# Effect of Tuned Spring on Vibration Control Performance of Modified Liquid Column Ball Damper
## 1 Introduction
Many researchers optimized the performance of TLCDs using a variable orifice opening [31], the variable cross-section area of the liquid column [32; 33; 34; 35], and different shapes of TLCDs [30; 36]. Some studies proposed that using denser liquid significantly improved the energy dissipation mechanism of TLCDs [37; 38; 39]. [PERSON] proposed optimal design parameters under harmonic loading [40]. [PERSON] et al. established design guidelines for TLCDs to obtain optimum results [41]. [PERSON] et al. proposed a multi-pressurized tuned liquid column damper (MPTLCD) to reduce wind-induced vibration of long-span cable-stayed bridges [42]. Recently, a newly modified version of TLCDs was proposed by [PERSON] et al., known as a tuned liquid column ball damper (TLCBD) [43]. In a TLCBD, the orifice is replaced with a rolling ball in the horizontal section of the tube that acts as a moving orifice and is more efficient in the dissipation of vibrations than a static orifice. Experimentally, it has been observed that the response reduction of the structure attached with a TLCBD is greater than that of TLCD under harmonic loading [43]. [PERSON] et al. studied TLCBDs for multi-degree freedom structures under harmonic and seismic loading [44; 45]. The effectiveness of TLCBDs was observed over TLCDs under both loading conditions. [PERSON] et al. proposed optimal design guidelines for TLCBDs under seismic loading [46]. [PERSON] et al. demonstrated constrained and unconstrained optimization of TLCBDs [47]. [PERSON] and [PERSON] replaced the solid ball of the TLCBD with a hollow ball and studied the performance of the modified version of the TLCBD [48]. In the current paper, a new modified version of a TLCBD is proposed, known as a tuned liquid column ball spring damper (TLCBSD). This proposed version is better than TLCBDs in terms of performance. For better performance of the TLCBD, a modification is made in the new version by introducing a spring system. The inspiration behind the introduction of a spring in the modified form of the TLCBD is to control the motion of the moving ball in the horizontal section of the tube, because in a TLCBD the ball acts as a free body and the movement of the ball is uncontrolled. This uncontrolled movement of the ball can cause damage to the damper in the case of excessively high amplitude vibration. Therefore, in this modified arrangement the ball inside the horizontal section of the TLCBD is attached to the spring that was previously free, and is called TLCBSD. Two different arrangements of the TLCBSD are suggested. The first arrangement is known as the tuned liquid column ball spring sliding damper (TLCBSSD). In a TLCBSSD, the rotational motion of the ball attached to the spring is restricted. It can only translate. The second arrangement is called a tuned liquid column ball spring rolling damper (TLCBSRD). In a TLCBSRD, the ball attached to the spring can translate as well as rotate. The newly proposed dampers are theoretically analyzed in the present work. The equations of motion of both arrangements are derived using [PERSON]'s equation. Optimal parameters are derived for both the arrangements, solving the equation of motion numerically. A damped single-degree freedom structure is investigated with a TLCBD, TLCBSSD, and TLCBSRD numerically using linear state-space representation for differential equations under harmonic, seismic, and impulse loadings. The performance of the TLCBSSD and TLCBSRD are compared with the TLCBD. It is found that the performance of the TLCBSD under harmonic, seismic, and impulse loading is better than that of the TLCBD. Design guidelines are also proposed for TLCBSDs and TLCBSRDs.