A study of the application of the pendulum tuned mass dampers in building floor vibration controls

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1 A study of the application of the pendulum tuned mass dampers in building floor vibration controls M. Setareh 1, J. K. Ritchey & T. M. Murray 1 Department of Architecture, Virginia Polytechnic Institute and State University, U.S.A. Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, U.S.A. Abstract Results of an analytical study of using a pendulum tuned mass damper (PTMD) to control the excessive vibrations of building floors are presented. An equivalent single degree of freedom (SDOF) model for the PTMD is developed to derive the equations of motion of the coupled PTMD-floor system. Closed form solutions for the floor and PTMD dynamic responses in terms of nondimensional parameters are developed. The optimal design parameters of the PTMD are found using an optimization algorithm. Reductions in the floor response using the PTMD along with the effects of off-tuning due to variations in the floor system parameters on the PTMD performance are presented. Keywords: pendulum tuned mass damper, tuned vibration absorber, floor vibration, vibration control. 1 Introduction Excessive floor vibrations due to human movements have become a major source of serviceability problem in buildings. This is mainly due to several factors such as: decrease in floor mass resulting from the use of higher strength materials, decrease in the floor natural frequency caused by longer floor spans, increase in the number of rhythmic human activities such as aerobics, in-door jogging, rock concerts, etc., and decrease in damping due to less friction between building components, fewer partitions, etc. Tuned mass dampers (TMDs) have 004 WIT Press, ISBN

2 88 High Performance Structures and Materials II been used since early twentieth century for the vibration control of various mechanical systems. The first extensive work on TMDs was by Den Hartog [1]. In general a TMD consists of a secondary mass connected by elements providing stiffness and damping to the mass to be controlled. The TMD mass is usually a small percentage of the main mass. TMDs have also been used for building floor vibration, with the first study by Lenzen [] in 1966 who used a TMD mass of about % of the floor mass. Allen and Swallow [3] used steel boxes loaded with concrete blocks to provide the TMD mass. Allen and Pernica [4] used TMDs in the form of wooden planks with weights on top for the reduction of the excessive vibrations due to walking. Setareh and Hanson [5,6] used TMDs made of steel boxes filled with steel plates, connected to springs, and dampers to control floor vibrations due to dancing in an auditorium floor. Webster and Vaicajtis [7] constructed TMDs made of a concrete filled steel box and steel plates suspended by springs and viscous dampers to reduce the annoying vibrations of a ball room floor. Shope and Murray [8] used a multi-celled tuned liquid damper to reduce the annoying vibrations of an existing office floor. These efforts had various levels of success in reducing the annoying floor motions. There are three main constraints as related to the application of TMDs to floor systems. First, the TMD dimensions are limited to the floor envelope. Second, the TMD has to be finely tuned to be effective, and third the variation of the floor parameters should not result in significant off-tuning of the TMD. In an attempt to comply with these requirements, a Pendulum Tuned Mass Damper (PTMD) was considered to control floor vibrations. This paper presents the results of an analytical study of the use of the PTMD to control floor vibrations. Experimental validation of these results can be found elsewhere [9]. Description of the PTMD The PTMD considered here is designed by the ESI, Inc. [10], which is shown in fig. 1. To reduce the vertical dimension of the PTMD, the weights in the form of steel plates are distributed along the PTMD arm. The springs (to provide stiffness) are movable along the PTMD arm, therefore, the PTMD natural frequency can be finely tuned. The dampers are attached to the end of the PTMD arm to maximize the damping force. 3 PTMD Dynamic Model To simplify the coupled PTMD-floor equations of motion, an equivalent singledegree of freedom (SDOF) model of the PTMD was developed. Fig. (a) shows a general model of the PTMD. In this representation the PTMD has a mass m(x) uniformly distributed along the arm. The arm stiffness is also assumed to be uniform (EI(x)=constant). For the PTMD to act as a single-degree of freedom system the rigidity of the arm has to be large compared to the stiffness of the spring (k). To establish a minimum arm bending stiffness for this assumption, the vibrational shape function of the PTMD was derived using the superposition of two cases: (1) arm 004 WIT Press, ISBN

3 High Performance Structures and Materials II 89 to be infinitely rigid (EI ) [see fig. (b)], and () spring stiffness to be very large (k ) [see fig. (c)]. The shape functions developed were based on the static deflection of the arm resulting from the self-weight (w) in each case. As shown in fig. (c), the self weight (w) was applied in different directions on the two sides of the arm. This was done to estimate the PTMD lowest mode (see fig. (a) for the combined mode). 4 Stacks of Plates Pin Support Base to Spring Connection 4x4x3/8 Angles Base Composed of 4x4x3/8 Angles Spring Figure 1: Pendulum Tuned Mass Damper (PTMD). Eqns (1) and () are the resulting shape functions: w( s ) x1 wx1 3 3 Rs x1 1 ) ( s x1 ) ( s sx1 + x1 ) 1EI s 4EI k s f ( x = (1) 3 [ 4( ) + 6( ) x 4( x ] wx f ( x ) = s s s s ) + x 4EI Rs Rs 3 + w + 3 x + k s k Rs x 4EI s k () 004 WIT Press, ISBN

4 90 High Performance Structures and Materials II In these equations, E=modulus of elasticity; I=moment of inertia of the PTMD arm; =total PTMD length; s =location of spring from the left end pin; k=spring stiffness, R s =spring reaction force; w=self-weight of the PTMD arm; x 1 and x are the distances from the pin and the spring, respectively. EI(x), m(x) k EI(x) s (a) w k x 1 x w (b) x 1 x w (c) k M (d) Figure : PTMD dynamic modelling: (a) continuous model; (b) shape function for rigid arm; (c) shape function for flexible arm; (d) equivalent SDOF model of the PTMD. Using the following non-dimensional parameters: x 1 x a =, b = 3 s k β =, γ = EI (3) 004 WIT Press, ISBN

5 High Performance Structures and Materials II 91 the shape functions f(x 1 ) and f(x ) are normalized and rewritten in terms of a,b, β,and γ as: (1 β ) a a 3 3 a f '( x1) = ( β a ) + ( β βa + a ) γ + (4) 1β 4 β b f ' ( x) = β b [ 4(1 β ) β + 6(1 β ) b 4(1 β ) b + b ] + γ 4 b 1 + (5) β β Assuming the spring is located at about the center of the arm (β 0.5), fig. 3 shows the shape functions along the PTMD with varying stiffness factor (γ). From these plots for γ <10, the shape function is within 10% of a straight line representing a rigid PTMD arm. For a rigid arm the shape function is reduced to: x f ( x) = (6) and the lumped mass (M) at the tip of PTMD (see fig. (d)) is: x m M = o m( x)[ f ( x)] dx = o m( ) dx = (7) 3 4 Equations of motion The PTMD is typically tuned to a single mode of the floor structure. Therefore, the floor can be represented by a SDOF dynamic system with an attached PTMD as shown in fig. 4. The equations of motion for the coupled floor-ptmd are: ( m1 + m) x1 + m θ + c1x 1 + k1x1 = F( t) (8) m x1 + m θ + c θ + s kθ = 0 where m 1 is the floor mass, m is the PTMD lumped mass found from eqn. (7), c 1 and c are the floor and PTMD dampings, and k 1 and k are the floor and PTMD stiffnesses, respectively. x 1, x 1 and x 1 are the floor acceleration, velocity and displacement, respectively. θ, θ, and θ are the rotational acceleration, velocity and displacement of the PTMD, respectively. The excitation force and the floor and PTMD responses are: F = F i t oe ω x Xe i t 1 = ω (9) θ = θ e iω t where ω is the excitation frequency. 004 WIT Press, ISBN

6 9 High Performance Structures and Materials II Figure 3: Variation of the shape function along the PTMD with different values of γ. In addition, the following parameters are defined: k1 ω floor natural frequency; 1 = = ω = s k = PTMD natural frequency m1 m c1 F ξ1 = = floor damping ratio; x = o st = floor static deflection m1 ω1 k1 c m ξ PTMD damping ratio; = = µ = = mass ratio mω m1 ω f = = frequency ratio of the PTMD to the floor ω1 ω g = = frequency ratio of the excitation to the natural frequency of the floor ω1 Substituting eqn. (9) and the above parameters into eqn. (8), the floor X displacement response factor is found as: x st 004 WIT Press, ISBN

7 High Performance Structures and Materials II 93 X xst = ( f g ) + (ξ gf) [( 1 g )( f g ) fg (4ξξ 1 + fµ )] + [(g )( ξ f (1 g g µ ) + ξ1 ( f g )] (10) l s l EI = m θ(t) k c x 1 (t) F(t) m 1 k 1 c 1 Figure 4: -DOF floor-ptmd system. Considering the floor acceleration amplitude, A=ω X, the floor acceleration response factor is: A F m X = g (11) o x st 1 Eqn. (11) is used to find the floor acceleration response factor. Typically, a mass ratio (µ) of is practical for the TMDs used in building floors. With the measured floor damping ratio (ξ 1 ), the PTMD design parameters f, and ξ have to be found such that the peak floor acceleration response factor is minimized. This is solved using an optimization algorithm. 5 Application of the PTMD on a floor 5.1 PTMD performance The floor used in this study had a damping ratio, ξ 1 =0.8%, with a natural frequency of 7.4 Hz and the PTMD had a mass ratio, µ = Using these 004 WIT Press, ISBN

8 94 High Performance Structures and Materials II parameters, the optimum frequency ( f ) was found to be 0.985, and the optimum PTMD damping ratio was 11%. Fig. 5 shows the floor acceleration response factors without and with the PTMD. As can be noted the peak floor response decreased from 6.5 to 7.1, resulting in a reduction of 89% after adding the PTMD to the floor Peak Response = A/Fo/m Frequency, f (Hz) (a) 8 7 Peak Response = A/Fo/m Frequency, f (Hz) (b) Figure 5: Floor acceleration response factor (a) without PTMD; (b) with PTMD. 5. Off-tuning of the PTMD As mentioned in the introduction, one of the problems with the application of TMD to floors is the off-tuning of the TMD due to changes in the floor parameters, in particular the floor mass due to the variations in the live loads. 004 WIT Press, ISBN

9 High Performance Structures and Materials II 95 Therefore, a parametric study was performed to investigate the effects of floor mass, stiffness, and damping changes on the performance of the PTMD. Each parameter was varied from ¼ (5%) of their values where optimized to twice (00%) of the values at the optimum. Fig. 6 shows the results. As could be expected, the floor response decreases with an increase in the floor damping. When the floor mass is reduced the amount of off-tuning (increase in the floor response) is less than when the floor mass is increased. This could also be expected as a decrease in the floor mass results in an increase in the mass ratio (µ) and therefore, the PTMD has more effect on the floor. The variation in the floor stiffness has an opposite effect of the changes in the floor mass. When the floor stiffness is increased the floor response is less offtuned that when it is decreased by the same amount. This is due to the fact that the floor response decreases with an increase in its stiffness as it reduces the floor static displacement. Figure 6: Variation of the peak floor response due to changes in floor parameters. 6 Summary and conclusions This paper presented a study of PTMDs to control excessive vibrations of floors. From the results presented here it can be concluded that PTMDs can provide a practical method of vibration control for such applications. Acknowledgements The research presented in this paper has been supported by the National Science Foundation under Grant No. CMS The PTMD was furnished by the ESI Engineering, Inc., Minneapolis, Minnesota. Their support and in particular technical assistance of Mr. Anthony Baxter are greatly appreciated. References [1] Den Hartog, J.P., Mechanical Vibrations, McGraw-Hill: New York, WIT Press, ISBN

10 96 High Performance Structures and Materials II [] Lenzen, K.H., Vibration of steel joist-concrete slab floors. Engineering Journal, American Institute of Steel Construction, 3, pp , [3] Allen, D.L. & Swallow, J.C., Annoying floor vibrations diagnosis and therapy. Sound and Vibration, pp.1-17, [4] Allen, D.E. & Pernica, G., A simple absorber for walking vibrations. Canadian Journal of Civil Engineering, 11, pp , [5] Setareh, M. & Hanson, R.D., Tuned mass dampers for balcony vibration control. Journal of Structural Engineering, American Society of Civil Engineers, 118, pp , 199. [6] Setareh, M. & Hanson, R.D., Tuned mass dampers to control floor vibrations from humans. Journal of Structural Engineering, American Society of Civil Engineers, 118, pp , 199. [7] Webster, A.C. & Vaicajtis, R., Application of tuned mass dampers to control vibrations of composite-floor system. Engineering Journal, American Institute of Steel Construction, 9, pp , 199. [8] Shope, R. & Murray, T.M., Using tuned mass dampers to eliminate annoying floor vibrations. Proc. of Structures Congress XIII, American Society of Civil Engineers: Boston, Massachusetts, 1, pp , [9] Ritchey, J., Application of Magneto-Rheological Dampers in Tuned Mass Dampers for Floor Vibration Control, Masters Thesis, Virginia Polytechnic Institute and State University, 003. [10] ESI Engineering, Inc., Minneapolis, Minnesota, WIT Press, ISBN

Use of Portable Tuned Mass Dampers for Vibration Control of Pedestrian Bridges

Proceedings of the National Conference On Undergraduate Research (NCUR) 015 Eastern Washington University, Cheney, WA April 16-18, 015 Use of Portable Tuned Mass Dampers for Vibration Control of Pedestrian