In-plane antisymmetric response of cables through bifurcation under symmetric sinusoidally time-varying load

Analysis and results for in-plane non-linear antisymmetric responses of a cable, supported at the same level, through bifurcation under in-plane symmetric sinusoidally time-varying load are presented. The non-linear equation of the in-plane motion of the cable is solved by a Galerkin method and the harmonic balance method. From the computed results the frequency range, where the antisymmetric response occurs, varies with the sag-to-span ratio of the cable and is broad in the particular sag-to-span ratios. The second unstable region is important compared with the principal unstable region. Strong coupling between symmetric and antisymmetric modes is observed in the unstable regions for the particular sag-to-span ratios.     

Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cables

This paper presents a model formulation capable of analyzing large-amplitude free vibrations of a suspended cable in three dimensions. The virtual work-energy functional is used to obtain the non-linear equations of three-dimensional motion. The formulation is not restricted to cables having small sag-to-span ratios, and is conveniently applied for the case of a specified end tension. The axial extensibility effect is also included in order to obtain accurate results. Based on a multi-degree-of-freedom model, numerical procedures are implemented to solve both spatial and temporal problems. Various numerical examples of arbitrarily sagged cables with large-amplitude initial conditions are carried out to highlight some outstanding features of cable non-linear dynamics by accounting also for internal resonance phenomena. Non-linear coupling between three- and two-dimensional motions, and non-linear cable tension responses are analyzed. For specific cables, modal transition phenomena taking place during in-plane vibrations and ensuing from occurrence of a dominant internal resonance are observed. When only a single mode is initiated, a higher or lower mode can be accommodated into the responses, making cable spatial shapes hybrid in some time intervals.     

The effect of cable loosening on seismic response of a prestressed concrete cable-stayed bridge

In conventional non-linear seismic analyses of cable-stayed bridges, the non-linear characteristics of the girders, stay cables and towers are considered. The non-linearity caused by cable loosening should also be considered because a large axial force fluctuation is generated in the cables of a prestressed concrete (PC) cable-stayed bridge that is subjected to strong seismic motion. In this paper, the possibility of the cable loosening in a PC cable-stayed bridge is discussed by using a cable model that can express the cable loosening. Furthermore, the effect of the cable loosening on the responses of the cables, girder and towers is evaluated using the mean value for three seismic waves. Numerical analytic results imply that the cable loosening appears in the bottom cables of the multi-cable system and the dynamic response of the bridge is slightly increased.     

Non-linear vibrations of cables in three dimensions,part I: Non-linear free vibrations

Analysis and results for non-linear free vibrations of both horizontal and inclined cables in three dimensions are presented. Sag-to-span ratios of the cables are not limited to being small. Computed results are presented for various geometrical and material parameters. The major findings are that the geometrical non-linearity may be of the stiffening type or the softening type, depending on the sag-to-span ratio, and the stiffness of out-of-plane vibrations is affected by the corresponding in-plane vibration near a resonant frequency due to non-linear coupling between out-of-plane and in-plane vibrations.     

A method of stability analysis for non-linear vibration of beams

In this paper, a method of stability analysis for the large amplitude, steady state response of a non-linear beam under periodic excitation is presented. The stability problem is investigated by studying the behavior of a small perturbation of the steady state response which results in a coupled Hill-type equation. The problem is transformed by the harmonic balance method into an eigenvalue problem of a non-symmetric matrix. The effectiveness and the accuracy of the proposed method for a Mathieu equation are examined and the application to the stability analysis of the non-linear vibrations of a beam is presented.     

Whirling motion of a shallow cable with viscous dampers

The paper deals with the non-linear dynamic analysis of cables with a pair of viscous dampers close to one support. Such cables are characterized by a sag-to-chord-length ratio below 0.02, for which natural frequencies for the vertical and the horizontal vibrations are pair-wise close. Under resonance the non-linear coupling of pairs of modes may cause whirling harmonic motions around the chord line. Whirling motion may occur after bifurcation from single-mode response for harmonic loads in either vertical or horizontal direction. The non-linear features are included in the two coupled modes, while all other modes are treated as linear. The motion is discretized by expansion in terms of the damped complex eigenfunctions. The applied base functions fulfil the transition condition at the damper, leading to fast convergence of the expansion. It is demonstrated that the behaviour of the whirling motion is controlled primarily by the damper acting in the direction of the unloaded mode, whereas the magnitude of the damper in the loaded mode is less important. If the dampers in the vertical and horizontal direction are close to the optimal value of the corresponding taut cable case, substantial reduction of the vibration level of the whirling mode as well as the frequency interval of its occurrence is attained.     

Response characteristics of local vibrations in stay cables on an existing cable-stayed bridge

This paper examines local parametric vibrations in the stay cables of a cable-stayed bridge. The natural frequencies of the global modes are obtained by using a three-dimensional FE model. The global motions generated by (1) sinusoidal excitations using exciter, (2) a traffic loading, and (3) an earthquake are analyzed by using the modal analysis method or the direct integration method. The local vibration of stay cable is calculated by using a model in which inclined cable is subjected to time-varying displacement at one support during global motions. This paper describes the properties of the local vibrations in stay cables under these dynamic loadings by using an existing cable-stayed bridge.     

Time-average hologram interferometry of periodic,non-cosinusoidal vibrations

In the present paper, the method of time-average hologram interferometry has been applied to study periodic, non-cosinusoidal vibrations represented by a Jacobian elliptic function. For such vibrations the characteristic fringe function has been evaluated by making use of an equation derived from considerations of the effect of motion on coherence. Techniques of holographic addition and subtraction and extended pulse stroboscopic holography have been applied to analyze these vibrations in detail. Graphical representation of the fringe irradiance distributions in the reconstructed image is given in all the cases.     

SOME ASPECTS OF THE INVESTIGATION OF RANDOM VIBRATION INFLUENCE ON RIDE COMFORT

Contemporary vehicles must satisfy high ride comfort criteria. This paper attempts to develop criteria for ride comfort improvement. The highest loading levels have been found to be in the vertical direction and the lowest in lateral direction in passenger cars and trucks. These results have formed the basis for further laboratory and field investigations. An investigation of the human body behaviour under random vibrations is reported in this paper. The research included two phases; biodynamic research and ride comfort investigation. A group of 30 subjects was tested. The influence of broadband random vibrations on the human body was examined through the seat-to-head transmissibility function (STHT). Initially, vertical and fore and aft vibrations were considered. Multi-directional vibration was also investigated. In the biodynamic research, subjects were exposed to 0·55, 1·75 and 2·25 m/s² r.m.s. vibration levels in the 0·5- 40 Hz frequency domain. The influence of sitting position on human body behaviour under two axial vibrations was also examined. Data analysis showed that the human body behaviour under two-directional random vibrations could not be approximated by superposition of one-directional random vibrations. Non-linearity of the seated human body in the vertical and fore and aft directions was observed. Seat-backrest angle also influenced STHT. In the second phase of experimental research, a new method for the assessment of the influence of narrowband random vibration on the human body was formulated and tested. It included determination of equivalent comfort curves in the vertical and fore and aft directions under one- and two-directional narrowband random vibrations. Equivalent comfort curves for durations of 2·5, 4 and 8 h were determined.     

Holographic interferometry of non-sinusoidal vibrations

In this paper, the techniques of extended pulse stroboscopic holography and holographic subtraction have been applied to the study of periodic, non-sinusoidal vibrations represented by a Jacobian elliptic function. Fringe irradiance distribution in reconstructed images has been evaluated for the two cases. For this purpose we make use of an expression for the characteristic fringe function derived from considerations of the effect of motion on the coherence. It is shown that the above-mentioned techniques are more advantageous for the measurement of periodic, non-sinusoidal vibrations than in the case of pure sinusoidal vibrations.     

Output response identification in a multistable system for piezoelectric energy harvesting

In this paper we examine in detail the multiple responses of a novel vibrational energy harvester composed of a vertical bistable beam whose complex non-linear behavior is tuned via magnetic interaction. The beam was excited horizontally by a harmonic inertial force while mechanical vibrational energy is converted to electrical power through a piezoelectric element. The bistable laminate beam coupled to the piezoelectric transducer showed a variety of complex responses in terms of the beam displacement and harvested power output. The range of vibration patterns in this non-linear system included single-well oscillations and snap-through vibrations of periodic and chaotic character. Harvested power was found to be strongly dependent on the vibration pattern with nonlinearities providing a broadband response for energy harvesting. Wavelet analysis of measured voltage, displacement and velocity time histories indicated the presence of a variety of nonlinear periodic and also chaotic phenomena. To measure the complexity of response time series we applied phase portraits and determine stroboscopic points and multiscale entropy. It is demonstrated that by changing parameters such as the magnetic interaction, the characteristics of the bistable laminate harvester, such as the natural frequency, bandwidth, vibration response and peak power can be readily tailored for harvesting applications.     

A novel sensitivity-based method for damage detection of structures under unknown periodic excitations

This paper presents a technique for damage detection in structures under unknown periodic excitations using the transient displacement response. The method is capable of identifying the damage parameters without finding the input excitations. We first define the concept of displacement space as a linear space in which each point represents displacements of structure under an excitation and initial condition. Roughly speaking, the method is based on the fact that structural displacements under free and forced vibrations are associated with two parallel subspaces in the displacement space. Considering this novel geometrical viewpoint, an equation called kernel parallelization equation (KPE) is derived for damage detection under unknown periodic excitations and a sensitivity-based algorithm for solving KPE is proposed accordingly. The method is evaluated via three case studies under periodic excitations, which confirm the efficiency of the proposed method.     

Non-linear control of torsional and bending vibrations of oilwell drillstrings

Drillstring dynamics is highly non-linear in nature and its model can only be described by a set of non-linear differential equations. In addition to this complexity, the drillstring dynamics are not linearly controllable and thus linear control methods are not suitable for suppressing the coupled torsional and lateral vibrations of a rotating drillstring. In this paper a non-linear dynamic inversion control design method is used to suppress the lateral and the torsional vibrations of a non-linear drillstring. It was found that the designed controller is effective in suppressing the torsional vibrations and reducing the lateral vibrations significantly.     

Generalised modal stability of inclined cables subjected to support excitations

Parametric excitation is of concern for cables such as on cable-stayed bridges, whereby small amplitude end motion can lead to large, potentially damaging, cable vibrations. Previous identification of the stability boundaries for the onset of such vibrations has considered only a single mode of the cable, ignoring non-linear coupling between modes, or has been limited to special cases. Here multiple cable modes in both planes are included, with support excitation close to any natural frequency. Cable inclination, sag, parametric and direct excitation and nonlinearities, including modal coupling, are included. The only significant limitation is that the sag is small. The method of scaling and averaging is used to find the steady-state amplitude of the directly excited mode and, in the presence of this response, to define stability boundaries of other modes excited parametrically or through nonlinear modal coupling. It is found that the directly excited response significantly modifies the stability boundaries compared to previous simplified solutions. The analysis is validated by a series of experimental tests, which also identified another nonlinear mechanism which caused significant cable vibrations at twice the excitation frequency in certain conditions. This new mechanism is explained through a refinement of the analysis.     

Effects of cables damage on vertical and torsional eigenproperties of suspension bridges

This paper presents a continuum model for the nonlinear coupled vertical and torsional vibrations of suspension bridges with arbitrary damage in one main cable and, after pursuing a suitable linearization of the equations of motion, an investigation of damage effects on modal parameters. Damage is modeled as a diffused loss of cross-section representing the typical effect of fretting fatigue and it is introduced in the formulation by enforcing relevant literature results providing analytical solution for the static response of damaged suspended cables. The coupled nonlinear equations of motion of the damaged bridge, including the effects of shear deformation, rotary inertia and warping of the cross-section of the girder, are derived by application of Hamilton?s principle. In this way, the equations of motion available in the literature for undamaged suspension bridges are generalized to the presence of arbitrary damage in one main cable and the resulting eigenfrequencies and eigenfunctions are derived in an analytical fashion. An extensive parametric investigation is finally presented to discuss damage effects on eigenfunctions and eigenfrequencies under variation of practically meaningful parameters.     

ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES

An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived.     

Three dimensional vibration generators with a single rotational input

This paper presents a novel device capable of generating three-dimensional vibrations with a single-axis of rotation. The device resembles a vibration motor with an eccentric weight, but the weight is allowed to move up and down in parallel to the axis of rotations. While spinning of the eccentric weight causes lateral vibrations, vertical vibrations are excited by superposing a periodic torque on the rotary shaft. The frequency and magnitude of vertical oscillations can be independently regulated. Since the vertical natural frequency is sensitive to rotational speeds, maximum vertical oscillations can be achieved by properly adjusting the rotational speed according to the excitation frequency. Relations between the excitation torque and the vertical shaking force are examined using the frequency response for the linearized system. Numerical simulations on the original nonlinear system are conducted to verify the performance of vertical oscillations.     

DYNAMIC MODELLING AND VIBRATION ANALYSIS OF A FLEXIBLE CABLE-STAYED BEAM STRUCTURE

In this paper, the non-linear vibration of a cable-stayed beam with time-varying length and tension in the cable is investigated. A set of non-linear, time-varying differential equations describing this coupling system is derived by Hamilton's principle and the finite element method. According to the results of numerical simulation, the tension of the cable is related to the cable length, which in turn is a function of the longitudinal and transverse displacements of the cable. Furthermore, it is shown that the tension and length of the cable can be considerably different by using linear and non-linear models.