Free Vibration Analysis of a Model Structure with New Tuned Cradle Mass Damper

Transcription

1 Proc. Schl. Eng. Tokai Tokai Univ., Univ., Ser. ESer. E 37 (0) 37(0) Free Vibration Analysis of a Moel Structure with New Tune Crale Mass Damer by Jitjinakun AMONPHAN * an Yoji SHIMAZAKI * (Receive on Mar. 4, 0 an accete on May 7, 0) Abstract Tune crale mass amers (TCMDs) use the motion of a swing mass on a curve surface to issiate structural vibration energy. In this stuy, we eveloe a TCMD that ha a constant swing see. We verify its erformance by erforming eeriments an numerical analysis for a structure unergoing free vibrations. To obtain a evice with a constant see, variable raii of the curve surfaces were calculate using simle enulum ynamics. In this stuy, the amer was installe in a simle one-story rigi-frame moel with a frequency of aroimately Hz an free vibrations was ecite. The numerical analysis results agree well with the eerimental results. Keywors: tune mass amer, free vibration, structural control. Introuction Vibrations are ecite in structures by eternal ynamic forces generate by natural henomena such as earthquakes an win an human activities such as traffic, civil engineering work, an construction work. A structure generally ehibits low aming when its natural vibration moes are ecite by an eternal ynamic force. Resonance may cause structural amage, which can otentially lea to significant loss of life an reuce liveability. Vibration control systems have recently been alie to overcome these vibration roblems. Vibration control systems can be broaly classifie into two tyes: active systems that require an eternal energy source to absorb vibrational energy an assive systems that o not require an energy source. Several kins of ractical assive controllers have been eveloe; these inclue a controller that uses laminate rubber or a coil sring to suort the weight of a amer, a susene amer that emloys the rincile of a enulum ), an imact amer that uses a steel ball ), a tune liqui amer that uses 3 5 ) 6, 7 ) liquis, an a tune rotary-mass amer consisting of a rolling mass an a container that ermits the mass to move freely along its inner arc. The tune crale mass amer (TCMD) 8 ) is another assive amer. It relies on the movement of the swing * Grauate stuent, Course of Civil Engineering * Professor, Deartment of Civil Engineering mass along a curve surface to issiate the vibrational energy of a structure. TCMDs have the avantages of being simle, comact, an easy to maintain. The swing mass has small wheels attache that allow the mass to move along the curve surface. A recently eveloe TCMD emloys a rail with a variable raius curve to realize a constant swing see for large oscillations. Simle enulum ynamics is use to etermine the variable raius curve. This stuy evelos a moel of the new TCMD an verifies its erformance through both eeriments an numerical analysis.. Device configuration. Eerimental moel A moel of a one-story structure is use to eerimentally eamine the TCMD. Figure shows the moel. It is mae of steel PL-60 8 (SS400) an uses four 300-mm-high columns to suort a floor. The lateral sring constant of the structure, k, is. kgf/cm. Figure shows the force islacement relationshi of the structure. The structural moel has an effective mass of 0 kgf (incluing the TCMD) an a natural frequency of aroimately. Hz. The iamon symbols ( ) in eerimental wavefrom are use to calculate the aming constant h (= 0.003) of the structure. Vol., 0 3

2 Jitjinakun AMONPHAN * an Yoji SHIMAZAKI * Jitjinakun AMONPHAN an Yoji SHIMAZAKI 700 mm 85 mm Here, h is the aming constant obtaine from free vibrations of the TCMD. The TCMD has a natural frequency of.9 Hz; it is constant for large amlitues of the mass swing. 300 mm Fig. Structure Fig. 4 Plot of aming constant ( h ) against number of magnets 3. Numerical analysis 3. Determination of constant swing mass see Fig. Force islacement line of the structure The equation of motion for a simle enulum with no aming force is given by:. Moeling a crale TCMD g sin 0 t () Figure 3 shows lan, front, an sie views of the TCMD moel use in the eeriments. The TCMD is mae of three coer lates. The swing mass has three wheels. The imensions of the mile late are mm an those of the sie lates are mm. The wheel has a iameter of 7 mm. The swing mass (incluing the wheels) is about 6 kg. The mass moves along three curve surfaces as the structure moves. To realize magnetic aming, neoymium magnets are attache to both sies of the swing mass. A magnetic fiel is then generate when the mass moves along the coer lates. The aming strength of the mass can be ajuste by varying the number of magnets attache to the swing mass. Swing mass Sie late Mile late where g is the acceleration ue to gravity, is the enulum length, t is the time an is the angular islacement. The erio T for the motion escribe in Eq. () is given by: T 4 () where K 0 (3) sin ( m ) sin In Eq. (3), m is the maimum swing angle of the enulum. To obtain a constant T for large oscillation amlitues, Eq. () can be rewritten for g K as Wheels Plan view Rail T g (4) 4K We then calculate the variable raius for each m. Front view Fig.3 TCMD Sie view Figure 4 shows the relationshi between the aming constant h an the number of magnets use for the TCMD. Simson s rule is use to integrate Eq. (3). To obtain a constant see for the TCMD, a variable raius use to esign a curve rail surface. is 4 Proceeings of the School of Engineering, Tokai University, Series E

3 Free Vibration Analysis of a Moel Structure with New Tune Crale Mass Damer Free Vibration Analysis of a Moel Structure with New Tune Crale Mass Damer 3. Numerical analysis of moel structure an TCMD Figure 5 shows a schematic iagram of the swing mass. v F u m m mg N F t = Fig.5 Free boy iagram of TCMD structure. We then obtain the following equation of motion for the structure. c () or F t k m c k m m sin g cos sin m m m m c cos m () We can now numerically calculate the coule equations (0) an (). Here, m is the mass of the swing. Because the mass of the wheels is small relative to the swing mass, alternatively, we ignore the rotational energy of the wheels. In aition, F t is the horizontal force given by the TCMD, is the horizontal acceleration sulie by the structure, F is the aming force, an N is the normal force. The equation of motion for mass m in the u an v irections can be written as u irection: mg sin F m (5) v irection: mgcos Nm where (6) F c (7) only viscous aming c is consiere here; for simlicity, we ignore frictional aming between the wheel an the curve surface. The equation of motion for the crale in the horizontal irection then becomes m F t where 0 (8) F N sin F cos (9) t or sin c cos gcossin (0) m Figure 6 shows a schematic iagram of the structure. Here, m is the mass of the structure, c is the viscous aming coefficient, k is the sring constant of the structure, an is the horizontal islacement of the Fig.6 Free boy iagram of structure 4. Eerimental results Eerimental measurements were erforme to clarify the issiation of the vibrational energy of the simle moel structure by the TCMD. Laser islacement sensors were use to measure the time islacement resonses of the structure an the TCMD. The measurement samling rate was 5 ms. An initial lateral islacement of mm was given to the structure. Beats are observe because of the weak aming of the TCMD. When the swing mass with magnets moves on the curve surface, the structural aming becomes much greater than that without the magnets. Figures 8 0 show the eamles of eerimental an analytical waveforms for the structure an TCMD; Figures 7-9 (a) an (b) show the eerimental waveforms obtaine when si, nine, an 3 magnets were use for the TCMD, resectively. 5. Analysis results The fourth-orer Runge Kutta metho was use to numerically solve Eqs. (0) an (); was uate in each iteration time ste. Figures 7-9 comare the eerimental an analysis results for the structure uner free vibrations. Vol., 0 5

4 Jitjinakun AMONPHAN * an Yoji SHIMAZAKI * Jitjinakun AMONPHAN an Yoji SHIMAZAKI Dislacement (mm) Time (s) a) Structure resonses (eeriment) b) TCMD resonses (eeriment) c) Structure resonses (analysis) ) TCMD resonses (analysis) Fig.7 Dislacement resonses for h = 5.48% (si magnets) a) Structure resonses (eeriment) b) TCMD resonses (eeriment) c) Structure resonses (analysis) ) TCMD resonses (analysis) Fig. 8 Dislacement resonses for h = 0.05% (nine magnets) a) Structure resonses (eeriment) b) TCMD resonses (eeriment) c) Structure resonses (analysis) ) TCMD resonses (analysis) Fig.9 Dislacement resonses for h = 5.0% (3 magnets) 6 Proceeings of the School of Engineering, Tokai University, Series E

5 Free Vibration Analysis of Moel Structure with New Tune Crale Mass Damer Free Vibration Analysis of a Moel Structure with New Tune Crale Mass Damer Figure 0 shows a lot of the eerimentally obtaine viscous aming constant h of the structure obtaine against the number of magnets. The vertical ais reresents h an the horizontal ais reresents the number of magnets use in the TCMD. The heights of the two oints inicate by the iamon symbols in the Figures (a) are use to calculate h. Figure shows the viscous aming constant h of the structure obtaine from the numerical analysis. The vertical ais reresents h an horizontal ais reresents the aming constant from Fig.4. The aming constant of the structure ( h ) obtaine by the eeriment agrees well with those obtaine by analysis. Fig.0 Daming constant versus number of magnets (eeriment) Fig. Daming constant versus aming ratio of TCMD (analysis) h References ) Teramoto, H., Sano, S., Nagai, M., an Okui, Y., 990. Design an Fabrication on Proceures of TMD in the Bannaguro Brige (In Jaanese). Journal of Structural Engineering, JSCE, Vol. 36A, ) Uno, K., Kitagawa, S., Tsutsumi, H., an Jo, I., 990. Isolation of Lighting Pole from Win Vorte Sheing by Imact Damers (In Jaanese). Journal of Structural Engineering, JSCE, Vol. 36A, ) Fujino, Y., Pacheco, B. M., Chaiseri, P., an Sun, L. M., 988. Parametric Stuies on Tune Liqui Damer (TLD) using Circular Tanks by Freeoscillation Eeriments. Journal of Structural an Earthquake Engineering, JSCE, No. 398, ) Fujino, Y., Sun, L. M., Pacheco, B. M., an Chaiseri, P., 99. Tune Liqui Damer (TLD) for Suressing Horizontal Motion of Structure. Journal of Engineering Mechanics, ASCE, Vol. 8, No. 0, ) Ree, D., Yu, J., Yeh, H., an Gararsson S., 998. Investigation of Tune Liqui Damers uner Large Amlitue Ecitation. Journal of Engineering Mechanics, ASCE, ) Obata, M. an Shimazaki, Y., 007. Vibration Control Effects an Alication Eamle of Tune Rotary Dame Mass Damer. The Structural Engineer, Vol. 85, No. 3, ) Obata, M. an Shimazaki, Y., 008. Otimum Parametric Stuies on Tune Rotary-Mass Damer. Journal of Vibration an Control, Vol. 4, No. 6, ) Takei, M. an Shimazaki, Y., 00. Vibration Control Effects of Tune Crale Dame Mass Damer. Journal of Alie Mechanics, Vol. 3, Conclusion This stuy roose a new vibration aming evice (TCMD) that consists of a swing mass an a curve crale with a variable raius. A constant swing mass see was attaine using a crale esigne by a simle enulum ynamics. The erformance of the new TCMD was emonstrate through eeriments an numerical analysis for a structure with a frequency of aroimately Hz uner free vibrations. The new TCMD is more comact than conventional evices. In the future, we inten to etermine the otimal arameters using the numerical moel eveloe in this stuy. Vol., 0 7