Vibration Control Effects of Tuned Cradle Damped Mass Damper

Journal of Applied Mechanics Vol. Vol.13, (August pp.587-594 2010) (August 2010) JSCE JSCE Vibration Control Effects of Tuned Cradle Damped Mass Damper Hiromitsu TAKEI* and Yoji SHIMAZAKI** * MS Dept. Civil Eng. Tokai University (Kitakaname, Hiratsuka, Kanagawa 259-1292) ** Member PhD Prof. Dept. Civil Eng. Tokai University (Kitakaname, Hiratsuka, Kanagawa 259-1292) This research introduces a new mechanical device for dissipating vibration, called the tuned cradle mass damper (TCMD). This device relies on the motion of a swing mass on a curved surface to dissipate structural vibration energy. The objectives of this study are to develop a model of TCMD and verify its performance through both experiments and numerical analysis when the structure is under free vibrations. The proposed device was developed by using simple pendulum dynamics that are applied to a structure with a frequency of approximately 1 Hz. For this study, the damper was installed in a one-story simple rigid frame model and excited in free vibration. Key Words : vibration control, tuned mass damper, passive damper, free vibration 1. Introduction The Great Hanshin-Awaji Earthquake of 1995 served as a reminder to Japanese citizens of how important it is to enforce seismic safety in buildings. In recent years, mechanisms for absorbing the large-amplitude rocking motions of earthquakes have been installed not only in numerous high-rises, but also in smaller buildings. However, it is difficult to ensure that these mechanisms will function properly as designs have become increasingly complicated and larger in scale, thus raising their manufacturing and maintenance costs and posing other problems. If less-expensive versions of these mechanisms can be made, building owners will become more motivated about improving the safety of their buildings, which in turn will further protect our cities from earthquakes. Vibration control systems for buildings can be classified into two types: active controllers, which require an exterior energy source to absorb the vibrational energy, and passive controllers, which do not require an energy source. Several kinds of passive controllers have been developed into practical products, including a type using laminated rubber or a coil spring to support the weight of a damper, a hanging type based on the principle of the pendulum 1), an impact damper using a steel ball 2), a tuned liquid damper (TLD) using liquid materials 3,4,5) and a tuned rotary-mass damper (TRMD) consisting of a rolling mass and container allowing free movement of the mass along its inner arc 6,7). This research introduces a new mechanical vibration-dissipating device: the tuned cradle mass damper (TCMD). This device relies on the movement of the swing mass on a curved surface to change the dynamic characteristics of a structure by dissipating its vibration energy. The TCMD utilizes simple driving force, which is developed in response to the structural motion. The benefits of TCMD are its simplicity, compactness and ease of maintenance. Small wheels, attached to the swing mass, are seated and they move along the curved surface. This configuration enables the TCMD to sustain a natural frequency closely in tune with the engineered structure. The objectives of this study are to develop a model of TCMD and verify its performance through both experiments and numerical analysis when the structure is under free vibrations. There are more than 20,000 ten-story slender buildings in the Tokyo metropolitan area. The natural frequency of these buildings is approximately 1 Hz. The proposed device was developed by using simple pendulum dynamics that are applied to a structure with a frequency of approximately 1 Hz. For this study, the damper was installed in a one-story simple rigid frame model and excited in free vibration. 2. Device configuration 2.1 Experimental model As stated above, to examine TCMD experimentally, a model of a one-story structure is used. Figure 1 shows the model, which is made of steel columns 1250 mm in length, 60 mm in width and 4.5 mm in thickness. Four columns support a floor. - 587 -

The lateral spring constant k 1 of the structure is 0.175 kgf/mm. Figure 2 shows the force-displacement relationship of the structure for k 1. The effective mass of the structure is approximately 50 kg. The natural frequency of the structure is approximately 1 Hz. 2.2 Modeling of TCMD Figure 3 shows the TCMD model. The swing mass is made of two steel plates with three small wheels. The size of the plate is 175 60 3 mm and the diameter of the wheel is 22 mm. The swing mass including the wheels is about 600 g. The mass moves along three curved surfaces (radii of 300 mm) as the structure moves. To obtain magnetic damping 6,7), an aluminum plate is placed on the side of the middle curved surface. The damping strength of the mass can be adjusted by varying the number of magnets attached to the swing mass. Neodymium magnets (size: 10 10 4 mm) are used to obtain the damping. Figure 4 and 5 show the free vibration of TCMD when zero and seven magnets are used for the TCMD respectively. Fig.4 Free vibration of TCMD (0-magnet) a)front view b) Side view Fig.1 Simple structure model Fig.5 Free vibration of TCMD (7-magnet) Fig.2 Force-displacement line of the structure Figure 6 shows the relationship between the number of magnets used for the TCMD and the corresponding its damping ratio h 2. Here, h 2. is the damping ratio obtained from the free vibration of TCMD. The natural frequency of the TCMD is 0.92 Hz when the amplitude of the swing is small. The swing speed can be adjusted by modifying the radii of the curved surfaces and/or the size of three wheels. h2 (%) Number of magnets Fig.3 TCMD model Fig.6 Number of magnets versus damping constant h 2-588 -

3. Numerical analysis model Figure 7 shows the free body diagram of TCMD. Here, is the mass of the swing. Because the mass of the wheels is small compared to the swing mass itself, we do not take the rotational energy by the wheels into account. In addition, F t is the horizontal force given by TCMD, x 2 is the horizontal Only the viscous damping c 2 is considered here. We do not take the frictional damping between the wheel and the curved surface into account for simplicity. Then, the equation of the cradle in the horizontal direction becomes mx 2 2 F t 0, F N sin F cos where t d, (4) or 1 x sin c cos 2 2 2 Fd g cos sin (5) Figure 8 shows the schematic figure of the structure. Here, m 1 is the mass of the structure, c 1 is the viscous damping coefficient, k 1 is the spring constant of the structure, x 1 is the v u g F t N horizontal displacement of the structure. We then obtain the following equation of motion for the structure. F c x k x m x (6) t 1 1 1 1 1 2 or c k m x x x sin 1 1 2 1 1 1 m1 m1 m1 m c 2 2 g cos sin cos (7) m1 m1 We can now calculate the coupled equations (5) and (7) numerically. x 2 Fig.7 Free body diagram of TCMD acceleration supplied by the structure, Fd is the damping force, and N is the normal force. The equation of motion of the mass m 2 in the u and v directions can be written as mg sin F m, (1) u direction: 2 d 2 v direction: where F d = 2 2 mg 2 cos N, (2) c. (3) F t kx cx 1 1 1 1 Fig.8 Free body diagram of structure 4. Results of the experiment Experimental measurements are made to clarify the dissipation of vibration energy of the simple model structure by TCMD action. The laser displacement sensors are used to measure both the time-displacement m 1 x 1-589 -

responses of the structure and the TCMD. In response to the vibration stimuli, TCMD shows that the free vibration motion of the model structure can be controlled to within a frequency range of approximately 1 Hz. Therefore, good energy dissipation is obtained. Figure 9 shows the displacement responses of the experimental structure in the uncontrolled condition (i.e., without a damper). The initial lateral displacement given to the structure is 24 mm and the frequency of the structure f 1 =0.92 Hz. The damping ratio h 1 of the structure is 0.12%. Figure 10 shows the displacement responses of the experimental structure in the controlled condition when no magnet is installed on the swing mass of the TCMD. The beats are observed because of the weak damping of TCMD. Fig.9 Free vibration of frame model (uncontrolled) When the swing mass with magnets moves on the curved surface, the damping ratio becomes much greater than that without the magnet. Figure 11 shows the ratio obtained by the experiment. The initial displacements given to the free vibration of the structure are 12 mm, 24 mm and 34 mm. It is observed that the structural damping h 1 becomes more than 5% when an appropriate number of magnets is attached to the swing mass. Figures 12 through 17 show the experimental wave form of the structure with TCMD when f 1 = 0.90, 0.92 and 0.95 Hz. In the figures, and (b) show the experimental wave form when 3 and 7 magnets are used for TCMD, respectively. 5. Results of Analysis To solve equations (5) and (7), a fourth-order Runge-Kutta method is applied. Figures 18 through 23 show comparisons of the results obtained by the experiments and those by the analyses. The initial lateral displacement and the natural frequency of the structure are 24 mm and 0.92 Hz respectively. In the figures, both the structure and the swing mass movement are shown. It is shown that the results obtained by the numerical analysis agree well with those of experiment. Figures 20 and 21 suggest the swing mass with seven magnets for TCMD gives the best damping effect on the model structure. a) Structure response for TCMD with no magnet Initial displacement = 12 mm b) TCMD response with on magnet Fig.10 Structure-TCMD response (controlled) (b)initial displacement = 24 mm - 590 -

(c) Initial displacement = 34 mm Three magnets Fig.11 Damping ratio obtained by the experiment Seven magnets Fig.13 Structure responses ( f 1 = 0.9 Hz, initial displacement = 34 mm) Three magnets Time ( sec ) Three magnets (b) Seven magnets Fig.12 Structure responses ( f 1 = 0.90 Hz, initial displacement = 12 mm) (b) Seven magnets Fig.14 Structure responses ( f 1 = 0.92 Hz, initial displacement = 12 mm) - 591 -

Three magnets Three magnets (b) Seven magnets Seven magnets Fig.15 Structure responses ( f 1 = 0.92 Hz, initial displacement = 34 mm) Fig.17 Structure responses ( f 1 = 0.95 Hz, initial displacement = 34 mm) Three magnets Experiment (b) Seven magnets Fig.16 Structure responses ( f 1 = 0.95 Hz, initial displacement = 12 mm) Fig. 18 Structure responses for h 2 =5.3% (three magnets) - 592 -

Experiment Experiment (c) Aanalysis Fig. 19 Cradle responses for h 2 =5.3% (three magnets) Fig. 21 Cradle responses for h 2 =8.9% (seven magnets) Experiment Experiment Fig. 20 Structure responses for h 2 =8.9 % (seven magnets) Fig. 22 Structure responses for h 2 =12.7% (11 magnets) - 593 -

Experiment Fig. 23 Cradle responses for h 2 =12.7% (11 magnets) 6. Conclusions A new mechanical vibration absorber, Tuned Cradle Mass Damper (TCMC) is suggested. This TCMD has a simple construction and is applicable to the structure which may vibrate in lower frequency modes. The equation of reciprocal motion between TCMD and the structure is derived using simple pendulum theory. It is shown both experimentally and numerically that the new tuned cradle mass damper can efficiently dissipate undesirable vibration energy in a structure within the range of elastic deformation. 2) Uno, K., Kitagawa, S., Tsutsumi, H., and Jo, I., Isolation of Lighting Pole from Wind Vortex Shedding by Impact Dampers, Journal of Structural Engineering, JSCE, Vol. 36A, 1990, 565-575, (In Japanese). 3) Dorothy, Reed., Jinkyu, Yu., Harry, Yeh., and Sigurdur, Gardarsson., 1998, Investigation of Tuned Liquid Dampers under Large Amplitude Excitation, Journal of Engineering mechanics, ASCE, 1956, 405-413. 4) Fujino, Y., Pacheco, B. M., Chaiseri, P. and Sun, L. M., Parametric Studies on Tuned Liquid Damper (TLD) using Circular Tanks by Free-oscillation Experiments, Journal of Structural and Earthquake Engineering, JSCE, No. 398, 1988, 177-187. 5) Fujino, Y., Sun, L M., Pacheco, B. M. and Chaiseri, P., Tuned Liquid Damper (TLD) for Suppressing Horizontal Motion of Structure, Journal of Engineering Mechanics, ASCE, Vol. 118, No. 10, 1992, 2017-2030. 6) Obata, M., Shimazaki, Y., Vibration control effects and application example of tuned rotary damped mass damper, The Structural Engineer, 85(13), 2007, 41-45. 7) Obata, M., Shimazaki, Y., Optimum Parametric Studies on Tuned Rotary-Mass Damper, Journal of Vibration and Control, 14(6), 2008, 867-884. (Received March 9, 2010) Acknowledgement The financial support of the Research for Promoting Technological Seeds is gratefully acknowledged. REFERENCES 1) Teramoto, H., Sano, S., Nagai, M., and Okui, Y., Design and Fabrication on Procedures of TMD in the Bannaguro Bridge, Journal of Structural Engineering, JSCE, Vol. 36A, 1990, 1129-1140, (In Japanese). - 594 -