Lever-Type Tuned Mass Damper for Alleviating Dynamic Responses

This study considers the structural vibration control by a lever-type tuned mass damper (LTMD). The LTMD has a constraint condition to restrict the motion at both ends of the lever. The LTMD controls the dynamic responses at two locations combining the tuned mass damper (TMD) and the constraint condition. The parameters of the LTMD are ﬁrstly estimated from the TMD parameters and should be modiﬁed by them to obtain from numerical results. The eﬀectiveness of the LTMD is illustrated in two numerical experiments, and the sensitivity of the parameters is numerically investigated. It is shown that the LTMD leads to the remarkable displacement reduction and exhibits more deﬁnite control than the TMD system because the LTMD controls the vibration responses at two DOFs. More displacement responses are reduced when the installation locations of the LTMD coincide with the nodes to represent the largest modes’ values at the ﬁrst and second modes. The application of the LTMD at the dynamic system of a few degrees of freedom (DOFs) is more eﬀective than the system of many DOFs.


Introduction
Many countries specify the seismic-resistant design to protect the building structures and the residents inside due to earthquakes. Bracing and shear wall must be seismicresistant members to reduce the structural responses by improving the lateral stiffness of a structure. e existing methods to improve the seismic performance of structures consist of controlling the plastic hinge occurrence, increasing the deformation capacity and dissipated energy, strengthening or changing the structural system, and enhancing the lateral stiffness. e dynamic control systems can be considered for more definite vibration control. e utilization of a seismic control system to dissipate earthquake energy has been raised to reduce loss of lives and property caused by seismic disasters. e dynamic control systems are divided into three different systems of passive, active, and hybrid controls.
A tuned mass damper (TMD) is a kind of passive control system installed on a structure for reducing the dynamic responses. e frequency of the TMD is tuned to the structural frequency, and the energy is dissipated. e Citigroup Center in New York City, Chiba Port Tower in Japan, John Hancock Tower in Boston, Canadian National Tower in Toronto, Crystal Tower in Japan, Taipei 101 in Taiwan, etc., are representative examples to install TMDs. e design theory for the TMD was initiated by Ormondroyd and Den Hartog [1] in 1928. Various theories have been developed for the designs of undamped and damped TMD installed on the undamped and damped single-degreeof-freedom (SDOF) system subjected to harmonic excitation or seismic excitation.
Tsai and Lin [2] proposed the optimum tuning frequency and damping ratio of the TMD through a sequence of curvefitting schemes. Abdulsalam et al. [3] suggested the optimum frequency ratio and damping ratio of the TMD installed on a structure subjected to an earthquake loading. Den Hartog [4] did not consider the damping effect of the primary structure. Abubakar and Farid [5] presented the optimum design parameters for the TMD considering the damping of the primary structure. Okhovat et al. [6] performed a parametric study to evaluate the e ectiveness for the TMD at Tehran Tower through the nite element analysis. Murudi and Mane [7] investigated the seismic e ectiveness of TMD and found that the TMD is not a ected by the intensity of ground motion. Warburton and Ayorinde [8] studied the e ect on the optimum parameter conditions of light damping in the primary system. Farghaly and Ahmed [9] discussed the design procedure and the applications through a case study of a symmetrical moment resistance frame twenty-story three-dimensional model. Nigdeli and Bekdas [10] investigated the control e ect depending on the location of a TMD on a seismic structure for an e ective response reduction. Bakre and Jangid [11] derived the optimum parameters of TMD installed on a viscously damped SDOF system for various combinations of excitation and response parameters. e strategies to improve seismic performance may be established by the structural type and assessment results. Stoica [12] provided a seismic retro tting method to consolidate conventional methods and seismic device such as TMD. Brendike and Petryna [13] studied the TMD to control the dynamic responses as a seismic retro t device of RC frame structures. Suzuki et al. [14] developed a seismic control device to increase damping of an old bridge for seismic retro t. Nawrotzki et al. [15] introduced the e ectiveness of the tuned mass control systems for the seismic retro tting of existing structures.
is study considers the e ectiveness of LTMD installed between the two nodes in the structure. e LTMD controls the structural responses and is designed using a constraint condition of the lever responses as well as the optimum parameters of the TMD. is work performs the numerical study according to the design parameters of the LTMD and compares the seismic e ect by the LTMD and TMD in the numerical experiment. It is shown that the LTMD is more e ective in controlling the dynamic responses than the TMD. More displacement responses are reduced when the installation locations of the LTMD coincide with the nodes to represent the largest modes values at the rst and second modes. It is shown that the application of the LTMD at the dynamic system of a few DOFs is more e ective than the system of many DOFs.

Formulation
2.1. TMD Design Parameters. A primary structure described by n DOFs can be idealized as a SDOF structure. Figure 1 represents a SDOF system consisting of a primary structure and a TMD. Many researchers provided the optimum parameter values of the TMD for reducing the responses of the undamped or damped system subjected to harmonic forces or earthquake load. e dynamic equation of motion for the systems can be written by where the subscripts "p" and "T" indicate the primary structure and the TMD, respectively, m, c, and k denote the mass, damping, and sti ness, respectively, and u and F are the displacement response and external force, respectively. e optimum design parameters for the TMD take di erent forms depending on the types of external forces and the presence of the damping in the primary structure. Considering the harmonic forces and the earthquake load as the external forces, F p F 0 e iΩt and F T 0 and F p − m p € u g and F T − m T € u g , respectively. F 0 and € u g represent the force magnitude and the acceleration of earthquake, respectively.
Tables 1 and 2 denote the optimum parameters suggested by various researchers according to the undamped and damped primary structures, respectively. In the tables, μ m T /m p : mass ratio ξ p c p /2ω p m p : damping ratio of the primary structure ξ T c T /2ω T m T : damping ratio of the TMD ω p k p /m p : natural frequency of the primary structure ω T k T /m T : natural frequency of the TMD f ω T /ω p : natural frequency ratio f OPT : optimal frequency ratio ξ OPT T : optimal damping ratio of the TMD It is observed that the optimal parameters in Tables 1 and 2 are deeply a ected by the mass ratio between the primary structure and TMD and the damping ratio of the primary structure. However, the numerical values of the parameters are very close despite the di erent mathematical forms. From the parameters of the TMD, the LTMD parameters can be designed, and their e ectiveness is investigated.

LTMD.
is section considers the parameter design of a LTMD based on the concept of the TMD. e LTMD shown in Figure 2 is installed between the adjacent two nodes in a structure; it is designed by modifying the TMD design parameters and controls the dynamic responses at the two nodes unlike the TMD. e LTMD consists of the massless lever, the masses, springs, and dampers at both ends, and the  Advances in Civil Engineering hinge to restrict the responses at both ends. e system is subjected to a constraint to restrict the interstory drift and provides the control forces at both ends of the lever. e control forces indicate the constraint forces required for satisfying the constraint condition. e dynamic equation for the LTMD can be written by where the subscripts "T1" and "T2" denote the DOFs at the both ends of the lever, respectively, and the corresponding displacement response and external force are represented by u and F. e system constrained by one constraint condition becomes a SDOF system. e constraint condition from the relationship of l a θ u T1 and l b θ u T2 can be written by where α l b /l a and θ denotes the rotational angle at the hinge of the lever. e dynamic equation of motion for the constrained system was proposed by Udwadia and Kalaba [18] in 1992. e equation was derived by minimizing the Gaussian function as a function by the di erence between the constrained and unconstrained accelerations. e dynamic equation for the LTMD system can be expressed by where € u c and € u a denote the acceleration vector for the constrained and unconstrained dynamic system, respectively, and the matrix A and the vector b represent the coe cients in di erentiating equation (3) twice with respect to the time as Substituting equations (2a), (2b), and (5) into equation (4), utilizing the linear algebra, and arranging the result, the dynamic equation at the upper DOF of the lever yields where . e coe cients β, c, and η denote the sti ness ratio, damping ratio, and mass ratio at the one end with respect to the sti ness, damping, and mass at the other end of the lever, respectively.

Control of a ree-Story Building Structure.
Consider the design of the LTMD installed between the second and third oors in a three-story building structure, as shown in Figure 4, and its dynamic control. e TMD is installed on the oor of the structure to exhibit the largest mode value in the rst mode. If we add another TMD for more displacement control, it should be installed at the location to exhibit the highest mode value in the second mode. Utilizing the concept of MTMD (multi-TMD), the LTMD is located at the third and second oor corresponding to the highest mode values in the rst and second mode, respectively. e parameters interdependently a ect the dynamic responses and control. e control by the LTMD is numerically evaluated by the design parameters and compared with the dynamic control by the TMD. e mechanical properties of the primary structure are assumed as m 1 m 2 m 3 10 kg, c 1 c 2 c 3 2 N·sec/m, and k 1 k 2 k 3 1, 000 N/m. e primary structure is transformed to a SDOF system using the rst natural frequency ω 1 and the corresponding mode shape φ 1 : e modal mass can be calculated by Utilizing the above modal and the optimal parameters presented by Den Hartog and various researchers in Table 1, the optimum parameters of the TMD, f OPT and ξ OPT T , are calculated using the prescribed mass ratio μ.
e TMD parameters are designed selecting the mass ratios of 0.02 and 0.03 for this study. e design parameters α, β, c, and η of the LTMD may be estimated by the TMD optimum parameters and numerical analysis.
is study numerically investigates the dynamic control and design values of those parameters. Firstly, assuming numerical values of α and η with the prescribed mass ratio, the other parameters β and c to minimize the square root of the sum of the squares (SRSS) by the dynamic responses during an external excitation are determined. And another SRSS is calculated using the predetermined parameters β and c, and the parameters α and η to minimize the SRSS are also selected. We assume that the half-scaled N-S acceleration components of the 1940 El Centro earthquake acted on the structure during the rst 30 seconds. Figure 5  Substituting these values into equation (6) and numerically integrating the second-order di erential equation, the SRSS responses are determined according to the parameters α and η and are plotted in Figure 5(c). ey exhibit the minimized values at α 0.7 and η 0.95. It is observed from the SRSS plots that the design parameters for the LTMD interdependently a ect the dynamic responses of the structure. Figures 5(d)-5(f ) compare three dynamic responses of the structure without any dynamic control system, with TMD or LTMD. It is shown that the control system remarkably reduces the dynamic responses. And the LTMD system is more e ective in controlling the dynamic responses than the TMD. It is due to the control of the dynamic responses of adjacent two oors unlike the TMD. e control forces or constraint forces necessary to satisfy the constraint condition of the lever act on the structure, and the dynamic responses are controlled by the forces. Figure 5(g) represents the forces acting on the upper mass of the lever, and the forces αF c T1 act on the lower mass. e dynamic control is accomplished by those forces.
Similar process is performed using the mass ratio of μ 0.03, and Figure 6 shows the numerical results.  indicates that the LTMD to be installed between two oors corresponding to the highest mode values of the rst and second modes is e ective in reducing the dynamic responses. It more de nitely controls the dynamic responses than the TMD system as shown in Figures 6(c)-6(e). And the LTMD is very e ective in reducing the story drift by the control forces as shown in Figure 6(f ). Figure 7 compares the dynamic responses and the control forces in utilizing the coe cient values to minimize the SRSS shown in Figure 5 (μ 0.02) and Figure 6 (μ 0.03). It is shown that the dynamic responses are reduced with the increase in the mass ratio. e increase in the mass ratio also leads to the increase in the control forces as shown in Figure 7(c). ough the optimum parameters of the LTMD cannot be explicitly established, it is shown that they can be obtained by numerical experiments and the LTMD system can more de nitely control the dynamic responses than the TMD system.

Control of a Simply Supported Beam.
e distance between the locations to represent the largest mode values at the rst and second modes increases with the increase in the number of DOFs. In the case of a xed-end beam shown in Figure 8, one end of the lever should be installed at the midspan to represent the largest mode value at the rst mode.
is example investigates the vibration control according to the location of the other end of the LTMD and compares the e ectiveness with the TMD system.
Assume that the external excitations of 10% magnitude of the earthquake accelerations in Figure 5 Figure 9(b) that the vibration can be more explicitly controlled with the increase in the parameters α and β in the given ranges rather than the parameters c and η. It indicates that the vibration is more sensitive to the length ratio of the lever and the sti ness ratio at both ends of the lever. e SRSS of the displacements is shown in Figures 9(c) and 9(d) when the lever is installed at nodes 10 and 13. Minimum SRSS is obtained when the values of parameters are as follows: α 1.15, β 1.15, c 0.9, and η 0.8. ese plots also show that the length ratio and the sti ness ratio are sensitive to the vibration control of the beam.  it is found in those plots that the LTMD system can reduce a little more dynamic responses than the TMD system because the LTMD makes an additional control at another node 15 unlike the TMD. us, it is shown in Figure 10(c) that the displacement difference between nodes 10 and 15 can be reduced owing to the control force in the satisfaction of the constraint condition of the lever motion. Figure 11 compares the displacement responses at nodes 10 and 15 depending on the installation locations of the LTMD of two cases. e response difference at two cases cannot explicitly be recognized but the LTMD of case 1 is a little more effective than that of case 2. It is shown that the LTMD is a little more effective when it is installed at the location of the highest mode value of the second mode. Figure 12 compares the control forces exerted by the LTMD and TMD. e constraint forces at node 10 of the LTMD corresponding to two cases are shown in Figure 12(a). It is shown that the constraint forces in case 1 are a little higher than those in case 2. And, it is observed that the control forces exhibited by the TMD are larger than the constraint forces by the LTMD. us, it is found that the displacement responses can be more explicitly controlled by distributing the control effect of the TMD system into two nodes of the LTMD.
It can be expected from the above two applications that the LTMD can be more effective in controlling the dynamic responses of a low-rise building structure with a few DOFs than those of a high-rise building structure with many DOFs.

Conclusions
is study illustrates the effectiveness of the vibration control by the LTMD.
e LTMD controls the dynamic responses combining the TMD parameters and the constraint condition. ough the optimum parameter values of the LTMD cannot be explicitly established, they can be estimated by numerical experiments. e numerical applications exhibit that the LTMD leads to remarkable         displacement reduction. And, the LTMD is more effective control system than the TMD because the LTMD controls the displacements between adjacent floors. e control effect by the LTMD is more sensitive to the length ratio of the lever and the stiffness ratio at both ends of the lever than the other parameters. e LTMD is a little more effective when it is installed at the location of the highest mode value of the second mode. It is found that the displacement responses can be more explicitly controlled by distributing the control effect of the TMD system into two nodes of the LTMD.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.