Academic literature on the topic 'Free and Forced vibrations'

Forces are measured at both ends of rigid cylinders with span 60 cm, performing transverse oscillations within an oncoming stream of water, at Reynolds number Re≈3800. Forced harmonic motions and free vibrations of uniform and tapered cylinders are studied. To study free motions, a novel force-feedback control system has been developed, consisting of: (a) a force transducer, which measures forces on a section of a cylinder moving forward at constant speed; (b) a computer using the measured force signal to drive in real time a numerical simulation of an equivalent mass-dashpot-spring system; (c) a servomotor and linear table which impose, also in real time, the numerically calculated motion on the cylinder section. The apparatus allows very low equivalent system damping and strict control of the parametric values and structure of the equivalent system.Calculation of the cross-correlation coefficient between forces at the two ends of the uniform cylinder reveals five distinct regimes as a function of the nominal reduced velocity Vrn: two regimes, for low and high values of Vrn, and far away from the value of VrS corresponding to the Strouhal frequency, show small correlation; two regimes immediately adjacent to, but excluding, VrS show strong correlation, close to 1; surprisingly, there is a regime containing the Strouhal frequency, within which correlation is low. Free vibrations with a 40[ratio ]1 tapered cylinder show that the regime of low correlation, containing the Strouhal frequency, stretches to higher reduced velocities, while lock-in starts at lower reduced velocities.When comparing the amplitude and phase of the lift coefficient measured for free and then for forced vibrations, we obtain close agreement, both for tapered and uniform cylinders. When comparing the cross-correlation coefficient, however, we find that it is much higher in the forced oscillations, especially for the uniform cylinder. Hence, although the force magnitude and phase may be replicated well in forced vibrations, the correlation data suggest that differences exist between free and forced vibration cases.

In the existing reports regarding free and forced vibrations of the beams, most of them studied a uniform beam carrying various concentrated elements using Bernoulli-Euler Beam Theory (BET) but without axial force. The purpose of this paper is to utilize the numerical assembly technique to determine the exact frequency-response amplitudes of the axially-loaded Timoshenko multi-span beam carrying a number of various concentrated elements (including point masses, rotary inertias, linear springs and rotational springs) and subjected to a harmonic concentrated force and the exact natural frequencies and mode shapes of the beam for the free vibration analysis. The model allows analyzing the influence of the shear and axial force and harmonic concentrated force effects and intermediate concentrated elements on the dynamic behavior of the beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate concentrated elements, an intermediate pinned support, applied harmonic force, left-end support and right-end support of Timoshenko beam are derived. After the derivation of the coefficient matrices, the numerical assembly technique is used to establish the overall coefficient matrix for the whole vibrating system. Finally, solving the equations associated with the last overall coefficient matrix one determines the exact dynamic response amplitudes of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force. Equating the determinant of the overall coefficient matrix to zero one determines the natural frequencies of the free vibrating system (the case of zero harmonic force) and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. The calculated vibration amplitudes of the forced vibrating systems and the natural frequencies of the free vibrating systems are given in tables for different values of the axial force. The dynamic response amplitudes and the mode shapes are presented in graphs. The effects of axial force and harmonic concentrated force on the vibration analysis of Timoshenko multi-span beam are also investigated.

Free and forced longitudinal oscillations of homogeneous rods of constant cross section are considered. Analytical and numerical methods for solving problems are used. With free vibration, numerical examples are shown for a rod with a jammed and free end and for a rod with a concentrated non-deformable mass at the end, due to which the mathematical model accordingly changes. Forced oscillations are considered for distributed and concentrated loads. The eigenmodes of oscillations characteristic for continually discrete sisites are obtained.

AbstractThe flow-induced vibrations of a circular cylinder, free to oscillate in the cross-flow direction and subjected to a forced rotation about its axis, are analysed by means of two- and three-dimensional numerical simulations. The impact of the symmetry breaking caused by the forced rotation on the vortex-induced vibration (VIV) mechanisms is investigated for a Reynolds number equal to $100$, based on the cylinder diameter and inflow velocity. The cylinder is found to oscillate freely up to a rotation rate (ratio between the cylinder surface and inflow velocities) close to $4$. Under forced rotation, the vibration amplitude exhibits a bell-shaped evolution as a function of the reduced velocity (inverse of the oscillator natural frequency) and reaches $1.9$ diameters, i.e. three times the maximum amplitude in the non-rotating case. The free vibrations of the rotating cylinder occur under a condition of wake–body synchronization similar to the lock-in condition driving non-rotating cylinder VIV. The largest vibration amplitudes are associated with a novel asymmetric wake pattern composed of a triplet of vortices and a single vortex shed per cycle, the ${\rm T} + {\rm S}$ pattern. In the low-frequency vibration regime, the flow exhibits another new topology, the U pattern, characterized by a transverse undulation of the spanwise vorticity layers without vortex detachment; consequently, free oscillations of the rotating cylinder may also develop in the absence of vortex shedding. The symmetry breaking due to the rotation is shown to directly impact the selection of the higher harmonics appearing in the fluid force spectra. The rotation also influences the mechanism of phasing between the force and the structural response.

The objective of this paper is to establish the formulation of the problem of nonlinear transverse forced vibrations of uniform multi-span beams, with several intermediate simple supports and general end conditions, including use of translational and rotational springs at the ends. The beam bending vibration equation is first written at each span and then the continuity requirements at each simple support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. The formulation is based on the application of Hamilton’s principle and spectral analysis to the problem of nonlinear forced vibrations occurring at large displacement amplitudes, leading to the solution of a nonlinear algebraic system using numerical or analytical methods. The nonlinear algebraic system has been solved here in the case of a four span beam in the free regime using an approximate method developed previously (second formulation) leading to the amplitude dependent fundamental nonlinear mode of the multi-span beam and to the corresponding backbone curves. Considering the nonlinear regime, under a uniformly distributed excitation harmonic force, the calculation of the corresponding generalised forces has led to the conclusion that the nonlinear response involves predominately the fourth mode. Consequently, an analysis has been performed in the neighbourhood of this mode, based on the single mode approach, to obtain the multi-span beam nonlinear frequency response functions for various excitation levels.

In milling operations, cutting edge impacts due to the interaction between the cutter and the workpiece excite vibrations. It is possible to distinguish between free, forced and self-excited vibrations. Chatter is a self-excited vibration that can occur in machining processes, and is considered to be a common limitation of productivity and quality. Stability lobes diagrams (SLDs) show the frontier between chatter-free milling operations, i.e. stable dominated by forced vibrations, and operations with chatter, i.e. unstable. These diagrams are usually obtained from impact hammer testing. However, this method requires trained personnel with advanced knowledge and it is not easily applied in engineering studies or operator training. This paper presents an experimental method that allows engineering students and operators-in-training to observe the chatter phenomenon and to distinguish between forced and chatter vibrations and identify process stability diagrams.

In this paper, free and forced vibrations of symmetric laminated composite plates are studied analytically by using a perturbation method where the analytical results for transverse displacement are compared with the numerical results. The external force is taken to be harmonic in time and having uniform amplitude.

The free and forced response of spinning, viscoelastic, Rayleigh shafts is studied. Viscoelasticity is included using the three parameter solid model. The closed form polynomial frequency equation and integral expressions for the response to a general forcing function are derived. A convenient decay parameter is described. Results are given for natural frequencies and decay rates as functions of shaft rotation speed, stiffness, and viscosity. It is found that shaft materials are possible which have desirable damping and natural frequency characteristics. A parameter case is discussed in which natural frequency and damping simultaneous increase, while stiffness is held constant. Also, the special case of forced response to a step load is derived and used to illustrate the combined effects of viscoelasticity and gyroscopic forces.

In working processes, mechanical systems are often affected by both internal and external forces, which are the cause of the forced vibrations of the structures. They can be destroyed if the amplitude of vibration reaches a high enough value. One of the most popular ways to reduce these forced vibrations is to attach tuned mass damper (TMD) devices, which are commonly added at the maximum displacement point of the structures. This paper presents the computed results of the free vibration and the vibration response of the space frame system under an external random load, which is described as a stationary process with white noise. Static and dynamic equations are formed through the finite element method. In addition, this work also establishes artificial neural networks (ANNs) in order to predict the vibration response of the first frequencies of the structure. Numerical studies show that the data set of the TMD device strongly affects the first frequencies of the mechanical system, and the proposed artificial intelligence (AI) model can predict exactly the vibration response of the first frequencies of the structure. For the forced vibration problem, we can find optimal parameters of the TMD device and thus obtain minimum displacements of the structure. The results of this work can be used as a reference when applying this type of structure to TMD devices.

A general theory for the free and forced responses of elastically connected parallel structures is developed. It is shown that if the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for , then the stiffness operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on . This leads to the definition of energy inner products defined on . When a normal mode solution is used to develop the free response, it is shown that the natural frequencies are the square roots of the eigenvalues of an operator that is self-adjoint with respect to the energy inner product. The completeness of the eigenvectors in is used to develop a forced response. Special cases are considered. When the individual stiffness operators are proportional, the problem for the natural frequencies and mode shapes reduces to a matrix eigenvalue problem, and it is shown that for each spatial mode there is a set of intramodal mode shapes. When the structures are identical, uniform, or nonuniform, the differential equations are uncoupled through diagonalization of a coupling stiffness matrix. The most general case requires an iterative solution.