STRUCTURAL CONTROL AND HEALTH MONITORING Struct. Control Health Monit. 2016; 23:349 368 Published online 28 July 2015 in Wiley Online Library (wileyonlinelibrary.com)..1775 Discussion of system intrinsic parameters of tuned mass dampers used for seismic response reduction Julio C. Miranda*, CH2M HILL, 1737 N. First Street, Suite 300, San Jose, CA 95112, USA SUMMARY Assimilating the structures incorporating tuned mass dampers to 2-degrees-of-freedom mechanical systems, this paper discusses the salient parameters defining the efficiency of these devices when affixed to structures for the purpose of seismic response reduction. Focusing on parameters that are intrinsic to the mechanical systems and independent of ground motions, numerical and analytical expressions are first obtained for the modal damping of the systems. Subsequently, it is proposed that the highest efficiency in terms of modal damping allocation is achieved at tuning that results in modal damping that is in the same proportion as the participation factors for the modes. Further, some properties of the frequencies, tuning, and participation factors are analytically demonstrated. Finally, limited numerical calculations using a spectrum-compatible accelerogram are offered to support the proposed method for modal damping allocation. Copyright 2015 John Wiley & Sons, Ltd. Received 19 October 2014; Revised 26 April 2015; Accepted 7 July 2015 KEY WORDS: passive control; energy dissipation; seismic design; tuned mass dampers; building technology 1. INTRODUCTION Current research indicates that a growing consensus is gaining momentum suggesting that the use of tuned mass dampers (TMDs) is a practical and efficient method to reduce structural responses because of strong ground motions. Subsequent to the pioneering work by Villaverde [1], Feng et al. [2], and Sadek et al. [3], studies have been presented showing the feasibility of implementing systems that can develop substantial damping [4 6]. Further, other studies have identified that using large mass ratio TMDs results in a more robust definition of the parameters leading to seismic response reduction [7 10]. This observation extends the applicability of TMDs to systems in which portions of the structure itself may be mobilized to protect the complete structural assembly. Along this line, Hoang et al. [7] report the retrofit of a large bridge in Japan, for which the deck is allowed to behave as a TMD, with an effective generalized mass ratio of 0.77. De Angelis et al. [8] present information on large mass ratio TMDs used by others and report on their own experimental work with mass ratios up to 1.049. Other researchers have explored other aspects of using TMDs, such as the conditions in which these could be useful for reducing the response to impulsive ground motion [11,12]. Overall, two approaches have been taken for studying such devices: (i) Calibration, usually referred to as optimization, of the TMD parameters such that given a defined ground motion, a certain aspect of the structural response is minimized, and (ii) intrinsic consideration of the TMD parameters, such that independent of any ground motion, certain properties of the structural system, such as modal damping, are calibrated with the expectation of a reduced structural response. *Correspondence to: Julio C. Miranda, CH2M HILL, 1737 N. First Street, Suite 300, San Jose, CA 95112, USA. E-mail: julio.miranda@ch2m.com Copyright 2015 John Wiley & Sons, Ltd.
350 J. C. MIRANDA Proponents of the first modality have assumed ground motions that could be assimilated to white noise random processes, like Feng and Mita [2], or that could be characterized by Kanai Tajima type of power spectra density, like Hoang et al. [7]. These authors have provided expressions corresponding to the optimum tuning and damping likely to minimize the response of structures provided with TMDs. While the procedures that use this modality are versatile, the calibrated TMD properties, namely, the tuning and damping, become a function of the ground motion. Thus, the possibility of important variations between the properties of the ground motions used in the design and those of the actual event realization has to be considered. Indeed, under such circumstances, the expectations numerically forecasted might be critically affected. Recently, Salvi et al. [13,14] performed extensive time history calculations using suites of real earthquakes to analyze buildings equipped with TMDs and identified optimum tuning with a rather wide range of values, which in the view of the author of the present paper indicates the sensitivity of such optima to ground motions. While the response of structures incorporating TMDs calibrated per the second modality is obviously also a function of the spectral characteristics of the ground motion in relation to the dynamic properties of the mechanical systems, the tuning and damping themselves are preset in a manner that is independent of the potential earthquake. Along these lines, Villaverde [1] performed early work suggesting to use small, but highly damped, resonant masses attached to the top of buildings in order to induce two complex modes with damping ratios approximately equal to the average of the damping ratios of the resonant mode of the building and the TMD. Following on this lead, Sadek et al. [3] suggested to install roof-mounted TMDs proportioned so as to induce two complex modes with equal frequencies and damping ratios. Accordingly, these authors made extensive studies of single-degreeof-freedom and multiple-degree-of-freedom structural systems with a wide range of natural periods, provided with TMDs calibrated in their prescribed manner, and subjected to 52 real earthquakes. Their results showed substantial reduction of the response, in some cases of up to 50%. Miranda [4] presented an energy-based numerical model and successfully used it to verify the optimum tuning and damping proposed previously by Sadek et al. [3]. While this model ignores coupling due to damping, it provides a very convenient platform to study the systems under consideration. Moutinho [5] considered systems with two complex modes having equal modal damping coefficients, but different frequencies, and with TMDs parameters in correspondence with the minimum value of their dynamic amplification factors for harmonic loading. Miranda [6] proposed an analytical methodology for the tuning of systems with two equal real mode damping coefficients, observing that this condition also implies equality of damping coefficients for the corresponding two complex modes. Additionally, such proposal was shown to contain the methods by Villaverde [1] and by Sadek et al. [3] as particular cases. The purpose of this paper is to discuss the parameters that define the structural dynamics of the mechanical systems composed of the TMDs and the structures to which they are affixed to. This is carried out in a manner that examines these parameters intrinsically, without consideration of ground motions. To gain insight into such parameters, three methods are used: (i) the modal energy-based method proposed by Miranda [4], which is approximated because it ignores coupling due to damping, (ii) comparison of the system s characteristic equations in terms of real modes, and (iii) comparison of the system s characteristic equations in terms of complex modes. The last two methods stem from the invariant properties of the system s frequencies. Emphasis is placed on discussing the damping properties, as this is the most significant reason to use TMDs under seismic excitation. The paper will then discuss a method to calibrate TMDs according to a newly proposed damping control strategy. Subsequently, the paper discusses some properties relative to the frequencies, and participation factors. Finally, limited numerical calculations using spectrum-compatible accelerograms are offered to support the proposed control strategy. The theoretical model used in this paper considers a 2-degree-of-freedom system, which, although simplistic, has nevertheless enabled many studies, including the classic works by Den Hartog [15] and Warburton [16]. The study of such systems lends itself to complex modal analysis, and as such, in this paper, modes, frequencies, and damping are as appropriate double subindexed to signify derivation through, or in correspondence with, that methodology. Likewise, modes, frequencies, and damping, derived utilizing real mode analyses, are affected with single subindexes to differentiate them from the parameters obtained with complex procedures. As long as it is clear, this paper will refer to real or complex parameters, with the understanding that they pertain to real or complex mode analysis correspondingly.
SYSTEM INTRINSIC PARAMETERS OF TMDs 351 2. THEORETICAL BACKGROUND Consider the 2-degree-of-freedom mechanical system depicted in Figure 1. The upper portion, which constitutes the TMD, is characterized by its mass M U, its damping constant C U, and its spring stiffness K U. The lower portion is characterized by a mass M L, a damping constant C L, and a spring with stiffness K L. The latter parameters represent the effective properties for the structure under consideration and are presumed to be known. The following parameters are defined in order to characterize the mechanical system: ξ U ¼ ω 2 U ¼ K U M U (1) ω 2 L ¼ K L M L (2) Ω ¼ ω U (3) μ ¼ M U M L (4) C U p 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi (5) K U M U ξ L ¼ p 2 ffiffiffiffiffiffiffiffiffiffiffiffi (6) K L M L Equation (1) provides the circular frequency for the upper portion of the system, ω U, when considered independently. Equation (2) provides the circular frequency for the lower portion,,when considered independently. Equation (3) represents the tuning ratio, Ω, between the circular frequencies of the upper and lower portions. In Equation (4), μ is the ratio of the upper portion mass to the lower portion mass, parameter that is also presumed to be known. Equation (5) furnishes the coefficient of damping for the upper part, ξ U, whereas Equation (6) furnishes the coefficient of damping for the lower part, ξ L. Because the approximated numerical platform proposed by Miranda [4] is being used, the reader is referred to that paper. Per this method and for selected tuning and mass ratios, any dynamic state of the 2-degree-of-freedom system, as represented by its modal frequencies, modal shapes, modal C L Figure 1. Two-degree-of-freedom mechanical system.
352 J. C. MIRANDA participation factors, and modal damping, is defined by determining just one of the four interrelated modal energy parameters α j and β j corresponding to the jth mode of vibration. Considering the system shown in Figure 1 while undergoing free vibrations, its characteristic equation, see Miranda [6], may be written as follows: λ 4 þ f 3 λ 3 þ f 2 λ 2 þ f 1 λ þ f 0 ¼ 0 (7) where λ represents a complex frequency. The factors f i are given in Appendix Equations (A.1.a) to (A.1.d) and are a function of the mechanical properties, as previously defined. Similarly, if the equations of motion under free vibration are transformed to modal coordinates using the real modes, their characteristic equation is still written per Equation (7); see Miranda [6]. However, the factors f i are now expressed in terms of real mode properties. Such factors are also provided in Appendix, Equations (A.2.a) to (A.2.d). In these equations, M j, C j, and K j are the generalized mass, damping, and stiffness, for the jth real mode, respectively. C C is the term responsible for the coupling of the modal coordinates, while ω j and ξ j represent the circular frequencies and coefficients of damping for the jth mode of vibration. Proceeding with the third method presented to obtain the desired parameters, it is recalled that in case of nonproportional damping, the equations of motion can be exactly resolved using complex modal analysis, through a state-space decomposition as described, for example, by Hurty and Rubinstein [17]. It may be shown then that it is possible to transform to complex modal coordinates the equations of motion of the system shown in Figure 1 while undergoing free vibrations and write the following two equations: M 11 q 11 ðþþc t 11 _q 11 ðþþk t 11 q 11 ðþ¼0 t M 22 q 22 ðþþc t 22 _q 22 ðþþk t 22 q 22 ðþ¼0 t (8:b) wherein the q jj are complex modal coordinates and M jj, C jj, and K jj are the generalized mass, damping, and stiffness, corresponding to the jth complex mode, respectively. The characteristic equation of the last two expressions can again be written per Equation (7), but with the f i factors written as a function of complex modes, as presented in Appendix, Equations (A.4.a) to (A.4.d). The parameters ω jj and ξ jj represent the circular frequencies and coefficients of damping for the jth complex mode of vibration. It is noted that because the eigenvalues of the system are invariants, the three sets of factors f i, derived from the mechanical properties, the real modal, or the complex modal equations, are by definition equal to each other, respectively. This fundamental property provides the base for the discussion to be established later. (8:a) 3. DISCUSSION OF PARAMETERS Numerical examples will be prepared to help discuss the salient parameters of the systems under consideration. While the procedures and equations to be examined are general for any 2-degree-offreedom system, to highlight TMDs with large mass ratio, values of μ from 0.25 to 1.0, with increments of 0.25 are considered. Because TMDs are usually applied to weakly damped structures, a structural coefficient of damping ξ L equal to 0.03 is considered, while the TMD coefficient of damping ξ U is set at 0.3. The latter number corresponds to a value than can be readily achieved in current practice. Tuning values of Ω ranging from 0.1 up to higher than 1.0 are chosen. Tables I IV show the modal energy coefficients α j and β j, calculated per the procedures shown in Miranda [4], for the dynamic states corresponding to each chosen tuning and mass ratio. It is noted that these tables also provide coefficients corresponding to equal modal damping, real or complex; see Miranda [6]. In addition, the tables provide coefficients corresponding to the conditions of perfect balance of real mode strain energy, of resonance between the TMD and structure, and perfect balance of real mode kinetic energy, respectively; see Miranda [4]. Tables V VIII use the energy parameters to calculate the real circular frequencies normalized with respect to the structural frequency, the modal participation factors, and modal damping. In these tables, the participation factors, γ j, are defined, as usual, equal to φ T j Mr=φT j Mφ j where M is the system s mass matrix, φ j is the vector corresponding to the jth real
SYSTEM INTRINSIC PARAMETERS OF TMDs 353 Table I. Modal energy parameters for μ equal to 0.25, and Ω variable as shown, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω α 1 α 2 β 1 β 2 (2) 0.25 0.1000 39404.24997 0.025E-03 392.059949 2.551E-03 (3) 0.25 0.2000 2354.249575 0.000425 92.239159 0.010841 (4) 0.25 0.3000 431.408176 0.002318 36.979903 0.027042 (5) 0.25 0.4000 122.991869 0.008131 17.944272 0.055728 (6) 0.25 0.5000 44.227390 0.022610 9.456756 0.105745 (7) 0.25 0.7826 4.913209 0.203533 1.804745 0.554095 (8) 0.25 0.8944 2.618034 0.381966 1.000000 1.000000 (9) 0.25 1.0000 1.640388 0.609612 0.609612 1.640388 (10) 0.25 1.1547 1.000000 1.000000 0.333333 3.000000 Table II. Modal energy parameters for μ equal to 0.5, and Ω variable as shown, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω α 1 α 2 β 1 β 2 (2) 0.5 0.1000 19800.49994 0.051E-03 196.029850 5.101E-03 (3) 0.5 0.2000 1202.499168 0.000832 46.158335 0.021665 (4) 0.5 0.3000 227.186956 0.004402 18.573382 0.053841 (5) 0.5 0.4000 68.110318 0.014682 9.110233 0.109767 (6) 0.5 0.5000 26.462210 0.037790 4.921823 0.203177 (7) 0.5 0.6429 9.260441 0.107986 2.255428 0.443375 (8) 0.5 0.8165 3.732051 0.267949 1.000000 1.000000 (9) 0.5 1.0000 2.000000 0.500000 0.500000 2.000000 (10) 0.5 1.4142 1.000000 1.000000 0.171573 5.828427 Table III. Modal energy parameters for μ equal to 0.75, and Ω variable as shown, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω α 1 α 2 β 1 β 2 (2) 0.75 0.1000 13268.74992 0.075E-03 130.699849 0.007651 (3) 0.75 0.2000 818.748779 0.001221 30.797530 0.032470 (4) 0.75 0.3000 159.278701 0.006278 12.435231 0.080417 (5) 0.75 0.4000 49.979992 0.020008 6.157599 0.162401 (6) 0.75 0.5000 20.701695 0.048305 3.392754 0.294746 (7) 0.75 0.5455 14.837955 0.067395 2.652719 0.376972 (8) 0.75 0.7559 4.791288 0.208712 1.000000 1.000000 (9) 0.75 1.0000 2.318729 0.431271 0.431271 2.318729 (10) 0.75 2.0000 1.000000 1.000000 0.071797 13.928203 Table IV. Modal energy parameters for μ equal to 1.0, and Ω variable as shown, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω α 1 α 2 β 1 β 2 (2) 1.00 0.1000 10001.99990 0.100E-03 98.029799 0.010201 (3) 1.00 0.2000 626.998405 0.0011595 23.116741 0.043259 (4) 1.00 0.3000 125.448819 0.0007971 9.364323 0.106788 (5) 1.00 0.4000 41.038132 0.024368 4.676149 0.213851 (6) 1.00 0.4737 21.817178 0.045835 3.023565 0.330735 (7) 1.00 0.5000 17.944272 0.055728 2.618034 0.381966 (8) 1.00 0.7071 5.828427 0.171573 1.000000 1.000000 (9) 1.00 1.0000 2.618034 0.381966 0.381966 2.618034 (10) 1.00 1.000000 1.000000 0.00000
354 J. C. MIRANDA Table V. Normalized frequencies, participation factors, and real mode damping coefficients, for μ equal to 0.25, and Ω variable as shown in Table I, with ξ U = 0.3 and ξ L = 0.03. (7) (8) (1) μ Ω ω 1 / ω 2 / γ 1 γ 2 ξ 1 ξ 2 (2) 0.25 0.1000 0.0999 1.0013 0.0025 0.9975 0.2989 0.0376 (3) 0.25 0.2000 0.1990 1.0056 0.0107 0.9893 0.2953 0.0460 (4) 0.25 0.3000 0.2964 1.0123 0.0263 0.9737 0.2888 0.0562 (5) 0.25 0.4000 0.3909 1.0233 0.0528 0.9472 0.2783 0.0696 (6) 0.25 0.5000 0.4808 1.0399 0.0956 0.9044 0.2622 0.0879 (7) 0.25 0.7826 0.6887 1.1363 0.3565 0.6435 0.1772 0.1772 (8) 0.25 0.8944 0.7435 1.2030 0.5000 0.5000 0.1358 0.2198 (9) 0.25 1.0000 0.7808 1.2808 0.6213 0.3787 0.1033 0.2532 (10) 0.25 1.1547 0.8165 1.4142 0.7500 0.2500 0.0714 0.2862 Table VI. Normalized frequencies, participation factors, and real mode damping coefficients, for μ equal to 0.5, and Ω variable as shown in Table II, with ξ U = 0.3 and ξ L = 0.03. (7) (8) (1) μ Ω ω 1 / ω 2 / γ 1 γ 2 ξ 1 ξ 2 (2) 0.5 0.1000 0.0998 1.0025 0.0051 0.9949 0.2977 0.0452 (3) 0.5 0.2000 0.1980 1.0104 0.0212 0.9788 0.2908 0.0618 (4) 0.5 0.3000 0.2929 1.0243 0.0511 0.9489 0.2784 0.0815 (5) 0.5 0.4000 0.3825 1.0458 0.0989 0.9011 0.2596 0.1059 (6) 0.5 0.5000 0.4644 1.0767 0.1689 0.8311 0.2339 0.1359 (7) 0.5 0.6429 0.5633 1.1414 0.3072 0.6928 0.1873 0.1873 (8) 0.5 0.8165 0.6501 1.2559 0.5000 0.5000 0.1292 0.2496 (9) 0.5 1.0000 0.7071 1.4142 0.6667 0.3333 0.0849 0.2970 (10) 0.5 1.4142 0.7654 1.8478 0.8536 0.1464 0.0434 0.3427 Table VII. Normalized frequencies, participation factors, and real mode damping coefficients, for μ equal to 0.75, and Ω variable as shown in Table III, with ξ U = 0.3 and ξ L = 0.03. (7) (8) (1) μ Ω ω 1 / ω 2 / γ 1 γ 2 ξ 1 ξ 2 (2) 0.75 0.1000 0.0996 1.0038 0.0076 0.9924 0.2966 0.0528 (3) 0.75 0.2000 0.1970 1.0155 0.0314 0.9686 0.2863 0.0774 (4) 0.75 0.3000 0.2895 1.0362 0.0744 0.9256 0.2686 0.1059 (5) 0.75 0.4000 0.3747 1.0675 0.1397 0.8603 0.2433 0.1394 (6) 0.75 0.5000 0.4499 1.1113 0.2276 0.7724 0.2116 0.1775 (7) 0.75 0.5455 0.4802 1.1358 0.2738 0.7262 0.1956 0.1956 (8) 0.75 0.7559 0.5877 1.2863 0.5000 0.5000 0.1254 0.2745 (9) 0.75 1.0000 0.6567 1.5227 0.6987 0.3013 0.0731 0.3329 (10) 0.75 2.0000 0.7321 2.7321 0.9330 0.0670 0.0278 0.3878 Table VIII. Normalized frequencies, participation factors, and real mode damping coefficients, for μ equal to 1.0, and Ω variable as shown in Table IV, with ξ U = 0.3 and ξ L = 0.03. (7) (8) (1) μ Ω ω 1 / ω 2 / γ 1 γ 2 ξ 1 ξ 2 (2) 1.0 0.1000 0.0996 1.0051 0.0101 0.9899 0.2955 0.0603 (3) 1.0 0.2000 0.1960 1.0206 0.0415 0.9585 0.2820 0.0928 (4) 1.0 0.3000 0.2863 1.0516 0.0965 0.9035 02595 0.1295 (5) 1.0 0.4000 0.3675 1.0886 0.1762 0.8238 0.2290 0.1707 (6) 1.0 0.4737 0.4199 1.1280 0.2485 0.7515 0.2030 0.2030 (7) 1.0 0.5000 0.4370 1.1441 0.2764 0.7236 0.1934 0.2146 (8) 1.0 0.7071 0.5412 1.3066 0.5000 0.5000 0.1229 0.2968 (9) 1.0 1.0000 0.6180 1.6180 0.7236 0.2764 0.0647 0.3647 (10) 1.0 0.7071 1.0000 0.0000 0.0212 0.4243
SYSTEM INTRINSIC PARAMETERS OF TMDs 355 mode, and r is a vector of 1s. It is noted that the mode components corresponding to the lower mass have been conveniently normalized to 1. Tables IX XII use a complex modal procedure to calculate the exact frequencies, once again normalized with respect to the structural frequency, and the exact coefficients of modal damping corresponding to the chosen tuning and mass ratio, and for the same damping of the lower and upper portions of the mechanical system as considered in the preceding tables. Figure 2 depicts the approximated normalized frequencies per columns 3 and 4 of Tables V VIII, and the exact normalized frequencies per columns 3 and 4 of Tables IX XII, as a function of the tuning ratios, and for various mass ratios. It may be appreciated that the exact and approximated frequencies, for all mass ratios, and for both modes, agree very well. The largest difference is about 4.7% for the second mode, and for a mass ratio of 0.25. It is noted that in Figure 2(d), the extreme values for tuning at kinetic energy Table IX. Complex mode normalized frequencies and complex modal damping coefficient, for μ equal to 0.25, and Ω variable as shown in Table I, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω ω 11 / ω 22 / ξ 11 ξ 22 (2) 0.25 0.1000 0.0999 1.0008 0.2990 0.0376 (3) 0.25 0.2000 0.1993 1.0033 0.2959 0.0459 (4) 0.25 0.3000 0.2977 1.0076 0.2902 0.0557 (5) 0.25 0.4000 0.3945 1.0140 0.2812 0.0681 (6) 0.25 0.5000 0.4888 1.0228 0.2674 0.0848 (7) 0.25 0.7826 0.7237 1.0815 0.1792 0.1792 (8) 0.25 0.8944 0.7782 1.1493 0.1276 0.2315 (9) 0.25 1.0000 0.8069 1.2394 0.0929 0.2663 (10) 0.25 1.1547 0.8313 1.3890 0.0645 0.2947 Table X. Complex mode normalized frequencies and complex modal damping coefficients, for μ equal to 0.5, and Ω variable as shown in Table II, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω ω 11 / ω 22 / ξ 11 ξ 22 (2) 0.5 0.1 0.0998 1.0016 0.2980 0.0452 (3) 0.5 0.2 0.1987 1.0066 0.2918 0.0616 (4) 0.5 0.3 0.2954 1.0155 0.2808 0.0808 (5) 0.5 0.4 0.3886 1.0292 0.2637 0.1045 (6) 0.5 0.5000 0.4763 1.0497 0.2391 0.1344 (7) 0.5 0.6429 0.5848 1.0994 0.1896 0.1896` (8) 0.5 0.8165 0.6731 1.2131 0.1239 0.2589 (9) 0.5 1.0000 0.7221 1.3849 0.0791 0.3054 (10) 0.5 1.4142 0.7696 1.8377 0.0418 0.3451 Table XI. Complex mode normalized frequencies and complex modal damping coefficients, for μ equal to 0.75, and Ω variable as shown in Table III, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω ω 11 / ω 22 / ξ 11 ξ 22 (2) 0.75 0.1 0.0998 1.0024 0.2970 0.0527 (3) 0.75 0.2 0.1980 1.0105 0.2878 0.0772 (4) 0.75 0.3 0.2930 1.0238 0.2717 0.1054 (5) 0.75 0.4 0.3826 1.0455 0.2476 0.1390 (6) 0.75 0.5000 0.4635 1.0788 0.2152 0.1787 (7) 0.75 0.5455 0.4963 1.0992 0.1983 0.1983 (8) 0.75 0.7559 0.6059 1.2475 0.1210 0.2834 (9) 0.75 1.0000 0.6671 1.4991 0.0691 0.3395 (10) 0.75 2.0000 0.7328 2.7293 0.0277 0.3883
356 J. C. MIRANDA Table XII. Complex mode normalized frequencies and complex modal damping coefficients, for μ equal to 1.0, and Ω variable as shown in Table IV, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω ω 11 / ω 22 / ξ 11 ξ 22 (2) 1.00 0.1000 0.0997 1.0033 0.2960 0.0603 (3) 1.00 0.2000 0.1973 1.0135 0.2839 0.0927 (4) 1.00 0.3000 0.2906 1.0322 0.2629 0.1294 (5) 1.00 0.4000 0.3765 1.0624 0.2329 0.1716 (6) 1.00 0.4737 0.4327 1.0947 0.2057 0.2057 (7) 1.00 0.5000 0.4509 1.1088 0.1954 0.2182 (8) 1.00 0.7071 0.5568 1.2699 0.1190 0.3056 (9) 1.00 1.0000 0.6258 1.5979 0.0615 0.3702 balance are not reported. Analytical expression for the approximated normalized frequencies are given in Miranda [18]. Later, analytical expressions for the product of the frequencies will be obtained. Figure 3 depicts the approximated modal damping per columns 7 and 8 of Tables V VIII, and the exact modal damping per columns 5 and 6 of Tables IX XII, as a function of the tuning ratios, and for various mass ratios. It is noted that in Figure 3(d), the extreme values for tuning at kinetic energy balance are not reported. It may be appreciated that the approximated exact damping, for all mass ratios, and for both modes, agree well. The largest difference is about 5.0% for the second mode, and for a mass ratio of 0.25. Later, analytical expressions for the approximated exact modal damping will be obtained, as well as an expression furnishing the product of the real mode damping coefficients. Figure 4 depicts the mode participation factors per columns 5 and 6 of Tables V VIII, as a function of tuning, and for the selected values of the mass ratio. These expressions were derived using the expressions provided by Miranda [4] based on modal energy parameters. Later, analytical expressions for the product of the participation factors will be obtained. 3.1. Discussion of modal damping The dynamic behavior of the 2-degree-of-freedom systems composed of a TMD affixed to a structure is obviously fundamentally different from that of the original single-degree-of-freedom structure. Modal damping aside and as a function of the spectral distribution of seismic energy, the response of the mechanical system may be, but not necessarily so, reduced by the addition of an undamped upper mass. Thus, it will be assumed in this paper that the effect of enhanced damping due to the use of TMDs will, at a minimum, offset any potentially negative effects that could arise from the modified dynamics of the mechanical systems. Other than this consideration, this paper will focus on the damping generating properties of TMDs. It may be shown, Miranda [4], that the coefficients of damping for the real modes can be written as a linear combination of the damping furnished separately by the TMD and the structure, per the following equation: ξ j ¼ A j ξ U þ B j ξ L (9) The factors A j and B j, corresponding to the jth mode, may be written as functions of the modal energy coefficients as follows: β 1 A 1 ¼ p Ω ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi (10) 1 þ α 1 1 þ β 1 1 B 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffi (11) 1 þ α 1 1 þ β 1 β 2 A 2 ¼ p Ω ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi (12) 1 þ α 2 1 þ β 2 1 B 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffi (13) 1 þ α 2 1 þ β 2
SYSTEM INTRINSIC PARAMETERS OF TMDs 357 (a) (b) (c) (d) Figure 2. Approximated versus exact normalized frequencies.
358 J. C. MIRANDA (a) (b) (c) (d) Figure 3. Approximated versus exact modal damping.
SYSTEM INTRINSIC PARAMETERS OF TMDs 359 Figure 4. Participation factors.
360 J. C. MIRANDA with Ω 2 ¼ 2 þ β 1 þ β 2 2 þ α 1 þ α 2 (14) Equation (9) indicates the fundamental reason for using TMDs: In case of structures with low damping ξ L, significant modal damping can be induced by implementing TMDs delivering high values of the coefficient ξ U. Alternatively, an analytical form of Equation (9), using real mode properties, can be obtained through the equality of the f 0, f 1, and f 3 factors, per equations (A.1.a), (A.1.c), and (A.1.d) with (A.2.a), (A.2.c), and (A2.d), as follows: ω 2 ð1 þ μþω ω 1 A 1 ¼ ω 2 2 2 (15) ω 1 B 1 ¼ Ω ω 2 ω 1 2 2 (16) ω 1 ω 2 A 2 ¼ ð1 þ μþω ω 2 ω 1 2 2 (17) ω 1 ω 2 ω 2 ω B 2 ¼ L Ω ω 1 2 2 (18) ω 1 ω 2 It is seen that the following expression applies: Ω 2 ¼ B 1B 2 (19) A 1 A 2 Further, for the complex modes, a corresponding analytical expression can be obtained for the exact modal coefficients of damping through the equality of the f 0, f 1, and f 3 factors, per equations (A.1.a), (A.1.c), and (A.1.d), with (A.4.a), (A.4.c), and (A.4.d), as follows: ξ jj ¼ A jj ξ U þ B jj ξ L (20) It can be therefore written that ω 22 ð1 þ μþω ω 11 A 11 ¼ ω 22 2 2 (21) ω 11 B 11 ¼ Ω ω 22 ω 11 2 2 (22) ω 11 ω 22 A 22 ¼ ð1 þ μþω ω 22 ω 11 2 2 (23) ω 11 ω 22 ω 22 ω B 22 ¼ L Ω ω 11 2 2 (24) ω 11 ω 22 The correctness of the three formats for modal damping equations furnished earlier may be verified using the data provided in Tables I XII, for both the real and complex modes. In Figure 3, it may be seen that for the first mode, damping decreases with increasing tuning. Conversely, it may be seen that the damping for the second mode increases with increased tuning. Further,
SYSTEM INTRINSIC PARAMETERS OF TMDs 361 in Figure 4, it may be seen that for the first mode, the participation factor increases with increased tuning, whereas for the second mode, the participation factor decreases with increased tuning. It is seen thus that as represented by the listed values of damping, tuning, and mass ratios, the allocation of damping to the modes, per Equation (9) or (20), is discordant with respect to the importance of the participation factors. In other words, there is an inefficient modal damping allocation with respect to the importance of the modes providing the seismic response. Tuning a TMD that excessively overdamps or underdamps the modes of a mechanical system in an opposite sense to the importance of the participation factors might result in systems that do not experience seismic response reduction or that require inordinate amounts of damping in order to experience a desired level of response reduction. Economically and technically, this is not a rational situation. An improvement to this condition occurs when tuning is achieved at a value that results in equal damping for both modes, as under such circumstance, both participation factors will be equally damped, and this provides some measure of control to the overdamping or underdamping of the modes. This equality of energy dissipation potential explains the success that tuning for equal mode damping has to decrease the seismic response for single-degree-of-freedom or multi-degree-of-freedom structures as demonstrated by Villaverde [1] and by Sadek et al. [3]. Further in-depth discussion of mechanical systems tuned in such manner can be found in Miranda [6]. There appears to exist a lingering perception that TMDs require tuning at, or near, resonance in order to be effective. Such idea is flawed, as can be inferred from the discussion earlier. However, as shown by Miranda [4], for systems with significant exchange of energy between the upper and lower parts, as the mass ratio decreases, say to a range of 0.01, then tuning in general approaches values closer to 1, resulting in quasi-resonant conditions. Likewise, for such small mass ratios, both modal damping coefficients are close to each other, and with a magnitude roughly equal to the average of the damping furnished independently by the TMD and the structure, as demonstrated by Villaverde [1]. Using the method proposed by Miranda [4], the following expression yielding the product of the real mode coefficients of damping can be obtained: ξ 1 ξ 2 ¼ ξ2 U þ Ωξ Uξ L ðβ 1 þ β 2 ÞþΩ 2 ξ 2 L (25) Ωð2 þ β 1 þ β 2 Þ Given defined mechanical properties, the right term of Equation (25) is a constant. This mean, for instance, that small fundamental mode damping requires correspondingly high second mode damping in order to maintain the equality, and vice versa. The discussion regarding discrepancy between modal damping allocation and modal participation factors is retaken next. 3.2. Discussion on modal damping allocation It was observed earlier that while high modal damping can be obtained per Equation (9) or (20), the manner in which such damping is effectively conveyed to the modes requires proper calibration of the system parameters for due efficiency. It can be argued that for a rational assignment, the ratio of the modal damping should be equal to the ratio of the participation factors, that is, the modes are to be damped in proportion to their corresponding importance for the response, as reflected by the participation factors. Using the approximated method proposed by Miranda [4], it can be readily demonstrated that this approach leads to the following: ξ 2 ¼ γ 2 ¼ β ξ 1 γ 1 (26) 1 It is understood that while this method of tuning involves parameters that are exact within the consideration of real modes, it becomes approximated when dealing with complex modes. Additional calculations are prepared to illustrate the results derived from Equation (26). This time, it will be considered that the mass ratio μ varies from 0.1 to 1.0, with increments of 0.1, with a TMD damping ξ U equal to 0.3 and to 0.6, and with a structural damping ξ L equal to 0.03. Tables XIII and XIV present the corresponding modal energy parameters for the chosen mass ratio and damping, whereas Tables XV and XVI contain the real normalized frequencies and damping coefficients, all calculated with the constraint
362 J. C. MIRANDA Table XIII. Energy parameters for varying μ as shown, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω α 1 α 2 β 1 β 2 (2) 0.1 0.929214 2.214999 0.451467 1.186126 0.843081 (3) 0.2 0.869627 3.034512 0.329542 1.267804 0.788766 (4) 0.3 0.818667 3.830100 0.261090 1.331210 0.751196 (5) 0.4 0.774525 4.630111 0.215978 1.384732 0.722161 (6) 0.5 0.735860 5.443809 0.183695 1.431792 0.698426 (7) 0.6 0.701670 6.274864 0.159366 1.474158 0.678354 (8) 0.7 0.671184 7.124913 0.140353 1.512919 0.660974 (9) 0.8 0.643803 7.994700 0.125083 1.548807 0.645658 (10) 0.9 0.619054 8.884297 0.112558 1.582310 0.631988 (11) 1.0 0.596556 9.793650 0.102107 1.613803 0.619654 Table XIV. Energy parameters for varying μ as shown, with ξ U = 0.6 and ξ L = 0.03. (1) μ Ω α 1 α 2 β 1 β 2 (2) 0.1 0.930242 2.198464 0.454863 1.177488 0.849265 (3) 0.2 0.871348 3.004046 0.332884 1.255650 0.796401 (4) 0.3 0.820871 3.784934 0.264205 1.316498 0.759591 (5) 0.4 0.777058 4.569677 0.218834 1.368072 0.730956 (6) 0.5 0.738616 5.367480 0.186307 1.413558 0.707434 (7) 0.6 0.704573 6.182103 0.161757 1.454627 0.687461 (8) 0.7 0.674183 7.015200 0.142548 1.492291 0.670110 (9) 0.8 0.646860 7.867459 0.127106 1.527221 0.654784 (10) 0.9 0.622141 8.739088 0.114428 1.559891 0.641070 (11) 1.0 0.599654 9.630041 0.103842 1.590652 0.628673 Table XV. Normalized frequencies, participation factors, and real modal damping coefficients with μ, and Ω per Table XIII, with ξ U = 0.3 and ξ L = 0.03. (7) (8) (1) μ Ω ω 1 / ω 2 / γ 1 γ 2 ξ 1 ξ 2 (2) 0.1 0.929214 0.8246 1.1269 0.4574 0.5426 0.1558 0.1848 (3) 0.2 0.869627 0.7497 1.1599 0.4410 0.5590 0.1545 0.1959 (4) 0.3 0.818667 0.6947 1.1784 0.4290 0.5710 0.1543 0.2054 (5) 0.4 0.774525 0.6508 1.1901 0.4193 0.5807 0.1546 0.2140 (6) 0.5 0.735860 0.6143 1.1979 0.4112 0.5888 0.1550 0.2220 (7) 0.6 0.701670 0.5832 1.2032 0.4042 0.5958 0.1556 0.2294 (8) 0.7 0.671184 0.5561 1.2069 0.3979 0.6021 0.1563 0.2364 (9) 0.8 0.643803 0.5323 1.2094 0.3923 0.6077 0.1570 0.2432 (10) 0.9 0.619054 0.5111 1.2111 0.3873 0.6127 0.1577 0.2496 (11) 1.0 0.596556 0.4921 1.2123 0.3826 1.6174 0.1584 0.2557 Table XVI. Normalized frequencies, participation factors, and real modal damping coefficients with μ, and Ω per Table XIV, with ξ U = 0.6 and ξ L = 0.03. (7) (8) (1) μ Ω ω 1 / ω 2 / γ 1 γ 2 ξ 1 ξ 2 (2) 0.1 0.930242 0.8251 1.1274 0.4592 0.5408 0.2992 0.3522 (3) 0.2 0.871348 0.7506 1.1609 0.4433 0.5567 0.2977 0.3738 (4) 0.3 0.820871 0.6958 1.1798 0.4317 0.5683 0.2980 0.3924 (5) 0.4 0.777058 0.6521 1.1917 0.4223 0.5777 0.2991 0.4092 (6) 0.5 0.738616 0.6157 1.1997 0.4143 0.5857 0.3006 0.4249 (7) 0.6 0.704573 0.5846 1.2052 0.4074 0.5926 0.3022 0.4395 (8) 0.7 0.674183 0.5576 1.2090 0.4012 0.5988 0.3039 0.4534 (9) 0.8 0.646860 0.5339 1.2117 0.3957 0.6043 0.3056 0.4667 (10) 0.9 0.622141 0.5127 1.2135 0.3906 0.6094 0.3073 0.4794 (11) 1.0 0.599654 0.4937 1.2147 0.3860 0.6140 0.3090 0.4915
SYSTEM INTRINSIC PARAMETERS OF TMDs 363 per Equation (26). Tables XVII and XVIII use a complex modal procedure to calculate the exact normalized frequencies and modal damping corresponding to the chosen mass ratios and constrained tuning. Using a coefficient of damping ξ U equal to 0.3, Figure 5 indicates that except for mass ratios smaller than approximately 0.15, the agreement between the constrained approximated normalized frequencies and the exact ones is very good for the first mode, and slightly less good for the second mode having a maximum difference of 9%. Likewise, for the same damping coefficient and tuning, Figure 6 indicates that except for mass ratios smaller than approximately 0.18, the agreement between the constrained real modal damping and the exact one is very good for the first mode, and good for the second mode, with a maximum difference of 9%. It is also observed that while the damping for the first mode remains fairly constant, the damping for the second mode is higher and increases with the mass ratio. Table XVII. Complex mode normalized frequencies, and complex modal damping coefficients, with μ, and Ω per Table XIII, with ξ U = 0.3 and ξ L = 0.03. (1) μ Ω ω 11 / ω 22 / ξ 11 ξ 22 (2) 0.1 0.929214 0.8979 1.0349 0.1399 0.2039 (3) 0.2 0.869627 0.7923 1.0976 0.1492 0.2048 (4) 0.3 0.818667 0.7275 1.1253 0.1508 0.2129 (5) 0.4 0.774525 0.6784 1.1418 0.1518 0.2210 (6) 0.5 0.735860 0.6385 1.1525 0.1527 0.2287 (7) 0.6 0.701670 0.6050 1.1598 0.1537 0.2361 (8) 0.7 0.671184 0.5761 1.1650 0.1546 0.2431 (9) 0.8 0.643803 0.5509 1.1686 0.1554 0.2499 (10) 0.9 0.619054 0.5285 1.1713 0.1563 0.2563 (11) 1.0 0.596556 0.5085 1.1731 0.1572 0.2626 Table XVIII. Complex mode normalized frequencies, and complex modal damping coefficients, with μ, and Ω per Table XIV, with ξ U = 0.6 and ξ L = 0.03. (1) μ Ω ω 11 / ω 22 / ξ 11 ξ 22 (2) 0.1 0.930242 0.9490 0.9802 0.0720 0.5872 (3) 0.2 0.871348 0.9002 0.9680 0.1133 0.5738 (4) 0.3 0.820871 0.8518 0.9637 0.1525 0.5607 (5) 0.4 0.777058 0.8035 0.9671 0.1873 0.5503 (6) 0.5 0.738616 0.7565 0.9763 0.2148 0.5452 (7) 0.6 0.704573 0.7136 0.9874 0.2346 0.5459 (8) 0.7 0.674183 0.6757 0.9977 0.2484 0.5511 (9) 0.8 0.646860 0.6428 1.0063 0.2585 0.5589 (10) 0.9 0.622141 0.6140 1.0133 0.2662 0.5682 (11) 1.0 0.599654 0.5886 1.0188 0.2725 0.5783 Figure 5. Approximated versus exact constrained normalized frequencies.
364 J. C. MIRANDA Figure 6. Approximated versus exact constrained modal damping. For a coefficient of damping ξ U equal to 0.6, a poor agreement between the constrained normalized and exact frequencies is obtained, as indicated in Figure 7. This is indicative of limitations in the application of Equation (26) for systems incorporating highly damped TMDs, because the exact frequencies cannot reliably be predicted using the approximate ones. Likewise, for a value of ξ U equal to 0.6, Figure 8 indicates that poor agreement between the constrained approximated and exact modal damping is achieved for such high value of the TMD coefficient of damping. This, again, is indicative of limitations in the application of Equation (26) for systems incorporating highly damped TMDs, because the exact modal damping cannot reliably be predicted using the approximated ones. Further to Equation (26), an efficiency factor, e, may be defined as follows: e ¼ ξ 2 ξ 1 γ 2 γ (27) 1 As an illustration, Equation (27) is applied to data shown in Table XIX, indicating that for tuning per Equation (26), the adopted efficiency factor e is 0, whereas for other tuning, there is either a surplus Figure 7. Approximate versus exact constrained normalized frequencies. Figure 8. Approximate versus exact constrained modal damping.
SYSTEM INTRINSIC PARAMETERS OF TMDs 365 Table XIX. Selected normalized frequencies, participation factors, and real mode damping coefficients, for μ equal to 0.5, and Ω variable, with ξ U = 0.3 and ξ L = 0.03. (7) (8) (9) (1) μ Ω ω 1 / ω 2 / γ 1 γ 2 ξ 1 ξ 2 e (2) 0.5 0.6429 0.5633 1.1414 0.3072 0.6928 0.1873 0.1873 0.3393 (3) 0.5 0.7359 0.6143 1.1979 0.4112 0.5888 0.1550 0.2220 0.0000 (4) 0.5 0.8165 0.6501 1.2559 0.500 0.5000 0.1292 0.2496 0.2408 or a deficit of modal damping in relation to the corresponding participation factors. With the use of Equation (26), the damping induced in each real mode will be strictly proportional to the corresponding participation factor. Technically and economically, it seems that this is an attractive rational alternative. 3.3. Discussion of system frequencies and participation factors Per equality of the three f 0 factors in Appendix, it may be written that ω 1 ω2 ¼ ω 11 ω22 ¼ Ω (28) Given a defined set of mechanical properties, the right term of Equation (28) is fixed. Hence, low fundamental frequencies will require correspondingly high second mode frequencies in order to maintain the equality, and vice versa. The frequencies of the system will bracket the original frequency from above and below, conforming to the following expression: ω 1 < ω 11 < 1 < ω 22 < ω 2 Using the method proposed by Miranda [4], it may be shown that the product of the participation factors yields (29) or alternatively ðγ 1 Þðγ 2 α 1 Þ ¼ Ω 2 ð1 þ α 1 Þ 2 ¼ α 2 Ω 2 ð1 þ α 2 Þ 2 (30:a) β 1 ðγ 1 Þðγ 2 Þ ¼ ð1 þ β 1 Þ 2 ¼ β 2 ð1 þ β 2 Þ 2 (30:b) Given sets of defined structural properties, the second terms of Equations (30) will be fixed. Then as a manner of illustration, a low fundamental mode participation factor will necessitate a high second mode participation factor in order to maintain the equalities in the last two equations, and vice versa. 3.4. Some numerical results for seismic excitation Assume a 300-t mass having a circular frequency of 5.1302 rad/s and a damping coefficient of 0.0245 is divided into a lower mass of 200 t having a circular frequency of 2π rad/s and a damping coefficient ξ L of 0.03, plus an upper mass of 100 t with a damping coefficient of ξ U of 0.3, and a circular frequency to be tuned so as to reduce the displacement of the lower mass during an earthquake represented by the response spectrum depicted in Figure 9. After generating a spectrum-compatible time history, the responses of the 2-degree-of-freedom systems shown in Table XX have been obtained. In this table, x U and x L represent the displacements of the upper and lower masses, respectively. The values for Ω that are considered correspond in increasing order to tuning for equal modal damping, tuning per Equation (26), tuning for perfect balance of modal strain energy, tuning at resonance, and tuning for perfect balance of modal kinetic energy. In Figure 10, the lower curve represents the displacements of the lower mass when excited by the ground motion implied by Figure 9. It may be seen that tuning using parameters per Equation (26) correctly leads to a minimum of the response of about 6.11 cm. It is noted that for this particular example, tuning for exact modal balance of strain energy results in a 1% lower displacement of about
366 J. C. MIRANDA Figure 9. Response spectrum. Table XX. Response 2-degree-of-freedom systems for tuning and mass ratio as noted, with ξ U = 0.3 and ξ L = 0.03. (1) (2) (3) (4) (1) μ Ω x U (cm) x L (cm) (2) 0.5 0.6429 12.32 6.24 (3) 0.5 0.7359 12.80 6.11 (4) 0.5 0.8165 13.46 6.03 (5) 0.5 1.0000 15.31 8.77 (6) 0.5 1.4142 16.75 12.39 Figure 10. Maximum response upper and lower masses. 6.03 cm, which for practical purposes would be the same response. As seen, for the range under consideration, the upper mass displacement monotonically increases as tuning increases. As tuning is increased, the lower mass displacement increases as shown both in Table XX and Figure 10. For the case of the large single mass, the maximum displacement is calculated at 17.06 cm, and therefore, the use of the TMD with the properties earlier may reduce that displacement by about 65%. 4. CONCLUSIONS This paper discusses the properties that render TMDs efficient for seismic response reduction. Focusing on damping, it is demonstrated that modal damping can be expressed as a linear combination of the damping furnished separately by the TMD itself and by the structure. One numerical expression of such modal damping is provided, along with two analytical forms based on real mode, and complex mode, methodologies. It is shown via examples that the exact frequencies and damping, for sufficiently low TMD damping, may be forecasted using real mode methodologies. It is shown also via examples, that the
SYSTEM INTRINSIC PARAMETERS OF TMDs 367 modal damping thus induced is discordant in that it is furnished in the wrong proportions to the participation factors. Under such circumstances, the modes are overdamped or underdamped, a condition that reduces the efficiency of the TMDs, and that could lead to no response reduction or to require inordinate amounts of damping in order to achieve some response benefit. This observation leads to the proposal of using modal damping in the same proportion as the participation factors. Providing damping along this modality is believed to be the most rational use of the damping provided by the TMD to the mechanical system. Additionally, several useful relationships between frequencies, tuning, and participation factors are demonstrated. Finally, the efficiency of the new tuning proposed in this paper is shown via limited calculations for systems under ground motion excitation. APPENDIX: CHARACTERISTIC EQUATION IN TERMS OF STRUCTURAL AND MODAL PROPERTIES The factor f i for characteristic Equation (7) written in terms of structural properties are as follows: f 3 ¼ 21þ ð μþξ U ω U þ 2ξ L (A:1:a) f 2 ¼ ð1 þ μþω 2 U þ ω2 L þ 4ξ Uξ L ω U (A:1:b) f 1 ¼ 2ξ U ω U ω 2 L þ 2ξ L ω 2 U f 0 ¼ ω 2 U ω2 L Alternatively, the factors f i may be written in terms of modal properties as follows: f 3 ¼ 2ξ 1 ω 1 þ 2ξ 2 ω 2 C2 C f 2 ¼ ω 2 1 þ ω2 2 þ 4ξ 1ξ 2 ω 1 ω 2 M 1 M 2 f 1 ¼ 2ξ 1 ω 1 ω 2 2 þ 2ξ 2ω 2 ω 2 1 (A:1:c) (A:1:d) (A:2:a) (A:2:b) (A:2:c) where f 0 ¼ ω 2 1 ω2 2 ξ 1 ¼ ω 2 1 ¼ K 1 M 1 ω 2 2 ¼ K 2 M 2 C 1 p 2 ffiffiffiffiffiffiffiffiffiffiffiffi K 1 M 1 (A:2:d) (A:3:a) (A:3:b) (A:3:c) C 2 ξ 2 ¼ p 2 ffiffiffiffiffiffiffiffiffiffiffiffi (A:3:d) K 2 M 2 Additionally, the factors f i may be written in terms of complex modal properties as follows: f 3 ¼ 2ξ 11 ω 11 þ 2ξ 22 ω 22 (A:4:a) f 2 ¼ ω 2 11 þ ω2 22 þ 4ξ 11ξ 22 ω 11 ω 22 f 1 ¼ 2ξ 11 ω 11 ω 2 22 þ 2ξ 22ω 22 ω 2 11 f 0 ¼ ω 2 11 ω2 22 (A:4:b) (A:4:c) (A:4:d)
368 J. C. MIRANDA where ζ jj and ω jj are the exact damping and circular frequency corresponding to the jth complex mode of the mechanical system, which are written as follows: ξ 11 ¼ ξ 22 ¼ ω 2 11 ¼ K 11 M 11 ω 2 22 ¼ K 22 M 22 C 11 p 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 11 M 11 C 22 p 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 22 M 22 (A:5:a) (A:5:b) (A:c) (A:5:d) ACKNOWLEDGEMENTS The findings, procedures, and opinions expressed in this paper are the sole responsibility of the author, and do not necessarily represent the practice of his employer, CH2M HILL. REFERENCES 1. Villaverde R. Reduction in seismic response with heavily-damped vibration absorbers. Earthquake Engineering and Structural Dynamics 1985; 13:33 42. 2. Feng M, Mita A. Vibration control of tall buildings using mega subconfiguration. Journal of Engineering Mechanics, ASCE 1995; 10:1082 1088. 3. Sadek F, Mohraz B, Taylor A, Chung R. A method of estimating the parameters of tuned mass dampers for seismic applications. Earthquake Engineering and Structural Dynamics 1997; 26:617 635. 4. Miranda JC. On tuned mass dampers for reducing the seismic response of structures. Earthquake Engineering and Structural Dynamics 2005; 34:847 865. 5. Moutinho C. An alternative methodology for designing tuned mass dampers to reduce seismic vibrations in building structures. Earthquake Engineering and Structural Dynamics 2012; 41:2059 2073. 6. Miranda JC. A method for tuning tuned mass dampers for seismic application. Earthquake Engineering and Structural Dynamics 2013; 42:1103 1110. 7. Hoang N, Fujino Y, Warnitchai P. Optimal tuned mass damper for seismic applications and practical design formulas. Engineering Structures 2007; 30:707 715. 8. De Angelis M, Perno S, Reggio A. Dynamic response and optimal design of structures with large mass ratio TMD. Earthquake Engineering and Structural Dynamics 2012; 41:41 60. 9. Chen M, Chase J, Mander CA. Semi-active tuned mass damper building systems: design. Earthquake Engineering and Structural Dynamics 2010; 39:119 139. 10. Chen M, Chase J, Mander CA. Semi-active tuned mass damper building systems: application. Earthquake Engineering and Structural Dynamics 2010; 39:69 89. 11. Salvi J, Rizzi E, Rustighi E, Ferguson N. 2013. Analysis and optimization of tuned mass dampers for impulsive excitation. 11 International Conference RASD 2013. 12. Domizio M, Ambrosini D, Curadelli O. 2013. Evaluation of the performance of tuned mass dampers for near fault ground motions. Argentinian Association of Computational Mechanics, Vol. XXXII, Pages 2115 2129. (In Spanish). 13. Salvi J, Rizzi E, Gavazzeni M. 2014. Analysis on the optimum performance of tuned mass damper devices in the context of earthquake engineering. Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014. 14. Salvi J, Rizzi E. Optimum tuning mass damper for frame structures under earthquake excitation. Structural Control and Health Monitoring 2014. doi:10.1002/stc.1710. 15. Hartog JP. Mechanical Vibrations (4th edn). McGraw-Hill: New York, 1956. (Reprinted by Dover, New York. 1985). 16. Warburton GB. Optimum absorber parameters for various combinations of response and excitation parameters. Earthquake Engineering and Structural Dynamics 1982; 10:381 401. 17. Hurty WC, Rubinstein MF. Dynamics of Structures. Prentice Hall: Englewood Cliff, New Jersey, 1964. 18. Miranda JC. System intrinsic, damping maximized, tuned mass dampers for seismic applications. Structural Control and Health Monitoring 2012; 19:405 416.