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The paper investigates the natural periods of symmetrically cracked semi-circular masonry arches. The cracks are present due to previous actions, additionally new cracks do not appear and the depth of the cracks do not change during motion, which assumptions are acceptable, if the amplitude of the vibration is relatively small. During vibration the states of the cracks can change from open to closed.
The motion of the structures consists of states with linear vibrations. In our work we deal with non-breathing and breathing cracks. We speak about non-breathing crack, if the state of the crack do not change during vibration. If the cracks can change their states during vibration, then we refer to the cracks as breathing cracks. A new approximate approach is presented, which is capable of handling the changes in the state of the cracks in the case of breathing cracks.
The motion is periodic if during a half-period the kinetic energy vanishes twice: at the beginning and at the end of a half-period. Between the two extremes the boundary and continuity conditions change, which result in the changing of the modal coordinates in various states of the cracks. In our solutions some parasitic results may appear as well, which are sorted out during analysis.
In the paper the position of a crack and the depth of the cracks are under examination. The goal of the work is to analyse the linear and the nonlinear normal modes and natural periods of cracked masonry arches, because these fundamental periods can refer to the magnitude of the degradation in the system, which may help to monitor the safety of structures.

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... The analysis of the free vibration of damaged arch is extensively investigated in [29,30], however, in these works the breathing effect of the cracks are not taken into account. The analysis of cracked continua regarding breathing cracks was done in [31,32]. In the latter work the symmetric, free vibration of a quasi-continuous cracked arch with small displacements and breathing cracks was analysed. ...

... In the current paper the symmetric vibration of cracked semi-circular masonry arches with large displacements is analysed, which was not analysed in the literature till now to the knowledge of the authors. In the literature of arches' free vibration with breathing cracks the mostly similar work is [32]. This paper analyse the problem with a quasi-continuous model, in which the small displacement theory is applied. ...

The paper investigates the energy- and amplitude-dependent natural periods of symmetrically cracked semi-circular arches, which are loaded by their self-weight and horizontal support displacement. The cracks appear due to previous static effects. Symmetric free vibration of the arches is under analysis. The symmetry condition results in an SDoF system, but the latter purpose of the authors is to analyse nonsymmetric modes with an MDoF model.
To analyse the problem an energy-based approach is developed. The motion of a structure consists of states with nonlinear vibrations, each of which has constant energy content. Between these phases inelastic impacts occur, which have effect only on the velocity field, but have no effect on the displacements, because the time duration of the impacts is assumed to be zero.
In the work the position of the cracks and other geometrical parameters are under examination and the energy-, amplitude- and support displacement magnitude-dependent natural periods are obtained. These results may help to monitor the historical buildings structural safety, because with the measured natural period and amplitude one can determine the safety of the structure.

... In this study, because of discussion about multi-frequencies we consider one of the combinational resonance secular terms. This term is: 12nn (28) To eliminating the secular terms, we assumed a perturbation in the following form: ...

In this study, under harmonic multiple frequency excitations, the dynamic response of a cracked cantilever beam is investigated. The breathing crack model is assumed to show the nonlinear behaviour of a transverse crack. The first mode of vibration and the single degree freedom lumped system is considered to simplify the case study. Because of applying the multiple frequency excitations, the analysis is applied in a combinational resonance. Multiple time scales method is employed to solve the motion equation of the crack, and the nonlinear vibrational responses are obtained. Then, by changing the crack parameters and frequency of the excitations, the different dynamic responses of the crack are demonstrated. The proposed model shows that the crack parameters analysis in nonlinear vibration of multiple excitations could be an appropriate method to recognise the crack and the depth of the damage. Results indicate that the beam analysis under multiple frequency excitations is more sensitive than the single frequency excitation to illustrate the impacts of the crack parameters on its vibrational nonlinearity responses.

... Neglecting the higher modes introduces a numerical damping into the solution [12]. In [13], a general method was introduced to find periodic orbits based on modal analysis. ...

In this paper, the dynamic behaviour of a family of piecewise linear structures, namely the vibration of beams on block-and-tackle suspension system is analysed. The regularity of the vibration modes in one of the linear states induces non-harmonic, yet periodic free vibration modes. The periodicity constraint of the continuous structure is formulated using modal analysis in the regular state. The required number of modes in the finite modal analysis is specified so that the numerical damping caused by the omitted modes does not change the periodic or non-periodic nature of the free vibration of the continuous structure. It is shown, that the application of five excess passive modes allows to draw conclusions about the behaviour of the continuous structure. The periodic behaviour depends on the number and position of the suspension points and the number of the active vibration modes. Analysis of the limits of the periodic behaviour reveals that suspension points close to the middle of the beam, or first few active vibration modes result in periodic vibration of the nonlinear system.

Modal analysis is an often-used method for the solution of dynamical problems in engineering practice. In the case of continuous systems and discretization methods (e.g., FEM) with a large number of degrees of freedom, the reduced modal analysis is used, where only a selected number of modes are applied, while the higher modes are neglected. A piecewise linear elastic structure has state-dependent stiffness and as such, state-dependent vibration modes. During the dynamic modal analysis of such systems, one has to switch between the modes. During the switch a part of the kinetic and potential energy is transferring into modes, which are neglected in the reduced modal analysis. Thus, using the reduced modal analysis for the numerical calculation of the dynamic behavior of piecewise linear elastic structures introduces the modal truncation damping. In this paper, we present on a continuous system, how this modal truncation damping can be characterized, what are the upper and lower limits of that damping and how the characteristics of the free vibration of a piecewise linear elastic structure can be predicted.

Practical vibration-based nano-devices are normally subject to disturbances where intense vibrations reveal significant nonlinear characteristics. Efficient implementation of nonlinear nano-systems requires comprehensive knowledge of the nonlinear dynamics. Nonlinear vibration characterization of elastic nano-bars with large vibration amplitudes is rigorously examined in the present study. The fascinating concept of simulating long-range interactions can be realized in the framework of the nonlocal elasticity theory, and thus, nano-scale effects are taken into account in the framework of the stress-driven nonlocal integral elasticity. The equivalent nonlocal differential condition equipped with non-classical boundary conditions of constitutive type is consistently detected. For proper comparison sake, the strain gradient elasticity theory is selected due to its similarities in revealing the stiffening structural response at nano-scale. In simple structural schemes of technical interest, the space-time decoupled formulation is constructed applying the weighted residual Galerkin method which results in a strongly nonlinear ordinary differential equation with cubic and quadratic nonlinearities. Analytical approach for the nonlinear analysis of the system dynamics provides an effective tool for optimized design of vibration-based nano-devices. The homotopy analysis method is accordingly employed to analytically study the nonlinear vibration response of nano-bars and its efficiency and accuracy is verified in comparison with the multiple scales method. The conceived approach for the nonlinear vibration analysis of elastic nano-bars, therefore, provides a consistent methodology to tackle nonlinear dynamic phenomena in nano-mechanics.

Solution of mechanical problems often requires the analytical or numerical calculation of equilibrium paths, while multidimensional solution sets are rare. From this requirement emerged numerous methods for the calculation of bifurcation diagrams. Two large groups of solution methods are the continuation methods and the scanning methods (however hybrid algorithm exists as well). The Simplex Algorithm is a robust approximative technique based on the Piecewise Linearization (PL)-algorithm, which has its application as a continuation and as a scanning algorithm as well. In this paper we will show the extension of the method for finding a 2-dimensional manifold (i.e. surface) with the scanning of the parameter space. We analyze the performance of the algorithm and its parallelization through two simple examples.

This paper presents an algorithm for parallel computers, which is suitable for the global (arbitrary displacements) computation of elastic bar structures subject to quasi-static loads. Our method is also capable to determine equilibria which are not connected to the initial, trivial configuration. The paper discusses the gains and the disadvantages of the method, comparing it with other techniques.

In this paper we develop an equilibrium method for the derivation of critical equilibrium configurations of a non-linearly elastic, discrete rod model, which is supported in a statically indeterminate way, and subjected to general loading. We construct a global computation scheme for the critical equilibrium configurations and demonstrate the effectiveness of the method via a clamped-pinned rod fabricated from linearly elastic, or specially hardening/softening material, loaded by either a horizontal or a follower force. We show some correspondence between the equilibrium states of these two load cases, and different snapping processes are revealed.

The paper presents a method which utilizes substructure normal modes to predict the vibration properties of a cantilever beam with a breathing transverse crack. The two segments of the cantilever beam, separated by the crack are related to one another by time varying connection matrices representing the interaction forces. The connection matrices are expanded in a Fourier series leading to a classical eigenvalue problem. Subsequently, the initial formulation is extended to avoid interference of the crack interfaces with a time domain formulation. The Lagrange multipliers, used to enforce the exact continuity constraints when the crack is closed, produce the interfaces forces needed for the modelling of interface dry friction.

This report contains a review of the technical literature concerning the detection, location, and characterization of structural damage via techniques that examine changes in measured structural vibration response. The report first categorizes the methods according to required measured data and analysis technique. The analysis categories include changes in modal frequencies, changes in measured mode shapes (and their derivatives), and changes in measured flexibility coefficients. Methods that use property (stiffness, mass, damping) matrix updating, detection of nonlinear response, and damage detection via neural networks are also summarized. The applications of the various methods to different types of engineering problems are categorized by type of structure and are summarized. The types of structures include beams, trusses, plates, shells, bridges, offshore platforms, other large civil structures, aerospace structures, and composite structures. The report describes the development of the damage-identification methods and applications and summarizes the current state-of-the-art of the technology. The critical issues for future research in the area of damage identification are also discussed.

Shaking table tests were conducted to investigate the response of rectangular wooden blocks and block assemblies of various sizes and slenderness to harmonic and earthquake base excitation. The shaking tests were followed by an analytical and a numerical study of response of single blocks and block assemblies. The analytical study was aimed at establishing criteria for the initiation of rocking and of overturning in response to harmonic base motion and consisted of solving numerically the differential equations of motion of a rigid block on a rigid foundation. The numerical study, in the course of which the response of both single blocks and block assemblies was examined, was implemented by means of the Distinct Element Method (DEM). Prior to the DE simulation of actual shaking tests, preliminary analyses were conducted to confirm numerical stability and to evaluate material and damping parameters. Comparing the recorded time histories with those given by the analytical study and the DE simulation, good agreement was found. The distinct element model in use appeared to follow the highly non-linear motion of rigid body assemblies faithfully to reality. On the basis of the results, provided that the necessary parameters are carefully estimated, the employed DE model can be regarded as an appropriate tool to simulate response of rigid body assemblies to dynamic base excitation.

This paper summarizes the main critical points that arise when the problem of modelling the dynamics of block structures is tackled. In the first sections, a rigorous formulation of dynamics and impact problem is presented for a single rigid block freely supported on rigid ground, in order to illustrate the basic difficulties concerning the modelling of more complicated structures. Then, a critical review is presented on the numerous researches performed on this subject and the results achieved, and the problems still open, are put in evidence.In questo lavoro, si illustrano i punti salienti e critici che devono essere affrontati nella modellazione del comportamento dinamico di structture costituite da grandi blocchi assemblati a secco. Nei primi paragrafi, viene presentato e discusso il problema generale della dinamica e dell' urto del blocco singolo semplicemente appoggiato su suolo rigido: questa la base necessaria per affrontare in modo rigoroso la modellazione di strutture pi complesse. Viene quindi presentata una rassegna critica di vari modelli proposti in letteratura, evidenziando i problemi risolti e quelli ancora aperti.

The present paper focuses on in-plane linear free vibrations of circular arches, in undamaged and damaged configurations. For the model herein utilized, the equations of motion, in terms of displacements and rotation, take into account shearing and axial deformations and rotary inertia. The cracked section of the arch is modeled with an elastic spring. An exact analytical method of solution and an approximate numerical one are presented. The first method solves the fundamental system in closed form, by means of a characteristic polynomial; the second one is based on a simple and efficient differential quadrature and domain decomposition technique. Natural frequencies and mode shapes are computed for some significant cases, showing very good agreement between the two approaches.

This paper studies the influence of concentrated damage on the eigen-properties of Timoshenko curved beams. Either single spatial arches or frame dome structures, obtained as assemblage of circular Timoshenko arches, are considered in presence of single or multiple damage and their exact dynamic stiffness matrices are evaluated. The natural frequencies and the corresponding modes of vibration are exactly calculated by means of a numerical strategy based on the Wittrick & Williams algorithm. The proposed procedure allows evaluating the effects of damage positions and intensities on the eigen-properties of the considered arch structures and to observe that, in the case of arches, the effects of damage severity and its location cannot be rigorously uncoupled. This latter result appears to be in contrast to what obtained in the literature for beams and rods. Since the adopted numerical approach leads to exact solutions, the obtained results can also be used as benchmarks for validating approximate numerical FEM strategies of Timoshenko damaged curved beams.

This paper reviews the recent advances in computational methods for nonlinear normal modes (NNMs). Different algorithms for the computation of undamped and damped NNMs are presented, and their respective advantages and limitations are discussed. The methods are illustrated using various applications ranging from low-dimensional weakly nonlinear systems to strongly nonlinear industrial structures.

This study employs an analytical model to describe the rocking response of a masonry arch to in-plane seismic loading. Through evaluation of the rate of energy input to the system, the model reveals the ground motions that cause maximum rocking amplification. An experimental investigation of small-scale masonry arches subjected to past earthquake time histories is used to evaluate the analytical model and to explore arch rocking behaviour. The results demonstrate that rocking amplification can occur, but is highly sensitive to slight variations in the ground motion. Thus, the accuracy to which the arch response can be predicted is brought into perspective. The concept that the primary impulse of an expected ground motion is fundamentally important in predicting arch collapse is evaluated in light of the developed energy approach. Finally, a statistical method is proposed for predicting the probability of arch collapse during seismic loading.

We investigate in-plane vibrations of cracked circular Euler-Bernoulli beams. Only bending and extension effects are included, and the curvature is in a single plane. In-plane vibrations are analyzed using the finite element method. In the analysis, elongation, bending and rotary inertia effects are included. Four degrees of freedom for in-plane vibrations are assumed. We calculate natural frequencies of beams with cracks at different locations and of different depths. Comparisons are made for different angles.

During the Chilean earthquakes of May, 1960, a number of tall, slender structures survived the ground shaking whereas more stable appearing structures were severely damaged. An analysis is made of the rocking motion of structures of inverted pendulum type. It is shown that there is a scale effect which makes tall slender structures more stable against overturning than might have been expected, and, therefore, the survival of such structures during earth-quakes is not surprising.

Small-amplitude motions of dynamic systems (structural, fluid, control, etc.) about an equilibrium state are modeled by linear differential equations which have constant coefficients. These are typically obtained by a Taylor series expansion of the forces about the equilibrium point. Under quite general circumstances these equations admit a set of special solutions, called normal mode motions, in which each system component moves with the same frequency and with a fixed ratio amongst the displacements of the components (for a conservative system; for a non-conservative system all displacements and velocities are linearly related to a single displacement/velocity pair).

This monograph is devoted to the study of targeted energy transfer (TET) phenomena in dissipative mechanical and structural systems possessing essentially nonlinear local attachments. We will show that the addition of a local attachment possessing essential (nonlinearizable) stiffness nonlinearity to a linear system, may significantly alter the global dynamics of the resulting integrated system. The reason lies in the lack of a preferential resonance frequency of the attachment, which, in principle, enables it to engage in nonlinear resonance with any mode of the linear system, at arbitrary frequency ranges (provided, of course, that the mode has no node in the neighborhood of the point of attachment). The actual scenario of single-mode or multi-mode nonlinear resonance interaction of the attachment with the linear system will depend on the level and spatial distribution of the instantaneous vibration energy of the integrated system. We will show that under certain conditions, passive TET from the linear system to the NES occurs, i.e., a one-way and irreversible (on the average) flow of energy from the linear system to the attachment,
which acts, in effect, as a nonlinear energy sink – NES. Moreover, in contrast to the classical linear vibration absorber whose action is narrowband, we will show that under certain conditions the NES can resonantly interact with the linear system in a broadband fashion, and engage in a resonance capture cascade with a set of structural modes over a broad frequency range; then the NES, acts in essence, as a
passive, adaptive, broadband boundary controller.

A continuous cracked beam vibration theory is used for the prediction of changes in transverse vibration of a simply supported beam with a breathing crack. The equation of motion and the boundary conditions of the cracked beam considered as a one-dimensional continuum were used. The eigenfrequency changes due to a breathing edge-crack are shown to depend on the bi-linear character of the system. The associated linear problems are solved over their respective domains of definition and the solutions are matched through the initial conditions. The changes in vibration frequencies for a fatigue-breathing crack are smaller than the ones caused by open cracks. The method has been tested for the evaluation of the lowest natural frequency of lateral vibration for beams with a single-edge breathing crack. Experimental results from aluminium beams with fatigue cracks are used for comparison with the analytical results.

This paper assesses the feasibility of vibration testing to detect structural damage caused by the settlement of buttresses in the Beverley Minster, a Gothic church located in the UK. Over the past eight centuries, the accumulated support settlements of the buttresses of Beverley Minster have pulled the main nave walls outward, causing severe separation along the edges of the masonry vaults. Bays closer to the main crossing tower have remained intact; however, at the west end of the Minster, the crack width between the walls and vaults has reached about 150 mm, leading to approximately 200 mm of sag at the crown of the vaults. Due to uneven settlement of buttresses along the nave of the church, the Minster now has ten nominally identical vaults at different damage states. In this work, two of these vaults representing the two extremes, the most damaged and undamaged structural states, are subjected to vibration testing with impact hammer excitation. From these vibration measurements, damage indicators are extracted in the modal, frequency, and time domains. In the modal domain, the differences between modal parameters are observed to be comparable to measurement uncertainty and hence insufficient to reach conclusions about the presence of vault damage. However, the amplitudes of frequency response functions in the frequency domain are observed to indicate a clear difference between the damaged and undamaged states of the structure. A time domain autoregressive model, support vector machine regression, is also found to be successful at indicating the differences between the two structural states of the vaults. We conclude that vibration measurements offer a practical solution to detect wall–vault separation in historic masonry monuments, provided that multiple damage indicators are evaluated.

A crack reduces the flexural rigidity of a column. The effects of the reduced rigidity on the load carrying capacity, the deflection, and the fracture load of a slender column with a single-edge crack have been studied based on the column theory together with the well-known relationship between the compliance and the stress intensity factor of a cracked beam. A crack reduces the load carrying capacity and increases the lateral deflection of a column under an eccentric compressive load. The calculated deflection agrees very well with the values measured by Liebowitz, Vanderveldt and Harris. The increased lateral deflection increases the bending moment at the cracked section. The bending stress intensity factor of a cracked column was also calculated, with the increased bending moment taken into consideration. The net stress intensity factor at the crack tip is the difference between the bending and the compression stress intensity factors. The fracture toughness value obtained from cracked columns agree reasonably well with the value obtained from cracked plate. This study indicates that the superposition of stress intensity factors of the same mode is valid.

In this study, exact closed-form expressions for the vibration modes of the Euler–Bernoulli beam in the presence of multiple concentrated cracks are presented. The proposed expressions are provided explicitly as functions of four integration constants only, to be determined by the standard boundary conditions. The enforcement of the boundary conditions leads to explicit expressions of the natural frequency equations. Besides the evaluation of the natural frequencies, neither computational work nor recurrence expressions for the vibration modes are required.The cracks, that are not subjected to the closing phenomenon, are modelled as a sequence of Dirac's delta generalised functions in the flexural stiffness. The Eigen-mode governing equation is formulated over the entire domain of the beam without enforcement of any continuity conditions, which are already accounted for in the adopted flexural stiffness model. The vibration modes of beams with different numbers of cracks under different boundary conditions have been analysed by means of the proposed closed-form expressions in order to show their efficiency.

This paper deals with the buckling analysis of cracked beam-columns. The crack is modelled with a unilateral elastic bending-stiffness behaviour, represented by a unilateral rotational spring. This model takes into account the crack closure effect associated with the phenomenon of breathing crack. The rotational spring may model a one-sided crack (or a unilateral joint for other applications in biomechanics or civil engineering). The unilateral elastic constitutive law of the crack is introduced from a variational procedure based on energy arguments. It is shown that this constitutive law can be formulated as a function of a physically-based damage parameter. A one-crack and a two-crack hinged-hinged beam-column are theoretically investigated to illustrate the unilateral effect of the crack behaviour on the buckling load. It is shown that the open crack analogy can be adopted for the one-crack beam-column, whereas this simplified assumption cannot be retained for the two-crack beam-column. In this last case, the crack closure effect strongly affects the buckling load, with respect to the open crack analogy of the same problem. More specifically, the crack-closure phenomenon tends to increase the buckling load, with respect to the open crack assumption mainly retained in the literature.

This paper presents an analytical approach, as well as a calculation method for determining the dynamic response of the undamped Euler–Bernoulli beams with breathing cracks under a point moving mass using the so-called discrete element technique (DET) and the finite element method (FEM). First, the standard DET formulation is modified to consider the effects of Coriolis and centrifugal forces. Next, the formulation is extended to be able to evaluate the cases with open and breathing cracks under moving masses. The results will be validated against those reported in the literature and also compared with results from the finite element method. The effects of the moving mass velocity, location, and size of the crack on beam deflection will be investigated. Natural frequencies of the beam under the effect of crack will also be studied to compare the results with those of a beam without crack. It is observed that the presence of crack results in higher deflections and alters beam response patterns. In particular, the largest deflection in the beam for a given speed takes longer to build up, and a discontinuity appears in the slope of the beam deflected shape at the crack location. The effects of crack and load depend on speed, time, crack size, crack location, and the moving mass level.

In this paper, the so-called Couplet–Heyman problem of finding the minimum thickness necessary for equilibrium of a circular masonry arch, with general opening angle, subjected only to its own weight is reexamined. Classical analytical solutions provided by J. Heyman are first rederived and explored in details. Such derivations make obviously use of equilibrium relations. These are complemented by a tangency condition of the resultant thrust force at the haunches' intrados. Later, given the same basic equilibrium conditions, the tangency condition is more correctly restated explicitly in terms of the true line of thrust, i.e., the locus of the centers of pressure of the resultant internal forces at each theoretical joint of the arch. Explicit solutions are obtained for the unknown position of the intrados hinge at the haunches, the minimum thickness to radius ratio and the nondimensional horizontal thrust. As expected from quoted Coulomb's observations, only the first of these three characteristics is perceptibly influenced, in engineering terms, by the analysis. This occurs more evidently at increasing opening angle of the arch, especially for over-complete arches. On the other hand, the systematic treatment presented here reveals the implications of an important conceptual difference, which appears to be relevant in the statics of masonry arches. Finally, similar trends are confirmed as well for a Milankovitch-type solution that accounts for the true self-weight distribution along the arch.

A discussion with regards on the recent tests conducted on several Gothic churches which aims on improving the necessary knowledge needed on finite element (FE) models of the structure is presented. A focus will be given on the illustration of the typical modal testing results from complex vaulted Gothic churches, as well as to indicate the typical problems, while giving suggestions on the solutions during the test of a specific type of large civil engineering structure. It has been realized that when a careful consideration and adjustment has been done on the test setup and the data acquisition, satisfactory data can possibly be obtained. More importantly, the testing program appears to be useful on large-scale modal testing on vaulted masonry structures for analytical model calibration.

The dynamic analysis of stone arches, made up of rigid voussoir laid dray, can be performed in two phases. First of all the value of the horizontal acceleration necessary to turn the structure into a mechanism and the corresponding mechanism must be determined. Then the dynamic behaviour of such a mechanism under a given acceleration time history can be studied. The first step is a static matter. The second one requires the solution of the non-linear equation of motion of the one-degree-of-freedom system in which the arch is turned. In this paper an iteration procedure is proposed to find out the mechanism. Then the structural behaviour of the mechanism is analysed. Both free and forced vibrations are investigated and the study is limited to the first-half cycle of vibration. Damping is not considered and sliding between the blocks at the hinge sections is not allowed. © 1998 John Wiley & Sons, Ltd.

a b s t r a c t The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companion, Part II. Because there is virtually no application of the NNMs to large-scale engineering structures, these papers are an attempt to highlight several aspects that might drive their development in the future. Specifically, we support that (i) numerical methods for the continuation of periodic solutions pave the way for an effective and practical computation of NNMs, and (ii) time–frequency analysis is particularly suitable for the analysis of the resulting dynamics. Another objective of the present paper is to describe, in simple terms, and to illustrate the fundamental properties of NNMs. This is achieved to convince the structural dynamicist not necessarily acquainted with them that they are a useful framework for the analysis of nonlinear vibrating structures.

This paper presents a dynamic identification analysis of a masonry construction, built to be tested in "Laboratório Nacional de Engenharia Civil" (LNEC), in Lisbon, on the scope of the European Project ECOLEADER-LIS – Enhancing Seismic Resistance and Durability of Natural Stone Masonry. The masonry model was built with limestone units and lime mortar joints with polymeric grid reinforcement placed on the horizontal joins. The dynamic identification analysis was divided in several tasks. For the calculation of the expected dynamic parameters a preliminary FEM analysis was carried out. Two types of operational modal analysis were used, the EFDD and SSI methods. The main purpose of the analysis was to compare the classical modal analysis with the ambient based modal analysis and to verify if the ambient vibration methods are able to assess the damage in an earlier stage in the structure. Finally, concluding remarks of the work carried out are given.

Recent seismic events have caused damage or collapse of invaluable historical buildings, further proving the vulnerability of unreinforced masonry (URM) structures to earthquakes. This study aims to understand failure of masonry arches—typical components of URM historic structures—subjected to horizontal ground acceleration impulses. An analytical model is developed to describe the dynamic behaviour of the arch and is used to predict the combinations of impulse magnitudes and durations which lead to its collapse. The model considers impact of the rigid blocks through several cycles of motion, illustrating that failure can occur at lower ground accelerations than previously believed. The resulting failure domains are of potential use for design and assessment purposes. Predictions of the analytical model are compared with results of numerical modelling by the distinct element method, and the good agreement between results validates the analytical model and at the same time confirms the potential of the distinct element framework as a method of evaluating complex URM structures under dynamic loading. Copyright © 2007 John Wiley & Sons, Ltd.

This paper investigates the in-plane linear dynamic behaviour of multi-stepped and multi-damaged circular arches under different boundary conditions. Cracked cross-sections are modelled as massless elastic rotational hinges. In damaged configuration, cracks can be located both at the interface between two adjacent portions as well as inside the portion itself. For each arch portion bounded by two cracks, the differential equations of motion have been established considering axial extension, transverse shear effects and rotatory inertia. The equilibrium equations of arch portions are combined in the coupled fundamental system in terms of radial displacement, tangential displacement and rotation. Analytical and numerical solutions for multi-stepped arches, in undamaged as well as in damaged configurations, are proposed. The analytical solution is based on the Euler characteristic exponent procedure involving the roots of characteristic polynomials, while the numerical method is focused on the Generalized Differential Quadrature (GDQ) method and the Generalized Differential Quadrature Element (GDQE) technique. Numerical results are shown in terms of the first 10 analytical and numerical frequencies of multi-stepped and multi-damaged arches with different boundary conditions. Finally, convergence and stability characteristics of the GDQE procedure are investigated. The convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical procedure is very good.

A damage in a structure alters its dynamic characteristics. The change is characterized by changes in the eigenparameters, i.e., natural frequency, damping values and the mode shapes associated with each natural frequency. Considerable effort has been spent in obtaining a relationship between the changes in eigenparameters, the damage location and the damage size. Most of the emphasis has been on using the changes in the natural frequencies and the damping values to determine the location and the size of the damage. In this paper a new parameter called curvature mode shape is investigated as a possible candidate for identifying and locating damage in a structure. By using a cantilever and a simply supported analytical beam model, it is shown here that the absolute changes in the curvature mode shapes are localized in the region of damage and hence can be used to detect damage in a structure. The changes in the curvature mode shapes increase with increasing size of damage. This information can be used to obtain the amount of damage in the structure. Finite element analysis was used to obtain the displacement mode shapes of the two models. By using a central difference approximation, curvature mode shapes were then calculated from the displacement mode shapes.

In this paper the dynamic behaviour of a cantilever beam with a breathing crack is investigated both theoretically and experimentally. The primary aim is to reveal the nonlinear behaviour of the system by using time frequency methods as an alternative to Fourier analysis methodology. A simple single-degree-of freedom lumped system is employed to simulate the dynamic behaviour of the beam. The time varying stiffness is modelled using a simple periodic function. Both simulated and experimental response data are analysed by applying empirical mode decomposition and Hilbert transform and the instantaneous frequency (IF) is obtained. It is shown that the IF oscillates between the frequencies corresponding to open and closed states revealing the physical process of crack breathing. The variation of the IF follows definite trends and therefore can be used as an indicator of the crack size. It allows an efficient and accurate description of the nonlinearities caused by the presence of a breathing crack. Consequently, it can be used to improve vibration based crack detection techniques.

A method of analysis of the effect of two open cracks upon the frequencies of the natural flexural vibrations in a cantilever beam is presented. Two types of cracks are considered: double-sided, occurring in the case of cyclic loadings, and single-sided, which is principle occur as a result of fluctuating loadings. It is also assumed that the cracks occur in the first mode of fracture: i.e., the opening mode. An algorithm and a numerical example are included.

The possibility to detect the structural damage affecting a narrow zone of a doubly hinged plane circular arch by means of a few measured natural frequencies is considered. Such localised damage induces a discontinuity in the bending stiffness of the arch, modelled as a torsion spring joining two adjacent sections and characterised by the location and the stiffness of the spring. The direct problem in the damaged and undamaged case is examined; the inverse problem is then considered. Two different procedures to identify the damage parameters are introduced: the first is based on the search of an intersection point of curves obtained by the modal equation; the second is based on the comparison between the analytical and experimental values of the variation of frequencies passing from the undamaged to the damaged state. In conclusion, the possibility of identifying the damage parameters by means of pseudo-experimental data is examined.

Harmonic balance (HB) methods allow for rapid computation of time-periodic solutions for nonlinear dynamical systems. We present a filtered high dimensional harmonic balance (HDHB) approach, which operates in the time domain, and provide a framework for implementation into an existing finite element solver. To demonstrate its capabilities, the method is used to solve a set of nonlinear structural dynamics problems related to the field of flapping flight. For each example, the HDHB approach produces accurate steady-state solutions orders of magnitude faster than a traditional time-marching scheme.

The paper develops a finite element scheme for computing the eigensystem for a cracked beam for different degrees of closure. Previous work in the authors' laboratories has indicated that the ability to extend the use of mode superposition to model breathing conditions in the crack zone would overcome the need to switch from a frequency-domain-based model to a time-stepping scheme which had caused both implementational and theoretical problems. In this study, the finite element method, the component mode synthesis method and the linear elastic fracture mechanics theory are integrated for modelling of the cracked structures. It is believed that this is a novel synthesis of methods. The method used by the authors is benchmarked against earlier results in the literature.

The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companion, Part I. One reason of the still limited use of NNMs in structural dynamics is that their computation is often regarded as impractical. However, when resorting to numerical algorithms, we show that the NNM computation is possible with limited implementation effort, which paves the way to a practical method for determining the NNMs of nonlinear mechanical systems. The proposed algorithm relies on two main techniques, namely a shooting procedure and a method for the continuation of NNM motions. The algorithm is demonstrated using four different mechanical systems, a weakly and a strongly nonlinear two-degree-of-freedom system, a simplified discrete model of a nonlinear bladed disk and a nonlinear cantilever beam discretized by the finite element method.

The aim of this article is to present a technique capable of evaluating the dynamic response of a beam with several breathing cracks perpendicular to its axis and subjected to harmonic excitation. The method described is based on the assumption of periodic response and that cracks open and close continuously. In this way, a non-linear system of algebraic equations can be defined and solved iteratively, with the advantage over direct numerical integration of the equation of motion of being easier and therefore faster to compute.In this article, the vibrational response to harmonic force of a cantilever beam with cracks of different size and location is analyzed using this “harmonic balance” approach and the results are compared with those obtained through numerical integration.

Exact solution of free in-plane vibrations of circular arches of uniform cross-section is given by considering axial extension, transverse shear and rotatory inertia effects. In contrast with Kirchhoff's beam theory the restrictions of perpendicular cross-section and inextensible arc length are removed. The principal axes of the cross-section are assumed to coincide with the principal normal and binormal vectors of the centerline of the beam. A solution procedure is applied to obtain the fundamental matrix for various end conditions. Natural frequencies and mode shapes are given in figures and tables.

This paper presents a direct method for locating normal modes in certain holonomic, scleronomous, conservative non-linear two degree of freedom dynamical systems. The method does not require that the system studied be close to a linear system.RésuméCet article présente une méthode directe pour situer les modes normaux de certains systèmes dynamiques, holonomes, scléronomes, conservatifs, non linéaires à deux degrés de liberté. La méthode n'impose pas que le système étudié soit proche d'un système linéaire.ZusammenfassungDiese Arbeit beschreibt eine direkte Methode zur Auffindung der Eigenschwingungen in bestimmten holonomischen, skleronomischen, konservativen, dynamischen Systemen mit zwei Freiheitgraden. Es ist nicht erforderlich, dass das untersuchte System nahezu linear ist.

A single-degree of freedom non-linear oscillator is considered. The non-linearity is in the restoring force and is piecewise linear with a single change in slope. Such oscillators provide models for mechanical systems in which components make intermittent contact. A limiting case in which one slope approaches infinity, an impact oscillator, is also considered. Harmonic, subharmonic, and chaotic motions are found to exist and the bifurcations leading to them are analyzed.

The aim of the paper is to check the capability of dynamic identification procedures, usually applied to reinforced concrete or steel buildings, in the estimation of the dynamic characteristics of masonry buildings. To this purpose, the dynamic behaviour at low vibration levels of an existing masonry building subjected to forced, sinusoidal or sweep, vibration test, was investigated. Possibly on account of weak nonlinearities of the building, the measurements obtained with sinusoidal tests seemed more suited than those obtained by sweep tests for the application of identification techniques. These concern both modal and physical models: the dynamical characteristics of the building, the frequencies and some eigenvector components obtained by modal identification, were assumed as experimental data to update suitably selected stiffness parameters of a finite element model of the structure. The updating was performed by means of a specially developed dynamic identification code based on an output error equation. Criteria for a rational choice of physical parameters and measurements were applied. A very good agreement between numerical and experimental frequency response functions was obtained with the modal identification, while the physical model showed some rigidity in fitting all the experimental data, albeit with an acceptable level of scatter. Taken as a whole, the results show that well-established identification techniques can furnish useful information concerning the dynamic properties of existing masonry structures.

The results of the ambient-vibration based investigations carried out to assess the structural conditions of a masonry bell-tower are presented. The tower, dating back to the XVII century and about 74 m high, is characterised by the presence of major cracks on the western and eastern load-bearing walls.The assessment procedure includes full-scale ambient vibration testing, modal identification from ambient vibration responses, finite element modelling and dynamic-based identification of the uncertain structural parameters of the model. A good match between theoretical and experimental modal parameters was reached for relatively low stiffness ratios in the most damaged regions of the tower. Furthermore, the model identification, carried out by using two different methods, provided consistent structural parameters which are also in close agreement with the available characterization of the materials.