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ConCh04v2.fm Page 217 Thursday, July 11, 2002 4:33 PMC H A P T E R 4Tuned MassDamperSystems4.1INTRODUCTIONA tuned mass damper (TMD) is a device consisting of a mass, a spring, and a damperthat is attached to a structure in order to reduce the dynamic response of thestructure. The frequency of the damper is tuned to a particular structural frequency sothat when that frequency is excited, the damper will resonate out of phase with thestructural motion. Energy is dissipated by the damper inertia force acting on thestructure. The TMD concept was first applied by Frahm in 1909 (Frahm, 1909) toreduce the rolling motion of ships as well as ship hull vibrations. A theory for theTMD was presented later in the paper by Ormondroyd and Den Hartog (1928),followed by a detailed discussion of optimal tuning and damping parameters in DenHartog’s book on mechanical vibrations (1940). The initial theory was applicable foran undamped SDOF system subjected to a sinusoidal force excitation. Extension ofthe theory to damped SDOF systems has been investigated by numerous researchers.Significant contributions were made by Randall et al. (1981), Warburton (1981, 1982),Warburton and Ayorinde (1980), and Tsai and Lin (1993).This chapter starts with an introductory example of a TMD design and a briefdescription of some of the implementations of tuned mass dampers in buildingstructures. A rigorous theory of tuned mass dampers for SDOF systems subjectedto harmonic force excitation and harmonic ground motion is discussed next. Various cases, including an undamped TMD attached to an undamped SDOF system, adamped TMD attached to an undamped SDOF system, and a damped TMDattached to a damped SDOF system, are considered. Time history responses for a217

ConCh04v2.fm Page 218 Thursday, July 11, 2002 4:33 PM218Chapter 4Tuned Mass Damper Systemsrange of SDOF systems connected to optimally tuned TMD and subjected to harmonic and seismic excitations are presented. The theory is then extended to MDOFsystems, where the TMD is used to dampen out the vibrations of a specific mode.An assessment of the optimal placement locations of TMDs in building structures isincluded. Numerous examples are provided to illustrate the level of control that canbe achieved with such passive devices for both harmonic and seismic excitations.4.2AN INTRODUCTORY EXAMPLEIn this section, the concept of the tuned mass damper is illustrated using the twomass system shown in Figure 4.1. Here, the subscript d refers to the tuned massdamper; the structure is idealized as a single degree of freedom system. Introducingthe following notationk2ω ----m(4.1)c 2ξωm(4.2)k2ω d ------dmd(4.3)c d 2ξ d ω d m d(4.4)and defining m as the mass ratio,mm ------dm(4.5)the governing equations of motion are given byp2·····Primary mass ( 1 m )u 2ξωu ω u ----- – mu dm(4.6)pkdkmdmccduFIGURE 4.1: SDOF-TMD system.u 1 ud

ConCh04v2.fm Page 219 Thursday, July 11, 2002 4:33 PMSection 4.2An Introductory Example2·····Tuned mass u d 2ξ d ω d u d ω d u d – u219(4.7)The purpose of adding the mass damper is to limit the motion of the structurewhen it is subjected to a particular excitation. The design of the mass damperinvolves specifying the mass md , stiffness kd , and damping coefficient cd . Theoptimal choice of these quantities is discussed in Section 4.4. In this example, thenear-optimal approximation for the frequency of the damper,ωd ω(4.8)is used to illustrate the design procedure. The stiffnesses for this frequency combination are related byk d mk(4.9)Equation (4.8) corresponds to tuning the damper to the fundamental period of thestructure.Considering a periodic excitation,p p̂ sin Ωt(4.10)the response is given byu û sin ( Ωt δ 1 )u d û d sin ( Ωt δ 1 δ 2 )(4.11)(4.12)where û and δ denote the displacement amplitude and phase shift, respectively.The critical loading scenario is the resonant condition, Ω ω. The solution for thiscase has the following form:p̂1û -------- --------------------------------------21km 1 2ξ- -------- ----2ξmd(4.13)1û d --------û2ξ d(4.14)2ξ1tan δ 1 – ------ -------m 2ξ d(4.15)πtan δ 2 – --2(4.16)

ConCh04v2.fm Page 220 Thursday, July 11, 2002 4:33 PM220Chapter 4Tuned Mass Damper SystemsNote that the response of the tuned mass is 90º out of phase with the response of theprimary mass. This difference in phase produces the energy dissipation contributedby the damper inertia force.The response for no damper is given byp̂ 1û --- ------ k 2ξ (4.17)πδ 1 – --- 2 (4.18)To compare these two cases, we express Eq. (4.13) in terms of an equivalentdamping ratio:p̂ 1û --- -------- k 2ξ e (4.19)m2ξ1 2ξ e ----- 1 ------ -------- m 2ξ d 2(4.20)whereEquation (4.20) shows the relative contribution of the damper parameters to thetotal damping. Increasing the mass ratio magnifies the damping. However, since theadded mass also increases, there is a practical limit on m. Decreasing the dampingcoefficient for the damper also increases the damping. Noting Eq. (4.14), the relative displacement also increases in this case, and just as for the mass, there is a practical limit on the relative motion of the damper. Selecting the final design requires acompromise between these two constraints.Example 4.1: Preliminary design of a TMD for a SDOF systemSuppose ξ 0 and we want to add a tuned mass damper such that the equivalentdamping ratio is 10%. Using Eq. (4.20), and setting ξ e 0.1 , the following relationbetween m and ξ d is obtained:m2ξ1 2----- 1 ------ -------- 0.1 m 2ξ d 2(1)The relative displacement constraint is given by Eq. (4.14):1û d -------- û 2ξ d (2)

ConCh04v2.fm Page 221 Thursday, July 11, 2002 4:33 PMSection 4.2An Introductory Example221Combining Eq. (1) and Eq. (2) and setting ξ 0 leads to2û dm----- 1 ----- 0.1 û 2(3)Usually, û d is taken to be an order of magnitude greater than û . In this case, Eq. (3)can be approximated asm û d ----- ----- 0.12 û (4)The generalized form of Eq. (4) follows from Eq. (4.20):1m 2ξ e ------------- û d û (5)Finally, taking û d 10û yields an estimate for m:2 ( 0.1 )m --------------- 0.0210(6)This magnitude is typical for m. The other parameters are1 û ξ d --- ----- 0.052 û d (7)k d mk 0.02k(8)and from Eq. (4.9)It is important to note that with the addition of only 2% of the primary mass,we obtain an effective damping ratio of 10%. The negative aspect is the large relative motion of the damper mass; in this case, 10 times the displacement of the primary mass. How to accommodate this motion in an actual structure is an importantdesign consideration.A description of some applications of tuned mass dampers to building structures is presented in the following section to provide additional background on thistype of device prior to entering into a detailed discussion of the underlying theory.

ConCh04v2.fm Page 222 Thursday, July 11, 2002 4:33 PM2224.3Chapter 4Tuned Mass Damper SystemsEXAMPLES OF EXISTING TUNED MASS DAMPER SYSTEMSAlthough the majority of applications have been for mechanical systems, tunedmass dampers have been used to improve the response of building structures underwind excitation. A short description of the various types of dampers and severalbuilding structures that contain tuned mass dampers follows.4.3.1 Translational Tuned Mass DampersFigure 4.2 illustrates the typical configuration of a unidirectional translationaltuned mass damper. The mass rests on bearings that function as rollers and allowthe mass to translate laterally relative to the floor. Springs and dampers are insertedbetween the mass and the adjacent vertical support members, which transmit thelateral “out-of-phase” force to the floor level and then into the structural frame.Bidirectional translational dampers are configured with springs/dampers in twoorthogonal directions and provide the capability for controlling structural motion intwo orthogonal planes. Some examples of early versions of this type of damper aredescribed next. John Hancock Tower (Engineering News Record, Oct. 1975)Two dampers were added to the 60-story John Hancock Tower in Boston to reducethe response to wind gust loading. The dampers are placed at opposite ends of thefifty-eighth story, 67 m apart, and move to counteract sway as well as twisting due tothe shape of the building. Each damper weighs 2700 kN and consists of a lead-filledsteel box about 5.2 m square and 1 m deep that rides on a 9-m-long steel plate. Thelead-filled weight, laterally restrained by stiff springs anchored to the interior columns of the building and controlled by servo-hydraulic cylinders, slides back andforth on a hydrostatic bearing consisting of a thin layer of oil forced through holesin the steel plate. Whenever the horizontal acceleration exceeds 0.003g for two consecutive cycles, the system is automatically activated. This system was designed andmanufactured by LeMessurier Associates/SCI in association with MTS SystemCorp., at a cost of around 3 million dollars, and is expected to reduce the sway ofthe building by 40 to 50%.SupportmdFloor beamDirection of motionFIGURE 4.2:Schematic diagram of a translational tuned mass damper.