Modal Analysis. Werner Rücker. 6.1 Scope of Modal Analysis. 6.2 Excitation of Structures and Systems for Modal Analysis

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1 Modal Analysis Werner Rücker 6 The numerical modal analysis constitutes a methodology for the examination of the dynamic behaviour of structures, systems and buildings. The mechanical characteristics of structural systems are described by the parameters frequencies, damping values and vibration modes. The knowledge of these modal parameters facilitates to interpret the vibration of a system on the basis of measurements. Furthermore, these parameters can be used to formulate requirements for retrofitting and modifying the structural system. Modal data can be used for the following tasks: Design verification, modification and optimisation Prototype development Improvement of numerical dynamic models Measures for vibration reduction Verification of the refurbishment and retrofitting success of structures Condition monitoring and damage identification. This chapter describes the principles and methods of modal analysis and presents examples showing the application variety of numerical and analytical case studies. W. Rücker (&) Bundesanstalt für Materialforschung und prüfung, Unter den Eichen 87, Berlin, Germany werner.ruecker@bam.de 6.1 Scope of Modal Analysis Dynamically loaded structures have to be designed so that the excitation frequencies do not overlap with their natural frequencies. Therefore, it is of outstanding importance to understand and to visualise the modal parameters of different structural types in the design process which also helps to identify weak structural areas. The knowledge of the modal damping is important as it often determines the service life of a structure. On the basis of the results of numerical modal analysis it can be decided whether a numerical modal analysis model complies with the existing structure. A damage assessment of dynamically loaded structures may be achieved by the determination of the natural frequencies, mode shapes and modal damping. A possible necessary reduction of non-tolerable dynamic loading caused by, e.g. changing operation conditions of production facilities is often only technically practicable after the knowledge of the modal parameters. 6.2 Excitation of Structures and Systems for Modal Analysis The appropriate methodology for the excitation of structures and systems has to be chosen in such a way that the required modal parameters can be determined uniquely, precisely and H. Czichos (ed.), Handbook of Technical Diagnostics, DOI: / _6, Ó Springer-Verlag Berlin Heidelberg

2 110 W. Rücker Fig. 6.1 Comparison of result of dynamic analysis with impact and ambient excitation of a reinforced steel bridge for pedestrians comprehensively. This leads to requirements for the scale and for the frequency range as well as for the spatial location and distribution of the excitation on the structure. In general, ambient excitations such as seismic activities, wind and wave loading, traffic, operation excitations and others or excitations such as an impact hammer, drop weights and oscillator systems can be applied for the excitation of structural systems. The first step for the design of a modal analysis is to define the frequency and mode shape range of interest. Generally, these ranges are determined on the basis of a simplistic analytical model or pre-measurements. Furthermore, the degrees of freedom of interest are to be defined. In general, it is sensible to use natural (ambient) excitation sources as they have substantial benefits in terms of costs because no excitation systems are required. The disadvantage of an ambient excitation might be the fact that the excitation frequencies are not uniformly distributed in the frequency range of interest (Fig. 6.1). Further, the excitation may contain signals which have their origin in a surrounding

3 6 Modal Analysis 111 Fig. 6.2 Eccentric cable stayed 60 t cargo barge on the Vasco da Gama Bridge; release of the loading and the corresponding signal [3] production facility. Also, other effects can cause a scatter in the modal parameters, such as different wind conditions for high rise buildings[4]. The possible excitation frequency range is of importance for the application of excitation systems. For instance, the first natural frequency for high rise buildings is usually below 1 Hz. For the excitation of such a system a so-called jumping excitation is sensible. A jumping excitation is performed by a deflection of the structural system (with a force applied by a cable) followed by a sudden release. Examples of such an excitation are depicted in Figs. 6.2 and 6.3. For a frequency range of 4 to about 1,000 Hz and more, shakers on the basis of servo hydraulic and electro dynamic principles are available (Figs. 6.4 and 6.5). In principle, any force time series can be imposed with the mentioned shakers on the structural system. The advantage of a harmonic excitation is that the whole excitation energy is concentrated in one frequency which can lead to an optimal response noise ratio. Furthermore, possible nonlinear structural effects can be detected. Often, the very long time for testing is a disadvantage for this excitation type. For application of dynamic loading, a limited noise signal which can excite in a short time period a significant frequency range can also be used. The limited size of the forces of this excitation methodology can lead to several repetitions of the measurements and necessitate averaging to achieve a reasonable signal-noise ratio. With relatively little technical effort, systems and structures can be dynamically loaded using so-called impulse excitation. It can be induced by impact hammers, drop weights, cartridge ignitions or vehicle impacts as well as by starting or braking manoeuvres. The performed force-time series can be measured or used as initial forces for additional free vibration tests to determine the system s damping. Standard impact hammers can be used for excitation of medium-sized

4 112 W. Rücker Fig. 6.5 Electro dynamic vibration exciter on a footbridge, max dynamic load =±1.5 kn, 4 B f B 1000 Hz Fig. 6.3 Tegel Harbour Bridge: excitation due to cable release with a ship Fig. 6.6 Impulse hammer for excitation of civil structures, max dynamic load 22 kn, f B 750 Hz Fig. 6.4 Servo-hydraulic vibration exciter for excitation of large civil structures, max dynamic load =±5kN, f C 2.3 Hz bridges to a length of about 30 m (Fig. 6.6). They feature an integrated load cell for simultaneous measurement of the induced forces. Targeted and/ or synchronised hopping of humans (Fig. 6.7) can either induce an impulse force or, by using a pulse clock, can also induce a periodic excitation force into the structure of examination. For all the described excitation systems it is important to consider that the additional mass or interconnection to the structure will not change the dynamic properties of the systems and structures under investigation to a significant amount.

5 6 Modal Analysis 113 Fig. 6.7 Excitation by hopping: people on the grandstand of the stadium Cologne-Müngerdorf 6.3 Applications Modal Analysis of Bridges For a prestressed concrete highway bridge in Berlin (the Westend Bridge), shown in Fig. 6.8, specific procedures for automated monitoring and damage detection had to be developed and tested. In a first step, an extensive experimental modal analysis was carried out. The aim here was to use the data for validation and improvement of a numerical finite-element model in a way that the structural behaviour of the numerical model equals that of the real bridge. With the aid of such validated numerical models of real structures and utilising measured data series, possible structural damage, its location, type and quantity can be determined. For performing modal analysis a measuring grid with five measurement lines and a total of 245 measurement points was installed on the bridge deck. Another 32 measuring points were installed at the bridge columns. The used sensors were velocity transducers and accelerometers of the type Bruel and Kjaer 8306 (Fig. 6.9). For excitation, an electro dynamic shaker of EMPA, Dübendorf, Switzerland, was used (Fig. 6.4). The location of the excitation was determined by preliminary analysis in a way such that as many vibration modes as possible could be excited within the frequency range up to about 20 Hz. For determination of the structures eigenfrequencies and mode shapes the simultaneously measured excitation and vibration responses of the bridge were used to identify the transfer functions. The relevant eigenfrequencies and mode shapes were then derived from the transfer functions utilising the so-called phase separation procedure. Figure 6.10 shows some of the experimentally determined mode shapes up to a frequency of Hz. In addition, an averaged power density spectrum of a bridge measurement point is shown. The Kronprinzen-Bridge (Fig. 6.11) spans the River Spree in Berlin-Centre and has a length of 75 m. The steel-tube construction with rigid crossbeams carries an orthotropic steel deck. For determining the bridge s modal behaviour, five measurement lines in longitudinal bridge direction were installed on the deck, a total measurement grid of 145 measurement points. The behaviour of the bridge bearings were recorded on 20 measurement points. For the determination of the vibration behaviour of the bridge, vibration velocity transducers of type HS1 were used. The aim of this investigation was to use a complete set of the structure s modal data for validation of a numerical model of the structure to reality. With the improved numerical model the assumptions of the structural analysis were revised. Among others, it included the assumptions regarding the fatigue damage prognosis. Figure 6.11 shows two characteristic mode shapes of frequencies 8.42 and Hz. The upper mode shape is a superposition of bending and torsion modes, in which the bridge bearings are also involved. The lower eigenmode is a superposition of bending modes of the longitudinal and transverse beams, which induces plate vibrations into the bridge. The excitation of this vibration test was induced by an impact hammer (Fig. 6.6).

6 114 W. Rücker Fig. 6.8 Highway bridge Westend in Berlin Fig. 6.9 Sensors for dynamic analysis: vibration velocity transducers (Typ HS1) (left and right) and accelerometer (left) with fastening elements, low frequency velocity sensor (right) For comparison, an additional experimental modal analysis was carried out under ambient excitation Dynamic Examination of Machine Foundations The aim of the dynamic examination of the machine foundation shown in Fig was to adjust the parameters of a numerical model to realistic values. Therefore, numerically calculated and experimentally measured modal parameters were compared. A major task was to determine the modal damping values which result amongst others from the air damped suspension of the foundation. Mainly accelerometers were used to capture the elastic mode shapes and the rigid body modes, which are also relevant for this system. The excitation of the system was carried out by hopping and ambient excitation. Because no excitation forces can be measured for these two excitation mechanisms, so-called output-only methods [1] were used to determine the modal values. In Fig the numerical model of the foundation is shown as well as a relevant mode shape which consists of two rigid body relaxation vibrations superimposed with elastic deformations of the foundation sidewalls. In the long-term monitoring of a prototype of an offshore wind energy plant, the vibrations were measured with accelerometers. Later, these data were used for numerical modal analysis. This was also to compare the experimental modal parameters with numerical parameters

7 6 Modal Analysis 115 Fig Experimental eigenfrequencies and mode shapes (top) of the 250 m long Westend Bridge determined during the structural design of the construction. The construction and some relevant mode shapes are given in Fig The excitation of the construction resulted from wind and operation. Using the so called peak-picking method the modal data was determined from the vibration responses of the construction. An additional analysis applying the results of a numerical modal analysis was done on the basis of autoregressive models [2]. Only the comparison to the numerical results allowed the interpretation of the vibration modes inherent to the dominant frequencies. Figure 6.13 represents the first two bending mode shapes (indicated by 1 and 2B), and an operational vibration mode (indicated by 3P) resulting from the blade passage frequency. In addition, a further vibration mode of the structure (1RB) is shown which is excited by the vibrations of the rotor blades in their first natural frequency by the dynamic coupling of the two plant components. Due to the limited number of measuring points higher vibration modes cannot be observed Safety of Roof Constructions Figure 6.15 shows the result of an experimental modal analysis of a football stadium in Braga, Portugal (Fig. 6.14). The dynamic behaviour of the roof construction which covers both tribunes with a span width of 202 m and is connected with cables was examined. To the roofs, which

8 116 W. Rücker Fig Bottom side of the bridge with orthotropic deck (top). Power density spectrum (left). Experimentally determined mode shapes at natural frequencies f N = and f M = 8.42 Hz of the Kronprinzenbrücke (right) have partial length of 60 m, grids of measuring points were attached, each with 15 points. The possible vulnerability of the roof with respect to resonance behaviour was to be examined by using the results of the numerical modal analysis. This kind of stress significantly influences the fatigue strength and thus influences the life cycle. In addition to the modal analysis, a monitoring system was installed to examine the influence of environmental factors on the modal parameters. The influence of wind loads on the damping of the roof was of high interest. Ambient excitation was used for this investigation. The according vibration responses were analysed using different output-only methods [1]. In Fig the determined natural frequencies and mode shapes are presented. Due to a frequency resolution of Hz during the analysis, approximately 12 natural frequencies and mode shapes could be identified for frequencies between 0 and 1.1 Hz. Modal damping for three modes are shown as well in Fig The mean value indicates a very small modal damping. As a consequence, relatively large vibration amplitudes at the corresponding operational excitations have to be expected Stability of the Brandenburg Gate In 1998 it was planned to open the Brandenburg Gate in Berlin (Figs and 6.17) to traffic completely. By previous visual investigations a

9 6 Modal Analysis 117 Fig Machine foundation for rotors of gas turbines (bottom). Torsion mode at frequency 14.4 Hz (top) number of cracks at the Brandenburg Gate and at its surrounding gatehouses were detected. Therefore, the loads due to traffic and other excitations as well as the static and dynamic structural behaviours of the gate should be monitored continuously over a period of slightly more than a year. The planned measurements should first serve as a basis of decision-making for the proposed opening of the gate. They should also create the basis for decisions to realise appropriate and evident reorganisation measures of the Brandenburg Gate and its surrounding gatehouses. The determination of suitable measuring points should be based on the results of the investigations carried out at the construction as well as on a global vibration analysis of the Brandenburg Gate, including its annexes. As a result of the

10 118 W. Rücker Fig M5000_2 Natural vibration modes (red) and operational vibration modes (blue, green) of the wind energy plant Fig Football stadium in Braga, Portugal [3] analysis the other measuring points should be determined at which the structural behaviour will be observed over a long period. At the same time the behaviour of the cracks should also be determined depending on various impacts. Due to traffic actions the dynamic loads cause deformations of the entire structure with its corresponding stresses and strains. These stresses and strains depend on the magnitude of vibration amplitudes and on the local distributed curvatures of the vibration modes. The vibration modes will have the highest amplitudes, if the excitation frequencies of the aforementioned vibration sources coincide with the natural frequencies of the structure (resonance excitation). The results of the modal analysis carried out are as follows: The frequency range of the relevant measured vibration amplitudes is between 1.4 B f B 25 Hz. The highest amplitudes of the traffic excitation are focused in a frequency range

11 6 Modal Analysis 119 Fig Results of the experimental modal analysis for a stadium roof in Braga, Portugal: identified eigenmodes (upper), associated distribution of the determined damping (lower) [3] Fig Brandenburg Gate, built by Langhans from 1788 to 1791, a landmark of Berlin and Germany Fig A sketch of the design by architect Carl Gotthard Langhans between 7.3 B f B 12 Hz. The highest vertical vibration amplitudes occur from the traffic excitation, while the horizontal motions are mainly the result of the natural vibration behaviour. The relevant natural frequencies of the structure are in a range of 1.77 B f B 7.26 Hz, where the first two mode shapes are depicted in Figs and 6.19.

12 120 W. Rücker f 1 = 1,77 Hz (rotation) f 2 = 2,44 Hz (rotation) f 3 = 3,17 Hz f 4 = 7,26 Hz in the direction to Reichstag in the direction to Unter den Linden in the direction to Unter den Linden in the direction to Unter den Linden The above-mentioned natural frequencies are mainly rigid body vibrations. Elastic natural vibrations (shear, torsion and bending vibrations) occur only at frequencies approximately above 7 Hz The motions of the mode shapes for the frequencies f 1 und f 2 are slightly damped (\1 %). This means that these vibration modes can be excited by relative small dynamic loads (e.g. by wind). The dominant natural frequencies of the buildings next to the gate (northern roof house and columned hall) do not coincide with the natural frequency of the gate and also not among themselves. It is not expected that the vibration modes of these frequencies are synchronous with those of the adjacent buildings, which will lead to greater stresses and strains at the transition zone. Hence, the buildings gate and gatehouses need to be departed constructively. Fig First mode shape at frequency 1.77 Hz References Fig Second mode shape at frequency 2.44 Hz 1. Farrar, C.R., James III, G.H.: System identification from ambient vibration measurements of a bridge. J. Sound Vib. 205(1), 1 18 (1997) 2. Rücker, W., Fritzen, C.-P., et al.: IMO-WIND integral system for monitoring and assessment of offshore wind turbines. Joint Final Report on the Projects InnoNet 16INO326 and -327, University of Siegen, BAM, Jan Magalhães, F., Cunha, Á., Caetano, E.: Installation of a continuous dynamic monitoring system at Braga Stadium suspended roof: initial results from automated modal analysis. In: Proceedings EVACES 09, Wroclaw, Oct Brinker, R., Zhang, L., Andersen, P.: Modal identification from ambient responses using frequency domain decomposition. In: Proceedings of IMAC- XVIII, International Modal Analysis Conference, pp , San Antonio, Texas, USA (2000)