischen Verhaltens/Vibration damping on industrial robots – A semi-active approach to improve the dynamic behavior

Von Industrierobotern wird zunehmend eine hohe Bahn genauigkeit und ein gutes Störunterdrückungsverhalten gefordert. Um dem gerecht zu werden, wird hier ein semiaktiver Dämpfungsansatz vorgestellt, der Antriebsstrangschwingungen aktorbasiert dämpft. Damit geht ein stärker gedämpftes Systemverhalten einher, wodurch sich die Regelverstärkungsfaktoren erhöhen lassen. Das Resultat ist ein verbessertes Gesamtverhalten, das an einem „Kr210–2“ von Kuka mit semiaktiver Dämpfung an Achse 1 nachgewiesen wird.   Industrial robots are increasingly required to provide high path accuracy and good disturbance rejection behavior. In order to achieve this, a semi-active damping approach is presented, which damps drive train vibrations actuator-based. This leads to a more damped system behavior, allowing control gains to be increased. The result is an improved overall behavior, which is demonstrated on a Kuka Kr210–2 with semi-active damping on axis 1.

On the Formation Mechanism of the Seasonal Persistence Barrier

The mechanism of the seasonal persistence barrier (SPB) is studied in the framework of an autoregressive (AR) model. In contrast to the seasonal variance, whose minimum is modulated mainly by the minimum growth rate or noise forcing, the SPB is caused primarily by the declining growth rate or increasing noise forcing, instead of the minimum/maximum of the growth rate or noise forcing. In other words, the SPB is caused by the declining signal-to-noise ratio (SNR) rather than the weakest SNR. In a weakly damped system, the phase of the SPB is delayed from that of declining SNR by about a season. The mechanism is further applied to explain the observed SST variability in the tropical and North Pacific. For the tropical Pacific, the spring SPB could be caused by the decreasing growth rate from September to March and weak annual mean damping rate, instead of the minimum growth rate in spring. Over the North Pacific, the increasing noise forcing from March to June may lead to the summer SPB. Our mechanism provides a null hypothesis for understanding the SPB of climate variability.

Single Degree of Freedom Forced Vibration System

This chapter concerns the study of forced vibration of a single degree of freedom system, treating undamped and damped system under harmonic, periodic, and arbitrary loading with different cases and examples. Passing by all components of the general solution of an undamped forced system, which are a transient solution, depends only on initial conditions, transient solution due to the load at the end the stationary solution. In this chapter, a study of the dynamic influence factor depends on the ration between load frequency and structure one is presented.

Multi-Degrees of Freedom System and Hydrodynamic Principle

A system with one degree of freedom is far from reality, because we do not take into account all the degrees of freedom. In order to be close to the reality, it is necessary to use a system with several degrees of freedom. Efforts in this chapter are concentrated to the study of multi-degrees of freedom system, whether for a free undamped and forced damped system, by detailing the modal superposition method as well as a coupled coordinates. We finish the chapter with hydrodynamic study using Hozner method as well as some applications.