WWW.DISSERTATION.XLIBX.INFO
FREE ELECTRONIC LIBRARY - Dissertations, online materials
 
<< HOME
CONTACTS



Pages:     | 1 |   ...   | 12 | 13 || 15 | 16 |   ...   | 20 |

«A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of ...»

-- [ Page 14 ] --

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.

Table 6.1.

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.

146 8

–  –  –

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.



Pages:     | 1 |   ...   | 12 | 13 || 15 | 16 |   ...   | 20 |


Similar works:

«RI Department of Environmental Management Local Agriculture and Seafood Act Grants Program 2016 GUIDELINES & INSTRUCTIONS The Rhode Island Department of Environmental Management, Division of Agriculture is accepting grant applications for the Local Agriculture and Seafood Act Grants Program. The goal of the program, which was established by the Local Agriculture and Seafood Act (LASA) of 2012, is to support the growth, development, and marketing of local food and seafood in Rhode Island. It is...»

«Tsakiri, Ioannidis, Carty 1 LASER SCANNING ISSUES FOR THE GEOMETRICAL RECORDING OF A COMPLEX STATUE Maria TSAKIRI1, Charalambos IOANNIDIS1, Alistair CARTY2 1 School of Rural and Surveying Engineering, National Technical University of Athens, Greece. 2 Archaeoptics Ltd, Glasgow, UK KEY WORDS: laser scanning, heritage applications, three-dimensional, close range, data capture ABSTRACT Recent advances in laser scanning technology allow for fast and efficient 3D documentation of cultural heritage...»

«Best Management Practices for Agricultural Erosion and Sediment Control SONOMA COUNTY AGRICULTURAL COMMISSIONER’S OFFICE TABLE OF CONTENTS INTRODUCTION REGULATORY REQUIREMENTS CALIFORNIA TIGER SALAMANDER MAP CHAPTER ONE LAYOUT AND SITE DEVELOPMENT ENVIRONMENTAL CONCERNS SITE EVALUATION BEST MANAGEMENT PRACTICES EXAMPLE VINEYARD LAYOUT EXAMPLE VINEYARD LAYOUT NEAR STREAM CHAPTER TWO ROADS ENVIRONMENTAL CONCERNS SITE EVALUATION BEST MANAGEMENT PRACTICES FOR AGRICULTURAL ROADS EXAMPLE OUTSLOPED...»

«ANDREW G. VAUGHN Gustavus Adolphus College 800 West College Ave. St. Peter, MN 56082 507–933–7475 (O); 507–934–1225 (H) email: avaughn@gustavus.edu http://www.gac.edu/~avaughn POSITIONS HELD Associate Professor of Religion: Gustavus Adolphus College, St. Peter, MN. Appointment with continuous tenure in area of Hebrew Bible. 2002 – present. Chair, Department of Religion: Gustavus Adolphus College, St. Peter, MN. 2003 – present (on sabbatical 2004–5 academic year). Supply Preacher:...»

«THE LATENT LANDSCAPE A Thesis Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Master of Fine Arts in The School of Art by May Ann Babcock B.F.A., University of Connecticut, 2008 May, 2011 TABLE OF CONTENTS LIST OF FIGURES ABSTRACT THE LATENT LANDSCAPE BIBLIOGRAPHY VITA 
 ii
 LIST OF FIGURES 1. May Ann Babcock, Cinclaire Study 1, Handmade Paper, Monotype, Paper Cast...»

«A STUDY OF CHILDREN’S MUSICAL PLAY AT THE LITTLE GYM A Thesis Submitted to the Graduate Faculty of Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Master of Music Education In The School of Music by Alison Elaine Alexander B.A, Mercer University, 2003 B.M.E., Armstrong Atlantic State University, 2005 August 2012 To Connor and Brady, my inspirations! ii ACKNOWLEDGEMENTS I would like to thank my committee for their...»

«This work is used with the permission of Joelle G. Zeitouny. © 2007, Joelle G. Zeitouny. Wild Edible Plant Consumption and Age-Related Cataracts in a Rural Lebanese Elderly Population: A Case control Study By Joelle Zeitouny School of Dietetics and Human Nutrition McGill University Montreal, Quebec, Canada A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for a Master of Science August 2007 © Joelle G. Zeitouny, 2007 i Abstract The...»

«Milk Fat Globule Stability Lipolysis with Special Reference to Automatic Milking Systems Lars Wiking Faculty of Natural Resources and Agricultural Sciences Department of Food Science Uppsala Doctoral thesis Swedish University of Agricultural Sciences Uppsala 2005 Acta Universitatis Agriculturae Sueciae 2005: 49 ISSN 1652-6880 ISBN 91-576-7048-X © 2005 Lars Wiking, Uppsala Tryck: SLU Service/Repro, Uppsala 2 Abstract Wiking, L. 2005. Milk Fat Globule Stability Lipolysis with Special Reference...»

«EXTRACTION AND CHARACTERIZATION OF PURPLE PIGMENT FROM Chromobacterium violaceum GROWN IN AGRICULTURAL WASTES AKRAM NESHATI A Dissertation Submitted To The Faculty Of Science In Partial Fulfillment Of The Requirement For The Award Of The Degree In Masters of Science (Chemistry) Faculty of Science Universiti Teknologi Malaysia APRIL 2010 EXTRACTION AND CHARACTERIZATION OF PURPLE PIGMENT FROM Chromobacterium violaceum GROWN IN AGRICULTURAL WASTES AKRAM NESHATI iii To my Beloved Mother and Father...»

«As. J. Food Ag-Ind. 2012, 5(02), 96-103 Asian Journal of Food and Agro-Industry ISSN 1906-3040 Available online at www.ajofai.info Research Article Effect of vacuum cooling on physico-chemical properties of organic coriander Apichart Sirinanuwat1, Danai Boonyakiat 2,3 and Pichaya Boonprasom 1,3* 1 Division of Food Engineering, Faculty of Agro-Industry, Chiang Mai University, Thailand, 50200 2 Department of Horticulture, Faculty of Agriculture, Chiang Mai University, Thailand, 50200 3...»

«52 Chimera 26: Geographical Journal, University College Cork Suburbia: social and spatial trends that emerged in Celtic Tiger Ireland. Matthew Williams Department of Geography, University College Cork, Ireland. Long after the roar of the “Celtic Tiger” has become inaudible; its effects remain in the form of ghost estates, incomplete rural development and inadequate service provision across the Irish landscape. This paper will give a brief account of suburban housing development in Ireland...»

«SOMALIA 2015 HUMAN RIGHTS REPORT EXECUTIVE SUMMARY The Federal Government of Somalia, formed in 2012, was led by President Hassan Sheikh Mohamud. Clan elders nominated the members of the House of the People of the Federal Parliament in 2012. Parliament elected Hassan Sheikh Mohamud as president later that year. Former Transitional Federal Government (TFG) president and presidential candidate Sheikh Sharif described the presidential vote as fair and conceded defeat. The regional governments of...»





 
<<  HOME   |    CONTACTS
2016 www.dissertation.xlibx.info - Dissertations, online materials

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.