Determination of modal damping ratio for a hybrid floor system

1.1. Introduction to topic

The increasing requirement to produce structural systems that meet, in addition to the essential conditions of strength and stability, specific demands of energy efficiency and comfort, led to the design of hybrid systems which present many advantages related to their use. Globally, economic demands and modern technologies in construction operation have solicited the structural designers to come up with optimal solutions that lead to building components with specific prefabricated form, with reduced costs of production and assembly, and with slimmed down execution times. These requirements are applied also for the structural solutions of building timber floors. The structural elements have, sometimes, lower stiffness characteristics and, consequently, a greater vulnerability to the vibrations produced by various operating activities, the phenomenon being commonly met in dynamic actions.

Most problems encountered in practice for floor system have, as a starting point, the resonance phenomenon of one or more vibration eigenmodes produced by dynamic actions occurring during use. The eigenmodes, whose frequencies are in the working frequency range, will always represent a potential hazard under resonance and hence, the need for modal analysis, which studies the vibration modes, modes that characterize the properties of the floor system. Modal analysis can be approached by analytical or experimental. The experimental method is based on experimentally measured modal data, and allows identification of vibration modes of the structure. The results of these experimental studies, allow verification of admission requirements of modal parameters in the limits of technical regulations.

The construction is found extending the use of hybrid structural solutions for floors made of timber and metal components. Timber floor systems using trusses type MiTek, fit in hybrid structural solutions group, Fig. 1, [1]. Trusses of this type are innovative structural elements with good resistance and low energy consumption, performance-enhancing of timber floors through ease of installation, prefabrication and a more rigorous control of the mechanical characteristics of the elements. These floor trusses allow running all services in any direction of the building also.

Fig. 1Floor made of hybrid timber-metal beams

Hybrid trusses made of timber elements fastened with metal joining system are shown in Fig. 2, [2]. PosiStrut systems are modern joining systems used in timber constructions, and are part of multipath metal connectors used to manufacture the floor trusses. The timber planks have cross-section sizes reduced, and the metal lattice strips are made of pre-galvanized steel sheet with a thickness of 1 mm.

Fig. 2Hybrid beams type MiTek with PosiStrut metal joining system

The presented hybrid system of floor, Fig. 1, is currently frequently used for timber construction due to the structural shapes and small sizes of the components, which provide the necessary strength and economic efficiency. The timber-metal hybrid trusses, due to the reduced stiffness, are susceptible to vibrations. The conditions imposed by technical regulations for checking the serviceability limit states (SLS), in particular, the regulations for vibration level, are correctly evaluated if it is used the actual values of the modal parameters in analytical models. These actual values are strongly influenced by the peculiarities of the structural elements (shape, sizes, materials used etc.) and, in the present, the standardised imposed values are overcome by the innovation speed of structural concepts. There are few research studies in identification of the modal parameters, therefore, these particular structural compositions gives no certainty of correct results obtained from analytical models stipulated in technical regulations, to satisfy the criterion of comfort to the vibration effect on users.

In the mechanical testing laboratory of the faculty of Civil Engineering and Building Services from the “Gh. Asachi” Technical University of Iasi, it was developed an experimental research program to proper define behaviour of a floor module made of MiTek hybrid trusses subjected to vibrations, [3, 4]. The lack of detailed experimental studies of these floor systems, both in Romania and internationally, requires the development of the research through laboratory tests, to identify the modal parameters and to develop the solutions of improving the vibration behaviour of floors made of hybrid trusses.

The research purpose presented in paper is to evaluate the real values of modal damping ratio using experimental determined resonance frequency plot. The methodologies of experimental test and theoretical evaluation of the modal damping ratio in the case of a module of floor hybrid system, supported along two edges, are presented.

1.2. State of the art for assessing the vibration behaviour of floor structures

In the literature are found case studies on vibrations behaviour of traditional timber floors that have as the main objectives, in general, verifying the accuracy of the formula given by European design code, [5], and the contribution of the stiffeners, the contribution of the structural plywood from the top, or the contribution of the ceiling plaster sheets from the underside, to the overall stiffness of the floor system, [6, 7]. Another series of experimental research investigates the dynamic response of the new types of floors, recently introduced in the construction market, [8-12]. The results of these experiments have focused on determining the fundamental frequencies, vibration modes and the modal damping ratio, [3].

The European standard EC 5 describes the principles and requirements for safe operation and durability of timber structures. The essential requirements on the limit states of normal operation, in the case of the timber floors, are expressed by the joint slip, limiting values for deflections of beams, and the vibration level. The vibration level should be estimated by measurements or by calculation taking into account the expected stiffness of the member, component or structure, and the modal damping ratio [5]. The chapter 7.3 – “Serviceability limit states. Vibrations” of the EC 5 presents the design rules for action on timber structures, which may reasonably be anticipated to cause vibrations which do not affect the activity in normal conditions or to cause discomfort to occupants. The calculation and verification methodology of requirements is applied to the timber floors whose fundamental frequency ($f_0$) is greater than 8 Hz. For timber floors with a fundamental frequency less than 8 Hz, a special investigation should be made. Frequency components above 40 Hz may be disregarded, [5]. Checks to be made, according to European design code are: deflection control to a concentrated static load and vibration speed control. Limiting speed, $v$, of the response to an ideal unit impulse requires knowing the values of modal damping ratio, $\xi$. For timber floors, unless other values are proven to be more appropriate, a modal damping ratio of 1.0 should be assumed.

In national technical regulations or in the National Annexes of the EC 5, each country can adopt different values for the modal damping ratio according to certain criteria. An example, [13], which the authors consider it to pinpoint, is norm SIA, [14], according to which the indicative values for the modal damping ratio may be different depending on the type of floor structure:

– $\xi =$ 1.0 – timber structures without metal connections;

– $\xi =$ 1.5 – timber structures with metal connections;

– $\xi =$ 1.0 – timber floor without floating floor;

– $\xi =$ 2.0 – timber floor consists of glued elements (tiles) on floating floor;

– $\xi =$ 3.0 – timber floor consists of precambered beams and floating floors of planks.

The evaluation methodology of EN 1995-1-1/2004, [5], and the limits of modal damping ratio values presented do not give the confidence that they fit, theoretically, the case study presented in this paper.

1.3. Analytical models to determine the modal damping ratio

Experimental determination of the modal damping ratio of structural systems can be achieved by: 1) Analysing the free vibrations of the system and 2) Studying the resonance response of the system.

1) In the case of free vibration, the modal damping ratio can be determined by measuring two successive amplitudes, $a_1$ and $a_2$, from the plot of the frequency response of the system, and then calculating the logarithmic decrement, $\mathrm{\Delta }$. The modal damping ratio, $\xi$, is determined using Eq. (1):

2) In the case of studying the resonance responses of the system, the forced vibrations are used for the modal damping ratio determination, by measuring the amplitude and the frequency of vibration at resonance, $a_r$. At resonance, the vibration amplitude is calculated using the Eq. (2):

where $f_0$ is the fundamental frequency (un-damped natural frequency), $k$ is the elastic stiffness of the system, and $\xi$ is modal damping ratio (also known as damping factor, or percentage from critical damping).

When the systems have small damping, the resonance amplitude is calculated using the Eq. (3):

To determine the modal damping ratio, $\xi$, a harmonic excitation with variable pulsations (angular frequencies) is applied to the system, and then it is measured the resonance amplitude from the plot of the frequency response of the system. The modal damping ratio is evaluated according to Eq. (4), by calculating or experimentally determining the elastic stiffness of the system, $k$, and the fundamental frequency of the system, $f_0$:

Alternatively if $\xi$ is not assumed small, the relationship becomes:

Fig. 3The plot of the frequency response of the system in resonance

Experimental determination of the resonance response of the system, Fig. 3, allows the identification of the points, A and B, (on the plot of the frequency response of the system in resonance), for which the amplitude of vibration is:

Between frequency band, $\mathrm{\Delta }f$, delimited by points A and B, the resonance characteristic frequency, $f_r$, and the modal damping ratio, $\xi$, there is the dependence expressed by Eq. (7), [15]:

The modal damping ratio is calculated then using the Eq. (8):

2.1. Hybrid floor model used in experiments

The experimental model is a hybrid floor, with the span of 4000 mm and a width of 2500 mm. The floor, Fig. 4, is supported on two edges. For compiling the floor, it were used five MiTek type beams made of spruce boards with dimensions of 98×48 mm, fastened with PosiStrut joining systems made of galvanized steel with thickness of 1 mm, giving a total truss height of 204 mm and a length of 4000 mm. The floor, at midspan, is reinforced with longitudinal stiffeners made of timber, with cross-section dimensions of 35×97 mm, fixed to each truss individually and perpendicular to them. Between the trusses ends were arranged, too, stiffeners made of timber boards. Over floor trusses were fixed OSB plates, having a thickness of 22 mm.

Fig. 4The experimental model of the hybrid floor

The distance between two consecutive hybrid trusses was 600 mm, the standard spacing for these structural elements. The floor was supported by bearing elements of timber lattice girders type. Timber lattice girders have had a width of 100 mm and overall height of 700 mm. They have also had bracings at the ends and at midspan. The hybrid trusses were supported by girders, directly or through thick rubber pads of 15 mm.

2.2. Description of experimental methodology

Determining the modal damping ratio of system was done by two methodologies. The first methodology was based on free vibrations (Method 1) and the second methodology was based on forced vibrations (Method 2). These are described below.

In the Method 1 (free vibrations), to capture signals were used six accelerometers type KD 35, with sensitivity of 50 mV/g, positioned vertically. With each of the six accelerometers, five measurements were performed, in all cases an accelerometer has been kept in the centre or in the marginal area as reference. Two sets of experimental measurements were made.

Data acquisition system has been type-National Instruments using LabVIEW software. Rubber hammer blows are the source of excitation, Fig. 5.

The recordings were made simultaneously for the 6 channels over a total period of 60 s with a sampling period (time difference between the two readings) of 0.001 s (1000 readings/s). ARTeMIS Test (version 2011) was used to model the experimental tests. ARTeMIS Extractor (v. 2011) was used for dynamic identification of the floor hybrid system which was done by applying the following techniques: EFDD and SSI-UPC (Enhanced Frequency Domain Decomposition and Un-weighted Principal Component). Fourier analysis was conducted in 1024 averaging points. Vibration modes of the hybrid system were analysed up to the frequency of 50 Hz. The modal damping ratio was calculated using Eq. (1) in logarithmic decrement technique from the logarithmic envelope of the correlation function. The estimation is performed by using a linear regression technique.

In the Method 2 (forced vibrations), vibrations capture was done with a seismic velocity transducer, Kinemetrics SS1 model, with a very high sensitivity (345 V/m/s), connected to a data acquisition system that converts the analogue signal into a digital signal, recorded and stored on the hard disk of a computer. The used excitation source was an electro-dynamic vibration generator, Fig. 6. The vibration generator is connected to an amplifier that is controlled by an oscillator of sine waves with variable frequencies. Step amending the curves of resonance frequency was 0.01 Hz. A total of seven resonance curves have been identified related to the positions of excitation source. The modal damping ratio was calculated based on the resonance curves using Eq. (4).

Fig. 5Experimental test in the first methodology (free vibrations)

Fig. 6Experimental test in the second methodology (forced vibrations)