DESARROLLO DE UNIDADES FORMATIVAS

Similarly to the theoretical model of the one car-bridge model in chapter 5, an FE model of the plate structure is built in ABAQUS to obtain the numerical frequencies and mode shapes of the structure. Shell elements (S4R) and 3D beam elements (B31) are used to model the plate and the rails, respectively. Point mass and spring elements are used to model the additional masses and stiffness provided by the actuators. The material properties of the plate and the rails are identified from measured natural frequencies by modal testing and the stiffness of an actuator is measured by tension and compression tests using an INSTRON machine (Yang et al., 2017). The damping coefficients of actuators are assumed to be the same and identified from a free vibration test of one actuator to be around 1.487 Ns/m. The detailed identification of the damping of the actuator was given in section 5.2.2. The beam elements are tied to the shell elements. The offset between connected beam and shell elements is often taken as a model updating parameter to validate the model by

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comparing theoretical frequencies with measured counterparts (Mottershead et al., 2011). In this chapter, the offset ratio defined in section 5.2.5 is updated. It is found that when the offset ratio is 1.885, the theoretical frequencies of the plate structure are closest to the measured counterparts. Table 6.1 shows the differences between the two sets of frequencies for the first eight modes of the plate structure.

Table 6.1 Comparison between theoretical frequencies and measured frequencies

Mode 1 2 3 4 5 6 7 8

Measured Fre. (Hz) 21.781 23.676 27.995 33.386 68.573 72.964 81.002 89.935 Theoretical Fre. (Hz) 20.218 22.465 28.069 34.919 65.921 71.591 83.429 96.343 Difference (%) -7.2 -5.1 0.3 4.6 -3.9 -1.9 3.0 7.1

Applying the MS method to the equation of motion of the plate structure leads to (Baeza and Ouyang, 2008)

wX3 6 5 diag9 %_{<w3 6 # : ∑}¥ _{"• , ‚ 3 6}

m$ : ∑J†m$ †"3 †, †6"v3 †, †6w43 6 (6.1)

where ( # 1, 2 … o) is the th natural frequency of the plate structure (which includes the mass and stiffness of all actuators), with n being the number of modes used; and the coordinates of the Sth wheel of the cars, _† and _† the coordinates of the th actuator; "3 , 6 is the vector of the analytical modal functions of the plate structure, w the vector of the corresponding modal coordinates; the contact force at the Sth wheel, _† the viscous damping coefficient of the th actuator; a dot over denotes a derivative with respect to . Please note that _{"3 , 6 is} approximated by the products of the element shape functions and the numerical modes of the plate structure which are obtained from the modal analysis of its FE model in ABAQUS.

Each car is treated as a rigid-body with two DOFs and only heave and pitch motions are considered, as shown in Figure 6.3.

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Figure 6.3. Theoretical model of two cars: (a) elevation view of the first car, (b)

cross-section view of the first car, (c) elevation view of the second car, (d) cross- section view of the second car

The contact force 3 6 between the Sth wheel of the cars and the rails can be expressed as

3 6 # 9: $3 6 5 • , , ‚ :_{% $}3 6<, S # 1, 2 (6.2)

3 6 # 9: $3 6 5 • , , ‚ 5_{% $}3 6<, S # 3, 4 (6.3)

3 6 # 9: %3 6 5 • , , ‚ :_{% %}3 6<, S # 5, 6 (6.4)

3 6 # 9: %3 6 5 • , , ‚ 5_{% %}3 6<, S # 7, 8 (6.5)

where _$ and _% are the vertical displacements of the first car and the second car, respectively and _$ and _% are the rotations of the first car and second car, respectively, as shown in Figure 6.3. 3 , , 6 is the displacement of the plate structure at the contact point between the th wheel and the rails ( and are the coordinates of the _{Sth wheel).}

The equation of motion of the first car can be described as

$ X $ # : $° : $9 $5_{% $}: 3 $, $, 6< : %9 $5_{% $}: 3 %, %, 6< : H9 $:_{% $}: 3 H, H, 6< : J9 $:_{% $}: 3 J, J, 6< (6.6)

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$ X$ # : $§ $5_{% $}: 3 $, $, 6¨_%: %§ $5_{% $}: 3 %, %, 6¨_%5

H§ $:_{% $}: 3 H, H, 6¨_%5 J§ $:_{% $}: 3 J, J, 6¨_% (6.7)

The equation of motion of the second car is similar as Eq. (6.6) and Eq. (6.7), but _$,

$, $ and $ to J are changed to %, %, % and £ to ¥, respectively. If the car

connector is used to connect the two cars, the following relationships for the two cars can be obtained

£ # H5 ª, ¤ # J5 ª (6.8)

where _H and _J are the coordinates of the rear wheels of the first car in direction;

£ and ¤ are the coordinates of the front wheels of the second car and Δª is the

distance between the the rear wheel of the first car and the front wheel of the second car in x axis as shown in Figure 6.4. As only the heave and pitch motions of the cars are considered, the two wheels of one wheel-set can be taken at the same coordinate, namely _{H # J} and _{£ # ¤}.

Figure 6.4. Lateral view of relative location of contact forces (unit: cm)

It should be noted that Eq. (6.1) to Eq. (6.7) are valid only when all the wheels of the two cars are on the plate structure. Actually, there are extra stretches of rails in front of and behind the plate structure, as shown in Figure 6.1. The stiffness of the extra rails or spans is taken to be infinitely large, which means 3 , , 6 # 0 when the th wheel of the cars is on the extra rails. Eq. (6.1) to Eq. (6.7) should be changed according to the locations of wheels of the cars. The equations of motion of the plate structure and the cars can be solved by the iterative method presented in section 5.1.

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