Duality between time and frequency domains for vibration serviceability analysis of floor structures

For vibration serviceability of floors, current design guidelines adopt different criteria to assess vibration levels due to human walking dynamic excitation. Whatever the adopted criterion is, it requires a quantified vibration response of the structure. This quantification could be achieved following either a time- or a frequency-domain approach to response analysis. Each approach has its advantages and disadvantages. For instance, when using the time-domain analysis, exact time-domain amplitudes of the response time histories could be quantified but the process could take time. On the other hand, a frequency-domain analysis approach could reduce the calculation time, but it is impossible to recover exact time-domain amplitudes of the response, which is essentially averaged by the process of calculation. In this paper, the theoretical duality between time and frequency domains is examined practically in the context of vibration serviceability of a floor structure. Weight normalised vertical ground reaction force (GRF) measured on an instrumented treadmill due to walking is used for that purpose because it has realistic distribution of energy in the frequency domain. This GRF is applied on a finite element model of a reinforced concrete high-frequency floor and the responses are calculated via both time and frequency domain analyses. Comparison of these two methods reveals that time- domain analysis could introduce significant errors in the calculated vibration responses. This is due to the errors in the numerical solution of equation of motion.

Prototype structure

A prototype reinforced concrete structure was modelled using the finite element package ANSYS ® Academic Research, Release 17.1. The floor was modelled as an orthotropic shell structure with a thickness of 150 mm. It has three spans of 4.0 m length, three bays of 6.0 m width, and is supported by 300 mm × 600 mm reinforced concrete beams spanning across the 6.0 m wide bays. The beams are supported by 4.2 m high reinforced concrete columns with a cross section of 300 mm × 300 mm. For vibration serviceability considerations, the floor was modelled following the recommended techniques available in the state-of-the-art design guidelines where columns above and below the floor were modeled with fixed supports introduced at the far end from the floor. An overview of the floor FE model is shown in Fig. Modal analysis was performed and 38 modes of vibration up to 60 Hz were calculated. The first six modes of vibration are shown in Fig. 1.

Analysis approach

Both time-domain and frequency-domain analyses were used to calculate the Auto Spectral Density (ASD) of the response, which is the basis for evaluating the criterion for assessing the vibration serviceability of floors. By utilising Discrete Fourier Transform (DFT). where f is the frequency, and df is the frequency resolution of the DFT. For all analyses in this paper, a real walking force measured using an instrumented treadmill was utilised. The walking force was sampled at 200 Hz and the test subject speed was controlled. The complete process of measuring the force is described in detail by Brownjohn. The sampling frequency of the time history governs the size of the integration time-step in the time-domain analysis. Hence, it is required to resample the measured force in order to increase or decrease the integration time-step, which effect on the response calculations is investigated in this paper.

Conclusions
A response analysis was carried out on a prototype FE floor model using both time- and frequency-domain analyses. It is shown that the time-domain approach may fail to predict the responses of a linear system near resonances when compared with the frequency-domain approach. It is recommended that analysts should take extra care when utilising time-domain analysis for vibration serviceability assessment, as it may introduce errors when calculating RMS acceleration response.