Experimental static and dynamic tests on a largescale freeform Voronoi grid shell mockup in comparison with finiteelement method results
Abstract
Grid shells supporting transparent or opaque panels are largely used to cover longspanned spaces because of their lightness, the easy setup, and economy. This paper presents the results of experimental static and dynamic investigations carried out on a largescale freeform grid shell mockup, whose geometry descended from an innovative Voronoi polygonal pattern. Accompanying finiteelement method (FEM) simulations followed. To these purposes, a fourstep procedure was adopted: (1) a perfect FEM model was analyzed; (2) using the modal shapes scaled by measuring the mockup, a deformed unloaded geometry was built, which took into account the defects caused by the assembly phase; (3) experimental static tests were executed by affixing weights to the mockup, and a simplified representative FEM model was calibrated, choosing the nodes stiffness and the material properties as parameters; and (4) modal identification was performed through operational modal analysis and impulsive tests, and then, a simplified FEM dynamical model was calibrated. Due to the high deformability of the mockup, only a symmetric load case configuration was adopted.
Keywords
Free form Grid shell Voronoi Polygonal pattern Experimental static test Operational modal analysis Impulsive testIntroduction
Contextualization of the work
As spatial lightweight structures, freeform constructions played a prominent role in recent years thanks to more reliable computational techniques, increased knowledge of building materials, and new architectural expressivity. Longspanned freeform roofs and building skins are realized as grid shells connecting a set of discrete (opaque or transparent) panels to a supporting grid structure. Before the structural aspects, the engineering process of the geometrical and constructional definition of the facetted architectural shape is related to the feasibility and the modalities of the basesurface subdivision (Adriaenssens et al. Adriaenssens et al. 2012, 2014; Baldassini et al. 2010; Pottmann 2013; Pottmann et al. 2007; Sassone and Pugnale 2010).
Grid shells are afflicted by some static problems as deformability, buckling, and imperfection sensitivity (Schlaich and Schober 1996, 1997) and by constructional problems such as nodes assembly, feasibility of the panels geometry and curvature, connectivity of the elements, etc. because of their threedimensional spatial complexity. Therefore, a large displacement analysis typically applies to a grid shell due to the intrinsic deformability caused by its lightness. As they carry loads mainly by compressive forces, buckling failures (local, global, snapthrough, or even worse combinations of previous) should be avoided. Stability analysis is carried out considering both secondorder effects and imperfections, and a strong abatement of the buckling multiplier with respect to the ideally perfect structure is usually detected (Bacco and Borri 1993; Bulenda and Knippers 2001; Cai et al. 2013; Dini et al. 2013; Gioncu 1995).
Typically, in the structural analysis of a grid shell, the influence of the panel is omitted as long as the simulation of the overall behavior is concerned, neglecting the interaction with the supporting structure in terms of strength or stiffness. The panel is considered essentially as a dead load weighing on the structural nodes.
Our contribution
Nowadays, the use of polygonal patterns in grid shells (Jiang et al. 2014, 2015; Pottman 2009) inspired by organic shapes is growing. Their structural lightness, their interesting geometrical properties, and aesthetic appeal favor them in comparison with their triangular and quad competitors. Optimization procedures are often iterative and provide for shape alterations to increase structural shell features. On the contrary, Pietroni et al. (2015) introduced an algorithm to mesh assigned freeform continuous shell surfaces by means of Voronoi polygonal patterns, whose procedure is affected by the membrane stresses acting on the initial shell. Practically, the tessellation, the density, and the anisotropy of the cells can be altered with respect to the statics of the initial continuous surface, driven by some control parameters. As stated, Voronoi staticaware grid shells manifested good static performances, better than the stateoftheart competitors (Tonelli 2015). Furthermore, since the propensity to buckle of grid shells, Tonelli et al. (2016a) devoted a specific study to the stability and imperfection sensitivity analyses of hexdominant freeform grid shells generated with the Statics Aware Voronoi algorithm. As a result, under some specific hypotheses (uniform loading, rigid joints, and pinned boundary), the outcomes confirmed the static efficiency. In particular, the bigger the irregularity of the initial continuous shell, the better the static performances of the Voronoi grid shell.
Even though such algorithm was tested from a computational viewpoint, up to now, feasibility studies and experimentations are missing. Moreover, a knowledge gap was noticed in the assessment of the node stiffness that can influence the theoretical stresses and displacement fields. The previous modelling assumptions concerning the connections are barely achievable. The stiffness of the joints plays a significant role; and to such an extent if as an extreme case, we remove the rigid joints condition, and the polygonal mesh becomes a mechanism. We referred to experimental observations.
Because of the mechanical and geometric complexity, as first goal, the research focused on the theoretical behavior of the case study vault to identify the displacement field, the main load paths, and the modal properties under the assumption of rigid joints, fixed supports, and characteristic material properties. This part of the study was led on an FEM model, in the following named ‘perfect’, whose metric is that one generated by the Voronoi algorithm. Such model, as well as the following, adopted the same mockup size.
In spite of the first modelling assumption of perfect geometry, the assemblage of the mockup produced small defects related to materials, dimensional tolerances, difficulties of the assembling operations, particular shape of the shell structure, and local nonlinearities (Pietroni et al. 2015; Tonelli et al. 2016a). We conveniently decided to consider the physical replica state introducing an inelastic deformation field in a further numerical model. The field was obtained from calibration of the modal eigenforms derived from the perfect geometry FEM model on the survey made on the prototype. Such model constituted the base geometry for numerical models analyzed in the following phase.
The second goal of this study was to execute nondestructive tests and consequently simulate the observed mockup behavior with simplified FEM models. We performed two sets of experimental campaigns. The first set intended to investigate the statics of the mockup, submitted to symmetrical loading patterns. The descending representative FEM static model did not consider the presence of the PET panels. Moreover, to approach the experimental behavior, we varied the beams rotational endrelease stiffness and reduced the characteristic material properties, simulating the nonrigid connectivity, the local nonlinearities, and structural imperfections.
The second experimental set aimed at the dynamic identification of the prototype. We acquired the modal properties to understand the dynamic behavior of such structure, so to obtain information about the distribution of mass, stiffness, and damping. In the accompanying representative FEM dynamic model, we considered the structural response of both the supporting grid structure and the panels.
Analysis method and tests
 1.
Theoretical behavior of the staticaware Voronoi case study. A numerical model was based on the geometry directly descended from the processes of staticaware form genesis. Since the FEM model and the mockup have the same scale, the numerical results correspond to the theoretical behavior of the prototype. We refer to this phase as ‘perfect geometry modelling’.
 2.
Study of the mockup state. The aim is to define a displacement vector field to simulate the deformed unloaded mockup state. Such displacement vector field was introduced as an inelastic deformation in the following numerical models.
 3.
Realization of static experimental tests. We processed, collected, and analyzed testobserved data, and finally, we a generated simplified FEM static representation in reverse fitting procedure.
 4.
Realization of dynamic experimental tests. The specific aims are the dynamic identification and the calibration of an FEM dynamic model.
The present study relied on three base assumptions regarding nonlinearity, size of the models, and symmetry. In the nonlinear static analyses, only geometrical nonlinearity was included. Because the field of proportionality between stress and consequent displacement was indeed not exceeded, material nonlinearity was omitted. Moreover, other nonlinearities were omitted such as contact and friction at the nodes. Concerning the size, FEM models conveniently represent the real size of the prototype (2.40 m side), avoiding to extend the structural test results to a larger scale model and to introduce further uncertainty. Among multiple loading configurations, we considered only symmetrical load cases in accordance with the symmetry of the vault. Other possible and probably more dangerous loading configurations may be asymmetrical loads. However, the Voronoi StaticAware algorithm is still limited to the optimization of only symmetrically loaded surfaces (i.e., under dead loads, Pietroni et al. 2015; Tonelli et al. 2016a; Tonelli 2015); consequently, the high deformability of the mockup excluded to extend the study to other loading scenarios.
Perfect geometry modelling (phase 1)
The first numerical model, named GS_{0}, intended to test the staticaware Voronoi algorithm on the case study surface and to simulate the theoretical behavior of the mockup. The properties were set as follows. The circular wooden beam sections were 8 mm diameter with Young’s modulus E = 11,000 N/mm^{2}. The PET panels were 1 mm thick with E _{p} = 3700 N/mm^{2}. The prototype was installed on an OSB base panel by means of four stiff FDM special nodes. The latter are fixed through four screws on it. All the internal nodes were threeway connections. The beams were fitted into a 10 mm deep nodal plastic casing. Neither bolted nor glued connection was practiced. The mutual mechanical interaction between nodes and beams was by contact in case of beam compression, while only the friction limited the sliding in case of traction. Shear and bending were fully transmitted from beams to nodes. At this stage, in the FEM model, all the nodes of GS_{0} were estimated as rigid, in a simplified approach. External constraints were supposed as all fixed joints, thanks to the stiffness of the OSB base panel. Material properties are defined as their characteristic values.
The PET panel is point supported at its vertices, which resulted clamped between the printed plastic node and a metallic washer screwed on the node itself. So that, given a nonlinear constraint condition, the panel can detach and slide in case of tension stress. No direct contact is between the panels and the beams. After the attempt to include the panels into the model, we considered only their dead load: gravitational masses weighing 50 g were loaded at each node as simulating the printed joint and the panel influence. Since the aim was to provide a global simplified representation of the mockup, requiring an indepth modelling the complexity of the panels, joinery system would not have been compatible with a less refined global modelling scale. Finally, the model consisted of 465 nodes and 697 beam elements.
Assessment of the unloaded mockup state (phase 2)
Numerous factors related to the assembly phase affected the shape of the mockup. The structure was not freestanding until the end of such operation, and as an effect of tolerances, fabrication, and dryassembly errors, the theoretical expected shape resulted altered. To consider this geometrical deviation, we introduced an inelastic representative vector field into the Voronoi perfect model.
Such field was known by calculating its shape and its relative amplitude. We built the shape of the vector field through a composition of modal eigenforms, given by the natural frequency analysis of GS_{0}. In this phase, other fields were also considered, such as the deformed shape for uniform loading or for asymmetric loading and the main buckling eigenforms. We divided the whole surface into four homogeneous areas, identified by the X and Yoriented symmetry axes (Fig. 2a); then, we searched for the best eigenforms that locally approximated the unloaded (geometrically imperfect) mockup. In this phase, the selection started from a visual analysis of the prototype and neglected local deformations.

Such that δ _{ z,r } is the vertical displacement measured by the geometric survey, and δ _{ z,mode} is the one extracted from modal shape for the corresponding nodes; we run the ratios δ _{ z,r } /δ _{ z,mode}.

The scale factor σ represented the average of all the previous ratios.

By multiplying all components of the modal shape by the factor σ, we defined the entire vector field: δ _{ i } = σ δ _{ i,mode,} where i = (x, y, z).

We deducted the survey rate from the final value as δ _{ z } = σ δ _{ z,mode} to estimate the error made at the control points; the square root of the sum of the squares value returned an indicator I of severity of the error.
Experimental static behavior (phase 3)
We studied the static behavior by means of tests performed under timeinvariant load configurations, where the ith load increment is maintained until the stabilization of the measure. The aim was to simulate the serviceability loading of the mockup, transforming a distributed load into equivalent nodal forces. Due to the feasibility of the test, 16 nodes of the vault were loaded. Organized in groups of four, the 16 nodes were symmetrically located with respect to the axes of the prototype. Hollow metal discs, each weighing 125 g, constituted the applied load and were hanged to each node by means of a ligature and a doublehookedend selfbalancing steel element. Such metallic extremity had the advantage of reducing vibrations transmitted to the structure.

Preparation, counting, and approaching of weights close to the nodes to be loaded.

Transducers installation and check (in particular, verification of verticality because of the only Zdisplacement monitoring); connection to control unit and verification of the correct operating of the measuring system.

Test phase, in which at every load step, a weight per node was hanged (total of 16 plates, one for each hook).
 For each step, the load was maintained until the structure settled (Fig. 8).

Control and verification of the continuous recording of the data.

Photographic survey at the end of each stage.
Dynamic identification through OMA and impulsive tests (phase 4)
The parameters derived from the dynamic experimentation are natural frequencies, damping ratios, and modal shapes (Ewins 2000; Maia and Silva 1997). The high deformability of the prototype suggested the use of an outputonly method, such as the operational modal analysis (OMA) (Zhang et al. 2005). In addition, we used impulsive excitation tests to give feedback to the OMA data. Fundamental assumptions for the dynamic tests are linearity, stationarity, and observability, and the test programming complied with these assumptions.

installation of the accelerometers, namely, checking, cleaning, numbering, positioning, and connection to cables;

connection of the cables to the acquisition unit to verify the correct efficiency of the measuring system in free run;

execution of the dynamic experimental test and data saving.
We performed three cycles of OMA tests, each for a sampling time of about 30 min. Two series of impulsive dynamic tests followed. The first series consisted of 21 measurements with the impact hammer PCB Piezotronics Model 086D20, equipped with a transducer tip of average hardness 084A62 Tip—medium plastic (Fig. 8). The second series consisted in a sudden removal of a prestress condition from the mockup. Practically, we constrained a gravitational mass, made of a group of metallic discs, to an FDM printed structural node by means of a link cable. Cutting such cable and letting the mass fall on a neoprene mat, the Voronoi vault resulted excited but unperturbed from the mass fall. The amount of measurements performed with this procedure was 4.
Results and modelling
Experimental static output and FEM static model
GS_{1} linear analysis simulated only the conditions under own weight. As GS_{0} analysis, it showed predominantly beams axial stresses and negligible bending moments. Due to the introduction of the unloaded deformed geometry, the displacement field became asymmetric.
Experiment dynamic output and FEM dynamic model
Consequently, it is possible to approximate the ‘almost real’ modal parameters as real, appraising the Voronoi vault as a classic civil structure, whose problem of dynamic equilibrium in the design phase is usually solved in notdamped vibrations condition. This consideration made also possible to compare testobserved data with an FEM model, in which the problem of dynamic equilibrium is also solved in notdamped free vibrations.
The comparison between the FEM GS_{2} modal shapes and the testobserved data is possible only if the experimental eigenforms are converted into ‘real’ vectorial fields. Since the modal shape is defined up to a constant, it is allowed to project the complex experimental amplitudes of modal coordinates on the least squares trend line (dashed line in Fig. 14).
Finally, the GS_{2} modal analysis showed the prevalence of the first three modes in relation with the higher participating masses (about 50% of the total vibrating mass in X and Y directions for the first three modes) and the presence of a global motion. The upper ones achieved small increments of vibrating mass, totalizing about 85% in X and Y directions and about the 75% in Z direction while considering the first 90 modes. Both the FEM and the prototype had a torsional first mode, the next two (second and third) mainly translational but also with little torsional components, and the fourth torsional again. Higher modes have local characters.
Discussion
The method used within this work concerns the Voronoi staticaware vault, but is extendable to other Voronoi staticaware case studies. Simulating the size and the state of the mockup, a disadvantage is that the findings are strictly related to the specific shape, internal tessellation, building technology, materials, and local errors of assembly.
The stiffness of the joints, the structural imperfections, and the nonlinear behavior of the connections played the most significant role in the present research. Indeed, modelling assumptions of joints and material properties in the GS_{0} perfect geometry were not representative of the physical evidence. Internal connections were in practice not rigid and the nodes showed significant anomalies: they reacted when receiving the beams in compression, while offered only scattered and unpredictable frictional resistance in response to tension forces. This nonlinearity would require a lot of efforts and expertise for the calibration of detailed local models or an adequate probabilistic model. Because the aim of this work is to produce simplified global FEMs, such extensive research is referred to further work. Considering the simplifications adopted in the construction of the phase 2 geometry, particularly the neglecting the localized deformation of the prototype and the use of a unique mean value of rotational stiffness for all the beams, the matching of the FEM static model with the experienced data can be viewed as acceptable.
As an assumption to this work, we considered only symmetrical loading cases. Two reasons did not allow to consider asymmetrical loading: first, the geometric imperfection at the unloaded state and second, the high deformability of the mockup. The deviation of the initial geometry from the theoretical model influenced the structural behavior, producing asymmetrical displacement fields and asymmetrical modal shapes although the symmetrical loading. Hence, asymmetrical loading would have complicated the structural identification of the mockup. Moreover, since the lowest collapse multiplier is usually associated with asymmetrically loaded grid shells, the Voronoi vault would have prematurely drawn its ultimate limit state for the same (or even lower) loading rates, being the vault not optimized for asymmetric loading conditions. A complete validation for the Voronoi tessellation in the presence of asymmetric loading is demanded to future works, as well as investigations on the effect of additional stiffening contribution given by bracing cables.
The assessment of the prototype state before the tests resulted in a base geometry used in the following GS_{1} and GS_{2} FEM models. The vector field denoted only geometrical imperfection of that specific mockup. Shape similar structures and scaled structures could not manifest the same deformation. Unfortunately, there is no evident rule on the selection of the vector field components. The only criterion was the best fitting of the imperfect real structure. The selected modal shapes of GS_{0} indeed are not even the most significant in the dynamic behavior.
Modelling parameters of the FEM models
Base geometry  E (N/mm^{2})  E _{p} (N/mm^{2})  k _{beams} (Nmm)  k _{eq.PETrods} (Nmm)  

Characteristic values  –  11,000  3700  –  – 
GS_{0}  Perfect  11,000  0  ∞  – 
GS_{1}  Imperfect as per phase 2  3700  0  210,000  – 
GS_{2}  Imperfect as per phase 2  11,000  1200  ∞  0 
The reproducibility of the static tests is linked essentially to nonlinearity problems and to their random distribution. Similar scheduled tests have shown similar trends but different amplitudes of displacement and inelastic behavior at the unloading. Finally, we also observed some small cracks at the nodes and little but important localized sliding of the beams from their nodal plastic casing. The mockup nonlinearities and the complicated internal mechanisms of load dissipation are topics on which focusing for future works and for improving the prototypes design. On the other hand, the dynamic tests are reproducible because of their minute loading and the linearity of the mockup.
The dynamic behavior expected from the prototype was a combination of complex motions, typical of an irregular structure, characterized by local excitations. The GS_{0} model manifested as the unique exceptions the first two translational and symmetric modes. The real dynamics of the mockup showed considerable differences due to the loss of symmetry for the superimposition of the initial geometrical deviation. The first eigenform was torsional indeed. The general growth of the torsional components implies the impossibility to simplify the behavior of the structure and indicates a notuniform distribution of stiffness and in all likelihood of damping too.
In the dynamic tests, the number of accelerometers was enough to determine frequencies and damping ratios but not to acquire highresolution modal shapes. A better test would have used miniaturized transducers, overlapping the acquisitions of different measurements. Such procedure would have complied with the observability hypothesis because of the neglecting of the instruments mass. Despite our assets, we concluded that it is possible to make a simpler dynamic analysis, considering our case study as a not dumped system. This result appears to be one of the most relevant and pragmatic findings.
Conclusions
This paper examined a staticaware Voronoi freeform grid shell vault through experimental tests. Numerical simulations followed. The obtained results can be used as first step to gain a better understanding of the behavior of such novel hexdominant grid shells.
In perfect geometry modelling, the generated stress field and the distribution of compression stresses confirmed the work of the staticaware Voronoi algorithm in providing promising design alternative to build grid shells. Nonetheless, some assumptions of this formgenesis phase were removed to describe the statics and dynamics highlighted during tests performed at the Structural Laboratory of University of Pisa. Two FEM models were calibrated on the testobserved data. FEM analyses were conducted on a base geometry acquired from scaling the modal eigenforms with respect to a geometrical survey.

Clearly nonlinear experimental behavior since the earliest loading phases (geometrical nonlinearity).

Experimental output influenced by initial shape, local nonlinearities and stiffness of the joints.

A static FEM model setting a beams secant elastic modulus (E/3) and semirigid internal nodes drew with an adequate approximation the experimental behavior. The panels are considered in the model as a dead load only.

Excellent agreement between OMA and impulsive tests.

‘Almost real’ testobserved modes of vibration, the advantage was that we could neglect the influence of damping in the problem of dynamic equilibrium and warrant comparison with numerical modes obtained in nondamped free vibrations.

A dynamic FEM model considering the PET panel influence in the dynamic response, modelled as starshaped equivalent rods, matched the experimental frequencies and modal shapes.
Notes
Acknowledgements
The authors would like to acknowledge Dr. Ing. Davide Tonelli for his valuable collaboration, Prof. Ing. Paolo S. Valvo, Dr. Ing. Giuseppe Chellini and Mr. Michele Di Ruscio for the competent support provided during the experimental tests.
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