Vibration Control of a Gym Floor Using Tuned Mass Dampers: A Numerical Analysis ()
1. Introduction
Human induced loads can cause excessive vibration leading to discomfort and even compromising structural safety. Real occurrences of this problem in structures subjected to loads originated from people movement are worldwide known, a famous example is the London Millenium Footbridge [1]. Floor vibration studies have intensified in the last years. A number of factors cause this effect; the more important are: the use of more resistant materials that lead to more slender and flexible slabs, and the occurrence of non-provided activities in the original structural design. Those things have contributed to increasing emergence of excessive vibration on structures, especially on slabs [2,3]. This problem is directly linked to the structural control devices study. Structural control provide changes on stiffness and damping properties of the system, installing external devices or applying external forces, aiming to reduce original structure vibrations.
Among several structural control devices there is the Tuned Mass Damper (TMD), a damper applied in many practical cases, such as high buildings, footbridges, towers, etc. [4]. It is a mass-spring-dashpot system connected to the structure in a way to vibrate out of phase with the main structure, if tuned to the appropriated frequency. In this way the energy is transferred to the TMD reducing amplitude vibration on the main system. The use of TMD to reduce floor vibration has been studied by many researchers lately [5-9]; however there are still many topics to further development, such as the determination of appropriate TMD parameters to optimize control performance. This work presents a floor vibration study to a practical case of a gym. The slab analyzed presented excessive vibration problems when subjected to typical human induced loading of rhythmic exercise practicing. A control system consisting of various TMD was proposed. This proposal was based on a parametrical study to improve the performance of a slab set with typical dimensions of structures like residential and commercial buildings.
2. Human Induced Vibrations
Nowadays the design of building floors with low natural frequencies, with values near of those from human induced loads, can cause undesirable excessive vibrations.
Thus the understanding of structural dynamic behavior and the characteristics of the dynamic load is very important, particularly to floor slabs. Dynamic loading can produce high vibration levels than can cause discomfort to users and even compromise structural safety. Occurrence’s of these problems are well known worldwide, like the already mentioned London Millenium Footbridge [1]. The stadium grandstands collapse in the 1920 years lead the American Standards Association (ASA, atual ANSI—American National Standards Institute) to nominate a committee to establish safety standards for this type of structure. Decades later, researchers described mathematically human loading arising from typical activities using a Fourier series. Faisca [10] performed an experimental study focusing on this type of load, describing human induced dynamic load considering loss of contact with the structure. According to her, these loads can change depending on structure’s flexibility, namely they depend on people-structure interaction. Varela [5] studied human induced loading correlating the mathematical model with experimental results.
3. Structural Control: Tuned Mass Damper (TMD)
Tuned Mass Damper reduces the energy dissipation request of structural members when subjected to dynamic loads. This reduction occurs by transferring a portion of the vibratory energy to the TMD that in its simplest conception is a mass-spring-dashpot system connected to the structure, like shown on Figure 1. Considering the main system like a single-degree-of-freedom model, with mass M, damping C and stiffness K, on which is applied a dynamic load f(t) and attached a TMD with mass m, damping c and stiffness k, the equations of motion are:
(1)
(2)
where:
ÿ(t): main system acceleration;
y(t): main system velocity;
y(t): main system displacement;
z(t): relative displacemente between the TMD and the main system.
When installed, the TMD tries to bring resonance peaks to lower values; this effect is desirable occurring on a wide frequency range.
For this to happen, TMD optimum parameters should be obtained, improving the control system performance [11]. A large number of these devices have been installed in high buildings, bridges, towers and industrial chimneys to control the dynamic response due to strong winds mostly [2]. In recent years studies were developed to
Figure 1. Structural system model with a TMD installed [4].
harness the potential of TMD on floor vibration control. Battista & Varella [12] presented a practical case of using TMDs in commercial building with metallic beams and composite floor decks that presented excessive vibration levels according to ISO 2631/1, ISO 2631/2 and NBR 8800 ABNT codes. Setareh et al. [8] describe the application of a passive pendulum working as a TMD for two cases with excessive slab vibrations. In this study they emphasize men-structure interaction on experimental results obtained.
4. TMD Parametric Study
The four slab structures studied in this work are shown in Figure 2. The loading applied simulated people on the move.In all cases analyzed it was verified vibration levels higher than those recommended on standard codes. Thus it was performed a study analyzing different TMD configurations in a way to verify the best control proposal for each one of the slabs. This study also analyzed the influence of TMD parameters on the control strategy performance. Because of the absence of design criteria to set parameters to TMD installed on floors, it was performed a parametrical study varying the mass ratio μ, the frequency ratio α and the damping ratio ξ, and verifying for each set of parameters the TMD efficiency. The four models presented natural frequencies near of values that characterize human dynamic loads. On the first slab studied, simply supported at the four edges, the best configuration obtained was a single TMD placed at the midspan, it reduced on 75% displacement amplitude.
Table 1 presents the bests reductions for each slab, applying harmonic loading and human-induced loading, associated frequency ratios are also presented. It can be verified that best reductions were obtained considering the theoretical harmonic loading compared to human-induced loading. In the case of Slab 1, the best verified value of α was 0.95, namely tuning the TMD very close to the corresponding slab natural frequency. For damping ratio it was concluded that increasing its value can worsen TMD