13 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 4 Paper No. 11 RHOMBUS MECHANISM WITH FLUID DAMPER Deh-Shiu Hsu 1, Ming-Che Hsu, and Yung-Feng Lee 3 ABSTRACT Structural control technique becomes much more popular in this decade in order to reduce the responses of structures when the structures are subjected to some of the abnormal excitations, such as strong earthquake. In addition to the development of active control techniques, passive control, with many of passive control devices, have being developed. Many of the developed devices have been installed to the structures and applied for the aseismatic engineering purposes. Due to the relatively small displacement of the civil structures, it has been tried to find some methods to obtain a greater and more meaningful displacement, and velocity as well, to lead the action of the passive dampers so as to reach the goal of energy dissipation. A rhombus shaped mechanism is proposed in the paper. The relative displacement of the structure in between the nodes in which the proposed mechanism is linked can be magnified evidently. Both of the theoretical derivation and some of the numerical examples are presented to show the positive energy dissipation function of the proposed rhombus mechanism provides. INTRODUCTION Fluid damper has been used as one of efficient energy dissipation devices for aseismatic engineering structures [1]. It is installed in-between the degree of freedoms where meaningful displacement will be induced when the structure subjected to the abnormal excitations, for instance, earthquake excitations. Meaningful displacement induced instantly gives meaningful relative velocity in-between the degree of freedoms. Fluid dampers, velocity dependent devices, installed in-between the degree of freedoms to provide functions of energy dissipation so as to reduce the responses of the structures. In most of the cases, for constructing the aseismatic systems are executed by installing the dampers at selected panels in the structures[], as depicted in Fig.1. Some of the problems we are going to face are: 1. In order to dissipate the energy efficiently, how the velocity (or the displacement) of the damper encountered can be magnified.. Due to the installation diagonally in the panels, utilizing space could be greatly reduced. How can it be improved? 1 Professor, Dept. of Civil Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C. Graduate Student, Dept. of Civil Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C. 3 Ph. D. Student, Dept. of Civil Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C.
Fig.1. Installation of damper in the structure. The issues induced above are trying to be discussed by introducing fluid damper in rhombus mechanism proposed herein. RHOMBUS MECHANISM A rhombus mechanism ACBD is plotted in Fig.. The rhombus mechanism with four equal length members, AC, CB, BD, and AD. If we set the x-y coordinate as shown in Fig., the original O(,) can be called as the instantaneous center of the mechanism. If point A and B moved in minus and plus x-direction to A and B by a displacement d/, while point C and D moved to C and D in plus and minus y-direction by a displacement fd/ as shown by the dotted line in the figure, the value of the magnification factor, f, with respect to the angle θ of the rhombus mechanism, can be calculated and plotted in Fig.3[3]. The relationship between two perpendicular displacements can be written in Equ. 1. d D = fd (1) where d D is nothing but the relative displacement in-between point C and D. y D fd/ D d/ d/ b O(,) θ x A B A B a C a C b fd/ Fig.. Typical rhombus mechanism.
1 11 1 Magnification factor, f 9 8 7 6 5 4 3 1 1 3 4 5 6 7 8 9 Angle, θ Fig.3. Magnification factor, f, plotted with respect to the geometric angle of the rhombus mechanism, θ. FLUID DAMPER IN RHOMBUS MECHANISM It is proposed that a fluid damper is allocated in the rhombus mechanism at diagonal position as shown in Fig.4. D A B F D d D = f d F = f F D, d/ F = f F D, d/ C Fig.4. Rhombus mechanism is allocated with fluid damper. If the resistance of the damper induced due to the relative displacement occurred, d D, in-between point C and D is denoted by F D, then the external force of the mechanism, F, can be described as F = ff D () Taking the derivative for both side of Equ.1, we obtain the relative velocity in-between points C and D. That is nothing but the velocity subjected to the damper. d & = fd & (3) D
Let s take the resisting force-velocity relationship of the damper be F = C d& (4) D D D where C D is the damping coefficient of the damper. Substituting Equ.3 into Equ.4, we have the resisting force induced by the damper be F D = C fd& (5) Substituting Equ.5 into Equ., the resisting force provide by the proposed mechanism can be obtained as D F= ffd = CDf d= Cd & & (6) where C stands for the damping coefficient of the proposed system of fluid damper in rhombus mechanism. APPLICATION OF FLUID DAMPER SYSTEM IN RHOMBUS MECHANISM In general, the proposed fluid damper system in rhombus mechanism can be used as energy dissipation device by using rhombus brace taking the place of single damper bracing as described in Fig.5. θ 1 (a)conventional single damper bracing system (b)proposed fluid damper rhombus bracing system Fig.5. Fluid damper rhombus bracing system According to the regulation NEHRP(National Earthquake Hazard Reduction Program)[4] of FEMA (Federal Emergency Management Agency) 73 and 74, the effective damping ratio of the structure can be estimated by the following formula: ξ eff =ξ +ξ d Wj ξ d = (7) 4 π W k where ξ eff stands for the effective damping ratio of the structure installed with dampers; ξ stands for the damping ratio of the raw structure; ξ d stands for the damping ration increased by the dampers; W j stands for the work done/dissipated by the jth damper in one cycle in the motion; and W k stands for the maximum elastic strain energy of the structure.
When the diagonal damping bracing system is applied, the effective damping ratio can be modified as suggested in the previous work[5]: T C ( φ cos θ ) ξ =ξ + j rj j j eff 4π miφi i (8) where T stands for the period of the 1st mode system vibration; m i stands for the mass of the ith degree of freedom; φ i stands for the displacement of the ith degree of freedom in the 1st mode vibration; C j stands for the damping coefficient of the jth damper; φ rj stands for the horizontal relative displacement of the jth damper; and θ j stands for the direction angle of the jth damper with respect to the horizontal direction. When rhombus mechanism damping bracing system is applied, in case of there is floor relative displacement of u, the bracing elongation will be ucos θ 1. It follows that the displacement of the damper d D and the resisting force of the rhombus damping system as described in Fig.5(b) will be and ud = f Cosθ 1u (9) F= f Cosθ 1FD (1) Where u stands for the floor horizontal relative displacement; u D stands for the displacement of the damper subjected; F stands for the resisting force induced by the rhombus mechanism damping bracing system; f stands the magnification factor as described in Equ.(1). The effective damping ratio can also be modified in the form of T C ( φ f Cos θ ) ξ =ξ + j rj j ij j eff 4π miφi i (11) In addition to install the rhombus mechanism system as panel braces, the rhombus mechanism system can be installed at the corners of panel structure as depicted in Fig.6. It is obvious that in addition to the magnifying effect, the reservation of space utility is much worthwhile to be mentioned. Fig.6. Rhombus mechanism damping system can be installed at the corners of panel structure.
NUMERICAL EXAMPLE A three story steel frame as shown in Fig. 7(a) is taken as the structure to be analyzed. There are four cases analyzed herein, they are: Case 1: without additional damper is installed, shown in Fig.7(a), i.e., raw structure; Case : bracing damper system is installed in all three stories, shown in Fig.7(b); Case 3: rhombus damper bracing system is installed in all three stories, shown in Fig.7(c); and Case 4: rhombus damper system is installed at floor corners of the structure, shown in Fig.7(d). 4.5m Z 3m 3m 3.3m beam: Hx4x1x1 column: Hx15x6x9 ω(natural frequency)=1.75hz floor thickness: 1mm unit weight of the floor: 6.5E-8 kn/mm 3 Y (a) raw structure, without damper installed (b) bracing damper system (c) rhombus damper system (d) rhombus damper system installed in panels installed at corners Fig.7. Three story steel frame analyzed in various cases. The cases mentioned above are analyzed with 194 El Centro earthquake as the subjected excitation. Responses obtained are displayed in different variables such as displacement, acceleration, column shear force, and column axial force, etc. Control effect is estimated by the index of control efficiency (C.E.) defined as following: (max. response in case 1)-(max. response in case i) (C.E.) i = 1% (11) (max. response in case 1) Maximum floor displacement, maximum floor acceleration, maximum shear force at each column,
and maximum axial force at each column, and their corresponding control efficiency are listed in Table 1, Table, Table 3 and Table 4, respectively. For the purpose of demonstration, time history of top floor displacement for the cases of structure installed with dampers (i.e., case, case 3, and case 4) with respect to the response of raw structure (i.e., case 1) are plotted in Fig.8, Fig.9, and Fig.1. Equivalent damping ratio of the structure for each case can be calculated, and listed in Table 5. Table1. Maximum floor displacement and its control efficiency for each case: floor Case 1 Case Case 3 Case 4 1 displacement (mm) 56.5.77 3.87 16.71 control efficiency (%) 63.5 93.15 7.4 displacement (mm) 16.8 38.4 6.6 9.68 control efficiency (%) 64.3 94.14 7. 3 displacement (mm) 136.1 48.47 7.45 36.49 control efficiency (%) 64.39 94.53 73. Table. Maximum floor acceleration and its control efficiency for each case: floor Case 1 Case Case 3 Case 4 1 3 acceleration(g).436..7.3 control efficiency (%) 54.13 38.7 47. acceleration (g).535.1.6. control efficiency (%) 6.75 51.4 58.87 acceleration (g).76.31.7.7 control efficiency (%) 57.3 6.81 6.8 Table3. Maximum column shear force and its control efficiency for each case: floor Case 1 Case Case 3 Case 4 1 Shear force (kn) 43.8 16. 3.1 1.3 control efficiency (%) 63 9.9 51.37 Shear force (kn) 37.1 13 1.6 11.8 control efficiency (%) 64.96 95.69 68. 3 Shear force (kn) 1.8 8..85 6.5 control efficiency (%) 6.4 96.1 7.
Table4. Maximum column axial force and its control efficiency for each case: floor Case 1 Case Case 3 Case 4 1 Axial Force (kn) 15 37 5.5 3.9 control efficiency (%) 64.76 94.76 7.6 Axial Force (kn) 54.1 19.8.6 15.8 control efficiency (%) 63.4 95.19 7.8 3 Axial Force (kn) 16.3 6.1 1 4.3 control efficiency (%) 6.58 93.87 73.6 Displacement (mm) 1-1 EL Centro Case 1 Case - 1 3 4 5 Time (sec) Fig.8 Displacement time history of control result for bracing damper system. Displacement (mm) 1-1 EL Centro Case 1 Case 3-1 3 4 5 Time (sec) Fig.9 Displacement time history of control result for rhombus damper system installed in panels.
Displacement (mm) 1-1 EL Centro Case 1 Case 4-1 3 4 5 Time (sec) Fig.1 Displacement time history of control result for rhombus damper system installed at corners. Table 5 Equivalent damping ratio for each case: Case Case 3 Case 4 Expected equivalent damping ratio (%) Calculated equivalent damping ratio (%) 5 18 4 9 CONCLUSION For the purpose of aseismatic engineering, rhombus mechanism with fluid damper installed is proposed herein. The proposed system can be used as the energy dissipation device. By the inherent capability of the rhombus mechanism, responses such as displacement and velocity of the fluid damper can be magnified so as the energy dissipated can be increased in great amount. The result of the analyzed example shows that when the single damper brace is replaced by the proposed rhombus fluid damper system, the control efficiency improved dramatically. In other words, when rhombus fluid damper system is used, the equivalent damping ratio of the structure increased greatly. For the purpose of panel space utility concern, the rhombus fluid damper system is allocated at corners of the panel. The control efficiency shows the capability of control for this arrangement is still acceptable. It might be questioned that the proposed rhombus system would be worried about instability problem. However, it is very easy to overcome the problem just by slight modification to the mechanism. For example, the mechanism can be manufactured by making a sliding roller at the link points as depicted in Fig.11. The out of plan instability problem would not be occurred any more.
Fig.11. 3-D plot of the proposed rhombus mechanism fluid damping system. REFERENCES 1. Soong, T.T., and Dargush, G.F., Passive Energy Dissipation Systems in Structural Engineering, John Wiley Sons, New York, 1997.. Pekan, G., Mander, J.B., and Chen, S.S., Fundamental Considerations for the Design of Nonlinear Viscous Damper, Earthquake Engineering and Structural Dynamics, 1999; 8: 145-145. 3. Hsu, Ming-Che, Design and Application of Fluid Damper in Rhombus Mechanism, Dissertation of Master Degree (in Chinese), Dept. of Civil Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., June, 3. 4. FEMA, NEHRP Guidelines and Commentary for the Seismic Rehabilitation of Buildings, Reports No. 73 and 74, October, Washington, D. C., 1997. 5. Constantinou, M.C., Tsopelas, P., Hammel W., and Sigaher, A.N., Toggle-Brace-Damper Seismic Energy Dissipation Systems, Journal of Structural Engineering, ASCE, 1; 17(): 15-11.