STABILITY OF RHOMBUS MECHANISM DAMPER SYSTEM

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1 4th International Conference on Earthquake Engineering Taipei, Taiwan October 12-13, 26 Paper No. 24 STABILITY OF RHOMBUS MECHANISM DAMPER SYSTEM Fu-Sheng Lin 1, Yung-Feng Lee 2, and Deh-Shiu Hsu 3 ABSTRACT Rhombus mechanism damper system is proposed to be installed in the building structures to reduce the responses of the structures by dissipation energy as the structures subjected to abnormal excitations. A 2-D system is possibly to be damaged by out-of-plane buckling. The system is modeled and analyzed by semi-hinge joint simulated to practical constructing joints to check the critical stiffness of the out-of-plane buckling. It is proved that the proposed mechanism is highly stabilized even it is theoretically an unstable structure. Keywords: rhombus mechanism, damper system, stabilize, out-of-plane buckling. INTRODUCTION Viscous damper is popularly used as an energy dissipation device for the structural passive control purposes. It is needed to induce meaningful relative displacement at the damper connected joints to bring the expected function efficiently. In order to magnify the mentioned relative displacement, rhombus mechanism damper system is proposed (Hsu et al., 24). It is sketched with its displacement diagram in Figure 1(a) and Figure 1(b). According to the rhombus mechanism, relative displacement of the damper ( CD ) to the relative displacement of the joint points ( AB ) of the system allocated on the structure would be magnified when θ< 45 degree. The magnification factor, f, with respect to the angle, θ, can be rationally calculated and plotted in Figure 2. Consequently, the damping force induced by the damper system, F, can be enlarged according to the following equation: θ 2 D 2 D D (1) F= F cot( ) = ff = fcd = f Cd Where F and F D stand for the resisting force induced by the proposed damper system and by the damper itself, respectively; d and d D stand for the relative displacement of the proposed damper system and of the damper itself, respectively; C stands for the damping coefficient of the damper; f stands for the magnifying factor as shown in Figure 2. The proposed rhombus damping system can be allocated in spans of the structures or at corners of structures as indicated in Figure 3(a) and Figure 3(b). 1 Graduate Student, Dept. of Civil Engineering, National Cheng Kung University, Tainan, Taiwan, fredlin@twce.org.tw 2 Ph. D. Student, Dept. of Civil Engineering, National Cheng Kung University, Tainan, Taiwan, lyf_926@hotmail.com.tw 3 Professor, Dept. of Civil Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., dshsu@mail.ncku.edu.tw

2 A D D d ' D θ/2 B θ/2 damper θ A A B B d d C d ' C C (a) The system (b) Displacement diagram for rhombus mechanism Figure 1. Rhombus mechanism damper system Magnification factor, f Angle, θ Figure 2. Magnification factor of the rhombus mechanism θ θ1 (a) Allocated in span (b) Allocated at corners Figure 3. Allocation of the rhombus damper system

3 Since all of the connections of the mechanism are easily to be considered as hinge joints, as in a 2-D mechanism, it is no doubt of being inquired about the out-of-plane stability problems. We tried to work the problem by analyzing practical structures allocated with the damper system. Result showing the possibility of the occurrence of the out-of-plane phenomena could be much smaller than we imagined. STABILITY OF THE RHOMBUS DAMPER SYSTEM Once the member connections of the rhombus damper system are considered to be hinge joints, the perpendicular view to the 2-D mechanism (in the direction along the damper) could be modeled as shown in Figure 4(a) (b) and (c). It gives us an unstable structure due to a mode of three hinges co-linear mechanism. If it is in this case, the proposed rhombus damper mechanism will bring nothing help to the structure because the mechanism can not take any load, i.e., can not induce any resisting force, due to the out-of-plane buckling problem. It can be understandable that the mechanism would be stable if the exterior joints are modeled as fixed boundaries as shown in Figure 4(b). Of course, most of the real case in practical applications could be in-between as shown in Figure 4(c), in which semi-hinge joints are represented by rotational springs. Stability capacities can be analyzed numerically in terms of the partial stiffness simulated by the rotational springs. (a) (b) (c) Figure 4. Stability of the mechanism from the view along the direction of the damper NUMERICAL ANALYSIS A joint angle of 3-degree (θ=π/6) is adopted for the rhombus mechanism considered in the analysis. The corresponding magnification factor is f=cos(15 )=3.73. The members for the rhombus mechanism system are designed according to the type of installation form of the system, whether it is a installed to the full span of the frame or it is installed at the corners of the frame. The selected member sizes are listed in Table 1. Table 1. Member size of the rhombus mechanism system. Type of installation To the full span At the corner Tube thickness Diameter Slenderness ratio 1 cm 6 cm ~ 16.5 cm 2.5 cm ~ 15

4 Three story steel frame as shown in Figure 5 is selected as the benchmark structure in the analysis. According to the numerical value provided by NCREE (NCREE, 22), the fundamental natural frequency of the frame is calculated as 1.75Hz with.12m thickness floors of unit weight 65856N/m 3. Acceleration time history of 194 El Centro earthquake is applied as the external excitation. The 1 st mode shape can be calculated and obtain as Φ rj = = (2) Figure 5. Three story benchmark steel frame The inherent damping ratio of the structure is assumed to be 2%. The equivalent damping ratio to be increased by the installation of damper system is assumed to be 18%. In addition, damper used in each individual case is assumed to be identical to each other. The damping coefficient of the damper can be calculated to be C 7324 N-sec/m for the type of installing to full span, and C N-sec/m for the type of installing at corners. For each case of numerical analysis, stiffness of the rotational spring is started from assuming a very big value to simulate a fixed joint condition. Under this assumption, it won t lead to any out-of-plane buckling phenomena. All of the damping resistance induced as expected ideally. The stiffness of the rotational spring is then reduced step by step gradually. At the critical point, it shows the damping resistance dropped suddenly to zero. That means the phenomena of out-of-plane buckling occurred. Special cases for the damper system installed to full span and installed at corners are plotted in Figure 6 and Figure 7 respectively. The spring constant k for the critical points are equal to 75 N/rad and 79 N/rad approximately for each case. When compared to the fix condition, the spring constants are about 14, N/rad and 12, N/rad, the spring constant of 75 N/rad and 79 N/rad is really a very small value. That means the semi-hinge joint even provide just a very little constrain, the system is still remain its working condition, free from out-of-plane buckling. Another example shows the effect of various damping coefficients to the critical spring constant. The calculation result is plotted in Figure 8 and Figure 9, which showing there is no difference to the critical spring constant even different dampers are used in the system. Generally, in most of the examples, even when the stiffness of the rotational spring becomes very small, out-of-plane un-stability won t be occurred for the application of 2-D rhombus mechanism damper system.

5 5 5 Fix 4 4 Damper force (N/m) (a) Damping force vs spring constant. (b) Magnifying depict near the critical point. Figure 6. Example for the damper system installed to full span. 6 6 Fix (a)damping force vs spring constant. (b) Magnifying depict near the critical point. Figure 7. Example for the damper system installed at corners C=15N-sec/m C=1N-sec/m C=7324N-sec/m C=5N-sec/m 2 C=15N-m/sec C=1N-sec/m C=7324N-sec/m C=5N-sec/m (a) Damping force vs spring constant. (b) Magnifying depict near the critical point. Figure 8 Example for the damper system with different dampers are installed to full span.

6 C=3N-sec/m C=25N-sec/m C=21925N-sec/m C=15N-sec/m 2 C=3N-sec/m C=25N-sec/m C=21925N-sec/m C=15N-sec/m (a) Damping force vs spring constant. (b) Magnifying depict near the critical point. Figure 9. Example for the damper system with different dampers are installed at corners. CONCLUSIONS It was raised doubts that a 2-D rhombus mechanism damper system would cause out-of-plane buckling problem. In order to avoid the doubt, 3-D rhombus mechanism damper system was proposed (Hsu et al., 24). However, the work done in this paper showing that even a 2-D rhombus mechanism damper system in practical connection condition, out-of-plane buckling can be occurred only when the connections were made almost idealized to be a hinge connection in the out-of-plane direction. Otherwise, out-of-plane un-stability won t be occurred for the proposed rhombus mechanism damper system applied practically. REFERENCES Deh-Shiu Hsu, Ming-Che Hsu, and Yung-Feng Lee (24), Rhombus Mechanism with Fluid Damper, 13 th World Conference on Earthquake Engineering, Vancouver, Canada, Paper No National Center for Research on Earthquake Engineering (22), Shaking Table and Benchmark Structure of 3D Three Story Steel Frame,

DYNAMIC TEST OF MULTIPLE TUNED MASS DAMPERS FOR VIBRATION CONTROL OF HIGH-RISE BUILDINGS

1NCEE Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21-25, 214 Anchorage, Alaa DYNAMIC TEST OF MULTIPLE TUNED MASS DAMPERS FOR VIBRATION CONTROL OF HIGH-RISE