Frequency response function ‐ based explicit framework for dynamic 1 identification in human ‐ structure systems

The aim of this paper is to propose a novel theoretical framework for dynamic identification 10 in a structure occupied by a single human. The framework enables the prediction of the 11 dynamics of the human-structure system from the known properties of the individual system 12 components, the identification of human body dynamics from the known dynamics of the 13 empty structure and the human-structure system and the identification of the properties of the 14 structure from the known dynamics of the human and the human-structure system. The 15 novelty of the proposed framework is the provision of closed-form solutions in terms of 16 frequency response functions obtained by curve fitting measured data. The advantages of the 17 framework over existing methods are that there is neither need for nonlinear optimisation nor 18 need for spatial/modal models of the empty structure and the human-structure system. In 19 addition, the second-order perturbation method is employed to quantify the effect of 20 uncertainties in human body dynamics on the dynamic identification of the empty structure 21 and the human-structure system. The explicit formulation makes the method computationally 22 efficient and straightforward to use. A series of numerical examples and experiments are 23 provided to illustrate the working of the method.


Introduction 27
Dynamic interaction between a human and a low-frequency structure supporting the human is 28 a well-recognised phenomenon that has become increasingly prominent over the last two 29 decades due to the increase in slenderness of modern structures [1][2][3][4]. Naturally, the dynamic 30 properties of the human-structure system are influenced by the interplay of dynamics of the 31 two subsystems and they differ from those of the structure itself [1][2][3][4][5][6][7]. When considering the 32 vertical flexural vibration modes of a structure, the human occupancy is known to cause a 33 shift in the natural frequency and an increase in the damping ratio [3,[8][9][10]. Knowledge of 34 the dynamic properties of both the occupant(s) and the structure is crucial for developing 35 better understanding of the extent of the human-structure interaction and its influence on the 36 dynamic response analysis and vibration control design for structures accommodating 37 humans. 38 In structural engineering applications, the dynamics of a human are usually described using a 39 single-degree-of-freedom (SDOF) mass-spring-damper model [3,6,[9][10][11][12][13][14]. The dynamics of a 40 structure are often described utilising a spatial model or a modal model (having, say, n DOFs) 41 that can be established using either finite element method or modal analysis [15]. The human-42 structure system can then be represented by a  When a stationary human occupies a structure, the structure and the human form a new joint 128 system whose dynamics are influenced by the dynamics of the two individual components. In 129 line with the previous research, the human is modelled as a SDOF system having mass h m , implemented in some previous studies [11,19,[27][28][29][30]. Without loss of generality, it is 132 This paper has been published under the following reference: Wei, X. and Živanović, S. (2018) Frequency response function-based explicit framework for dynamic identification in human-structure systems. Journal of Sound and Vibration, Vol. 422, pp. 453-470, https://doi.org/10.1016/j.jsv.2018.02.015 assumed that the human is located at the -th n degree of freedom of the structure. Therefore, 133 the forced-vibration of the human-structure system can be described by 134 Eq. (3) may be expressed in Laplace domain as 139 where 141 This paper has been published under the following reference: Wei, X. and Živanović, S. (2018) Frequency response function-based explicit framework for dynamic identification in human-structure systems.
In this expression,  

143
For the sake of clarity, dimensions of some matrices are stated in the equation. 144 The receptance matrix of the human-structure system is 145 from which the eigenvalues (and therefore natural frequencies and damping ratios) of the 180 human-structure system can be calculated.

194
The magnitude and phase of the -th pn receptance of the joint system may be expressed as 195 Re Im and 212 This paper has been published under the following reference: Wei, X. and Živanović, S. (2018) Frequency response function-based explicit framework for dynamic identification in human-structure systems. Journal of Sound and Vibration, Vol. 422, pp. 453-470, https://doi.org/10.1016/j.jsv.2018.02.015 2.3. Numerical example: 2DOF human-structure system 214 Let us consider a SDOF structure occupied by a SDOF human shown in Fig. 1. The dynamic 215 properties of the individual human and the structure as well as those of the human-structure 216 system (obtained from eigenvalue analysis of the 2DOF model) are given in Table 1 generates the eigenvalues of the human-structure system 1  The synthesised direct receptance of the human-structure system described by Eq. (23), 237 shown as the thick dashed line in Fig. 2, accurately reproduces the actual receptance of the 238 human-structure system shown as the thin solid line. Fig. 2 also shows that the presence of 239 the human shifts the frequency from 4.85 Hz for the empty structure (dash-dotted line) to 240 4.51 Hz for the human-structure system. It also significantly increases the damping ratio of 241 the mode dominated by structural motion (from 2.0% to 7.3%). The mode dominated by 242 human motion is heavily damped which is the reason that it cannot be observed in the 243 receptance graph for the human-structure system. The structure in this example is an actual 244 16.9 m long glass FRP composite bridge [34]. The example demonstrates that the presence of 245 a single human can significantly modify the dynamics of the empty structure. 246  , estimated using the second-order perturbation method 252 [32]). By using the proposed uncertainty estimation method in Section 2.2, the expectation 253 and standard deviation of the direct receptance of the human-structure system can be obtained, 254 as plotted in Fig. 3. The coefficient of variation (CoV) of 10% for both the damping and 255 stiffness of the human body led to the maximum CoV of 9% for the magnitude and phase of 256 the predicted FRF of the human-structure system. The predicted expectations and standard 257 deviations shown in Fig. 3 were verified using Monte Carlo simulations (sample size = 1000). 258

Theoretical derivations 265
Let us assume that the direct receptance at the -th n DOF s nn h of the empty structure is 266 available (for example, through modal testing). In addition, let us assume that the direct 267 receptance at the -th n DOF or the cross receptance between the -th p output and the -th n input 268 of the human-structure system is also available resulting in identifying the complex conjugate 269 As can be seen from Eqs. (25) and (27), the identification of human body dynamics relies on 287 the quality of the curve fitting of the FRFs of the empty structure and also of the FRFs around 288 the modes dominated by structural motion of the joint system. The strategies for performing 289 curve fitting have been investigated elsewhere, e.g. [35,36], and have not been elaborated in 290 this paper. 291

Experimental case study: Dynamic properties of a human in a standing posture 292
The use of the proposed method is demonstrated on an example of identifying the dynamic 293 properties of a human standing on a steel-concrete composite bridge situated in the Structures 294 Laboratory at the University of Warwick (Fig. 4). The bridge has a mass of 16,500 kg whilst 295 its deck is 19.9 m long and 2 m wide. It sits on two meccano frames that span 16.34 m. The The dynamic properties of the human body were identified at three different force levels. The 321 induced maximum accelerations at the driving point on the empty bridge and the human-322 bridge system ranged from 0.36 m·s -2 to 0.65 m·s -2 and from 0.34 m·s -2 to 0.62 m·s -2 , 323 respectively. The frequencies and damping ratios of the empty structure showed negligible 324 variation with the response level. The same conclusion was drawn for the human-structure 325 system. These findings suggest that the empty bridge and the human-bridge system exhibited 326 relatively linear behaviour at the three different force levels and they all resulted in almost the 327 same properties of the human body. The force level chosen for presentation in this paper is 328 shown in Fig. 7 whilst the corresponding vibration response at TP1 for the unoccupied bridge 329 This paper has been published under the following reference: Wei, X. and Živanović, S. (2018) Frequency response function-based explicit framework for dynamic identification in human-structure systems. Journal of Sound and Vibration, Vol. 422, pp. 453-470, https://doi.org/10.1016/j.jsv.2018.02.015 21 is shown in Fig. 8. The direct and cross accelerances for the empty bridge and the bridge 330 occupied by the test subject are shown in Fig. 9 and Fig. 10, respectively. The two figures 331 show that the presence of the test subject affects the dynamics of the system slightly. 332

335
The measured accelerance ,11 s a h of the empty structure was curve fitted using the rational 336 fraction polynomial method [35]. Good agreement between the curve-fitted (CF) 337 accelereance and its measured counterpart is demonstrated in Fig. 11. The analytical 338 expression of the curve-fitted accelerance is 339    Fig. 12 and Fig. 13 show that the synthesised accelerances (thick dashed curves) of the 352 human-structure system agree well with their measured counterparts (thin solid curves This paper has been published under the following reference: Wei, X. and Živanović, S.

Experimental case study: Dynamic properties of the structure 438
The use of the proposed method is demonstrated on an example of identifying the properties 439 of the bridge from Section 3.2, by utilising the measured accelerances while the test subject 440 was standing on the structure and known properties of the human. The measured direct and 441 cross accelerances were curve fitted using the rational fraction polynomial method [35].

Numerical example: correcting multiple modes of a bridge 476
A three-span continuous bridge (Fig. 18) Table 2. 502 By using the proposed method, the known human body dynamics, Eqs. (38) and (29), the 503 direct receptance of the unoccupied bridge in the frequency range from 2 Hz to 8 Hz can be 504 synthesised, shown by the thick dashed line in Fig. 19. It can be seen that the synthesised 505 receptance of the unoccupied bridge is coincident with the actual receptance (denoted as Act 506 in Fig. 19) of the unoccupied bridge (the thin solid line in Fig. 19), which was obtained 507 numerically. The natural frequencies and damping ratios of the unoccupied bridge were then 508 determined by solving the characteristic equation obtained from the synthesised direct 509 receptance of the unoccupied bridge and they are presented in Table 2.  The occupancy of the human increases the damping ratios of the first three structural motion 513 dominated modes by 443%, 413% and 145%, respectively (Table 2). By contrast, it decreases 514 the frequencies of the first three modes by 5.1%, 5.1% and 0.3%, respectively. This example 515 demonstrates that had the measured human-structure receptances not been corrected, the 516 modal properties of the structure would be erroneous. 517 Fig. 19 shows that the proposed method corrects the dynamic properties of the first three 518 modes (two of which are relatively closely spaced) simultaneously which is advantageous 519 compared with the methods that rely on SDOF models of the structure. 520