«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»
The deflection components of the bridge are represented in terms of the mode shapes and generalized coordinate as (Jain et al. 1996)
where n1 = total number of natural modes considered; ξ i ( t ) = generalized coordinate; r = h, p or α; h(x), p(x), and α(x) = vertical, lateral, and torsional mode shape, respectively and
where ξ = generalized coordinate vector; the superscript prime “`” represents a derivative with respect to dimensionless time s = Ut / b ; I = identity matrix; Q b = excitation force vector normalized to the generalized mass inertia; FTMD = reaction force vector of TMD on
where Ii = generalized mass inertia for the ith mode; ρ = air density; l = bridge length;
H*, Pi*, A* ( i = 1 − 6) = experimentally determined flutter derivatives for the bridge deck; and i i the modal integrals ( G ris j ) are computed as
where ωTMDp, ζ TMDp and m TMDp = circular frequency; damping ratio; and mass for the pth TMD, respectively.
Similar simplification to that by Linda and Donald (1998) is followed below. The ith equation is extracted from Eq. (6.12) and rewritten as
The superscript “un” for ξiun (K ) stands for the uncoupled single-mode solution that is obtained from Eq. (6.19) by ignoring all the coupling effect. Eq. (6.21) thus represents the ratio of coupled and uncoupled solutions.
Eq. (6.19) can be rewritten considering Eq. (6.21) as
According to the definition in Eq. (6.21), ξi (K) and ξ j (K) are equal to 1 if the modal coupling effect is entirely omitted. Otherwise, for weak coupling system, there exists the condition ε 1. When the ith mode is under study (defined as current mode thereafter), its reduced frequency Ki is different from Kj. For ε 1, following equation can be derived (Linda and Donald 1998).
In the above derivation process, coupling effects with the order of O(ε2) are ignored.
Physically, this is to say that when the response of the ith mode is determined, the coupling effects between the ith mode and other modes (jth mode) are included in the solution.
However, the coupling effects between jth mode and the other modes (except for ith mode) are deemed negligible as indicated in (6.25). Since only the high order small terms are ignored, the accuracy of the solution is not significantly scarified (Linda and Donald 1998).
Converting (6.26) back to the original form (versus dimensionless form) gives
If the cross-modal buffeting spectrum is omitted (Simiu and Scanlan 1996), the power spectral density (PSD) for the generalized displacements of ith mode, ξi, is derived from (6.27) as
(6.31) 137 Eq.(6.28) indicates that the coupled response of each mode mainly consists of two parts. The first part (the first term) is the uncoupled response of the current ith mode, namely the resonant component of the ith mode buffeting. The second part (the second term) is due to the modal coupling between the ith mode and other modes and is called the coupled component of the ith mode buffeting here. In Eqs. (6.29), (6.30), and (6.31), the term associated with term R represents the contribution of the TMDs to the bridge vibrations. It can be seen from these equations that including TMDs may affect not only the first term of Eq. (6.28) the resonant component, but also the coupling component of the second term. The traditional control approach of resonant-suppression that targets at the first term is hardly able to control the coupling component directly. A new control approach may be naturally inspired to optimize the control efficiency by reducing the total response, not just the resonant vibration. This new control approach will be discussed below with numerical examples.
6.3 Coupled Vibration Control with a Typical 2DOF Model
As discussed above, conventional control strategy is to suppress resonant vibration that is essentially represented by the first term of Eq. (6.28). If the modal coupling among the current ith mode and the other modes is very weak, the second term of Eq. (6.28) will be trivial. In that case, conventional single-mode-based control analysis without considering the effect from the second term of Eq. (6.28) could lead to acceptable results. However, for the modes with strong modal coupling, the contribution of the second term to the total response can be significant. It becomes necessary to consider both the resonant vibration and that from coupling effects to achieve the optimal performance.
To examine this concept and verify the closed-form derivation conducted above, a simple 2DOF system attached with two identical TMDs was considered as shown in Fig. 6.1, where the parameters associated with masses M1, M2 and Mp represent the 1st DOF, the 2nd DOF, and the TMD DOF, respectively. The parameters for this 2DOF model are defined in Table 6.1.
Two coupled modes are the most typical and easiest example whose closed-form results can be more conveniently derived. Using two identical TMDs makes it easy to distinguish clearly the control effect on any part of the vibrations. For simplicity but without losing generality, it is assumed that the external excitation is white noise with a power spectral density of S0.
According to Eqs. from (6.28) to (6.31), the solution of the 2DOF model (n1 = 2) may reduce to
It is shown in Eq. (6.32) that the closed-form solution for this simple case (with 2DOF model and two identical TMDs) can be conveniently derived. By assuming that the structural damping ratios of both the 1st DOF (M1) and 2nd DOF (M2) are as low as 0.5% (a typical value for aerodynamic analysis of long-span bridges), the response power spectra were calculated with above formulas and shown in Figs. 6.2 and 6.3. In these figures, the top half is the spectra for the 1st DOF and the bottom half is for the 2nd DOF.
Parameters for the 2-DOF System Attached with Two Identical TMDs
Two identical TMDs are still considered here. In Fig. 6.2, the two identical TMDs are conventionally designed to suppress the resonant vibration of the 1st DOF. For comparison, both coupled and uncoupled analyses were conducted. It can be seen that when uncoupled vibration analysis is conducted, the vibration power spectrum for each DOF has only one peak from resonant vibration. However, there exist two peaks when coupled analysis is conducted. One peak is induced by resonant vibration corresponding to its modal frequency, while the other is due to the modal coupling effect between the 1st DOF and the 2nd DOF. The modal coupling effects are significant to the dynamic response.
It is shown in Fig. 6.2 that the TMDs designed for the 1st DOF have good control efficiency for the resonant vibration of the 1st DOF (the first peak of Fig. 6.2(a)), and also has some effect on the first peak of Fig. 6.2(b) that is the contribution of the 1st DOF to the 2nd DOF due to modal coupling. However, this design of TMDs doesn’t help reduce the vibrations due to the modal coupling from the 2nd DOF (the second peak of Fig. 6.2(a)) and the resonant vibration of the 2nd DOF (the second peak of Fig. 6.2(b)).
Fig. 6.3 shows the vibration power spectra when the TMDs are designed for the 2nd DOF. Similarly, the TMD helps reduce only the second peak values that are caused by the 2nd DOF, but not the peak values that are caused by the 1st DOF (the first peak of both Fig. 6.3(a) and Fig. 6.3(b)). Figs. 6.2 and 6.3 suggest that the TMDs should be optimally designed to suppress either the resonant vibration (first part in Eq. (6.28)), or the vibration due to modal coupling (second part in Eq. (6.28)) through weakening the modal coupling. When the overall response of the structure other than any single mode is considered, multiple TMDs can be designed to achieve the best control performance under any particular condition.
As stated before, wind-induced vibration results in aeroelastic damping so that the total vibrational damping of some modes may be large in strong wind. To simulate such a case that is common for modern long-span bridges, it is arbitrarily assumed that the damping ratio of
It can be found that when the 1st DOF vibrates with high damping ratio, the TMDs designed for the 1st DOF (Fig. 6.4) have less control efficiency for its resonant component (the first peak of Fig. 6.4(a)) than that of its counterpart when the TMDs are designed for the 2nd DOF (the second peak of Fig. 6.5(b)). The component of the 2nd DOF due to coupling even increases slightly as observed from the first peak of Fig. 6.4(b). In comparison, it can be seen from Fig. 6.5 that when TMDs are designed for the 2nd DOF with low damping ratio, the control efficiencies of its resonant component (the second peak of Fig. 6.5(b)) and the component of the 1st DOF due to coupling (the second peak of Fig. 6.5(a)) are still high, even though the 1st DOF has very high damping ratio.
For coupled vibrations, these observations have confirmed that the total modal vibration consists of mainly one portion from resonant vibration and another portion caused by coupling effects with other modes. The frequency of conventionally designed TMDs is tuned to that of the targeted mode to control the resonant vibrations and they may not achieve an efficient control especially when the coupling effect is significant. An optimal control strategy should aim at not only the resonant vibration, but also the vibration from modal coupling. Especially for some strongly-coupled modes vibrating in high wind velocity with high damping ratios, there exists a possibility that the vibration can be optimally suppressed even the TMD is not designed around the natural modal frequency of the targeted mode. For example, to control the vibration of the 1st DOF in strongly coupled vibration, the TMD frequency needs to be tuned to the natural frequency of the 2nd DOF rather than that of the 1st DOF. In other words, weakening the coupling effects may sometimes be more efficient than reducing the resonant vibrations when strong modal coupling exists (for maximum efficiency, both resonant and coupling components should be suppressed, but certainly that will be also more costly). To further understand this new control approach of the coupled buffeting response with TMDs, a prototype long-span bridge is studied in the next section.
6.4 Analysis of a Prototype Bridge
The Yichang Suspension Bridge located in the south of China has a main span length of 960 m and two side spans of 245 m each. The height of the bridge deck above water is 50 m.
Its main parameters are shown in Table 2.1. The four modes considered in the present study are shown in Table 4.1. Wind tunnel studies have shown that the 1st symmetric bending mode (Mode 2) and the 1st symmetric torsional mode (Mode 3) are the two key modes for buffeting and flutter analyses (Lin et al. 1998). Meanwhile, strong modal coupling between these two modes was observed at high wind velocity due to aeroelastic effects.
Complex eigenvalue approach was used to analyze the modal properties considering modal coupling. Fig. 6.6 shows that the modal damping ratios of the two vertical bending modes increase with the wind velocity. This increase is more significant when the wind speed surpasses 40 m/s. In contrast, at high wind velocity, the modal damping ratio of the symmetric torsional mode decreases with the increase of wind speed and eventually reaches zero. The critical wind flutter velocity is identified as 73 m/s by using the condition of zero total damping.
143 100 Response power spectrum (mm2/Hz) 10 1
As discussed before about Eq. (6.1), the higher vibration damping of the mode may cause the lower control efficiency of a given TMD. If the two vertical bending modes of the Yichang Bridge are deemed as two single modes omitting modal coupling with any other modes, Eq. (6.1) can be applied and implies that the control efficiencies of the two vertical bending modes should decrease at high wind velocity due to their high existent total damping ratios. In other words, it will be more difficult to suppress the vibration of vertical bending modes by using conventionally designed TMDs that essentially add supplemental damping to the concerned modes. Relatively, the control efficiency of torsional modes will be higher due to their low vibration damping ratios.
Fig. 6.8 Modal Frequency Versus Wind Velocity, Yichang Suspension Bridge 6.4.1 Buffeting Analysis with Conventional TMD Control For information, Fig. 6.7 shows the change of vibration frequencies with the wind speed. Similarly to the pattern of damping change, the modal frequencies of vertical bending modes increase while the torsional frequencies decrease with the increase of wind velocity, due to the effects of aeroelastic forces.
Fig. 6.8 shows the RMS of displacement response at the mid-span of the main span versus wind speed for the 1st symmetric bending mode and the 1st symmetric torsional mode using single mode analysis and multiple coupled mode analysis, without considering the TMDs. In this figure, the torsional response represents the vertical displacement at the edge of the cross section due to the torsional vibration. Differences between the results of singlemode analysis and coupled analysis are obvious in high wind speed, which also indicates strong modal coupling between these two modes.