This article presents the design of a tuned mass damper with a conical spring to enable tuning to the natural frequency of the system at multiple values, as may be convenient in case of a system with fluctuations in the mass. The principle and design procedure of the conical spring in the context of a varying mass system are presented. A passive feedback control mechanism based on a simple pulley-mass system is devised to cater to the multi-tuning requirements. A design example of an elevated water tank with fluctuating water content, subjected to ground excitation, is considered to numerically illustrate the efficiency of such a tuned mass damper associated with the conical spring. The conical spring is designed based on the tuning requirements at different mass conditions of the elevated water tank by satisfying the allowable load bearing capacity of the spring. Comparisons are made with the conventional passive tuned mass damper with a linear spring tuned to the full tank condition. Results from time history analysis reveal that the conical spring-tuned mass damper can be successfully designed to remain tuned and thereby achieve significant response reductions under stiffening conditions of the primary structure, whereas the linear spring-tuned mass damper suffers performance degradation because of detuning, whenever there is any fluctuation in the system mass. © The Author(s) 2021.