## 1 Introduction  
The design of modern vehicles, such as aircraft, helicopters, and ground vehicles, incorporates crashworthiness goals to enhance occupant safety and to protect avionics or electronics during survivable crash events [1, 2]. A common method to increase crashworthiness is to utilize energy absorbers (EAs) that can dissipate and absorb the impact energy [3]. Most commercial EAs employ stroking elements, such as plastic deformation [4] or collapsible materials [5, 6] in crumple zones with a suitable stroking load to dissipate kinetic energy through material deformation. Passive or deformable EAs are typically designed based on a worst-case or a specified crash scenario so that they have a single and fixed load-stroke profile [7]. They cannot be used in various crash conditions because the crashworthiness performance cannot be adaptively adjusted for a wide range of crash events [8]. To cope with this, in this study, a drop-induced shock mitigation system based on a magnetorheological energy absorber (MREA) is investigated because its load-stroke profile can be adjusted in a rapid, electronic, and continuous way [9, 10]. Semi-active MREAs have been investigated widely for shock mitigation. A magnetorheological grease buffer was designed, modeled, and tested [11]. A hybrid model based on the support vector regression and adaptive differential evolution algorithm for MREA was proposed. It has the optimal values for all the performance indices and the largestoverall desirability of 0.6274 [12]. The controllability of an MREA using a four-stage electromagnetic coil for gun recoil mitigation was presented [13]. An optimal controller was also developed to find a trade-off between the firing rate and recoil load [14]. The dynamic response of a rotorcraft skid landing gear system equipped with an MREA was designed and evaluated experimentally [15], and a nonlinear analysis and control algorithm were developed to ensure a soft landing [16]. Subsequently, the MREA landing performance was analyzed by considering the rotor lift factor [17]. A new magneto-rheological aircraft main landing gear (MRAMLG) system was proposed, and the landing efficiency was evaluated by a drop test. An MREA applicable to the landing gear system of airplanes was proposed and simulated [18]. A hybrid controller was developed for an MR landing gear system to enhance the landing efficiency by using a fail-safe function [19]. The adaptive aircraft landing gear equipped with an MREA was designed and simulated. The MR landing gear had better performance when compared to a conventional passive landing gear in different landing scenarios [20]. A neural network controller with supervised learning was proposed for the MREA model in an MR landing gear and accounted for aircraft mass [21; 22]. A lunar lander equipped with an MREA was designed, modeled, and simulated [23]. The results showed that the MR lunar lander can effectively decrease the largest acceleration in complex and unexplored areas [24]. A hybrid controller was proposed and designed for a seat suspension system to reduce both vibration and shock payload that transmit to the occupants [25]. The objective of MR shock mitigation system is to achieve a soft landing, that is, for the payload to fully utilize its stroke without any end stop impact, thus maximizing energy absorption with no end stop impact and thus minimizing injury to occupants or damage to equipment. In our prior studies, a soft landing was defined as the condition when the payload comes to a rest after fully utilizing the available stroke, and this was achieved using an optimal Bingham number (BN) control algorithm, which achieved such a soft landing [26]. In prior work, the effects of the MREA time lag as well as the payload maximum allowable threshold were considered, and a BN control algorithm incorporating a time lag [27] as well as minimum duration deceleration exposure (MDDE) control [28] were developed.