By making use of a mass normalised scaling for the mode shapes at the chosen locations of the third bending mode, i.e. [ ] [ ][ ] [ ] Φ Φ = Ι M T , the mass and stiffness matrices of the physical co-ordinates can be derived from Eqs. (1) and (2). Equation 3 is often used to obtain convergence i.e. diagonalisation of the mass and stiffness matrices of the physical co-ordinates. ( ) ( ) 1 1 1 − − − = Φ Φ = ΦΦT T M ( ) T M = ΦΦ −1 (1) ( ) 1 2 1− − = Φ Ω Φ T K (2) [ ] ( ) r T r r T r T ε φ ε φ ε φ ε δφ δφ δφ δφ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ = −1 22 12 21 11 (3) By applying Eqs. (1), (2) and (3) to the mass-normalised mode shapes of the laboratory structure derived from a Finite Element model, the following physical parameters of the 2-DOF lumped mass model in Eqs. (4) and (5) are obtained. .3 ; 5596 2 1 Kg M M= = e N m e N m K K K 7 / 2.9876 7 / ; 0.4565 2 3 1 + = + = = . The assumed Rayleigh damping matrix is as shown in Eq. (6). Mass matrix (Kg) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = 0 5596.3 5596.3 0 M (4) Stiffness matrix (N/m) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + − + + − = 7 7 3.4441 2.9876 7 7 2.9876 3.4441 e e e e K (5) Damping matrix (Ns/m) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − = 616.7 1896.5 1896.5 616.7 C (6) 2.2 Input Forces for Analytical Studies Human movement is often characterised by rhythmical body motions such as walking and running. Such motions induce dynamic loads into the structures they occupy and may result in significant resonant, transient, steady-state or impulsive responses (Bachmann et al 1995). The disturbance forces considered here are walking force time histories obtained from treadmill walking tests and a random excitation signal with a frequency span of 0 – 40 Hz. These are shown in Figs. 2a and 2b for 10s durations. a) Walking force time history (2.25 Hz) (b) Random excitation force (0 – 40 Hz) Fig. 2 Walking force time history and random excitation force time history for analytical simulations 217
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