## 1 Introduction
A comprehensive assessment of commercial vehicle electrification, particularly for medium- and heavy-duty vehicles, is crucial for guiding industry development and policy decisions. This assessment should encompass the evaluation of existing electrification architectures, component technologies, and factors influencing performance and cost [10]. Active vibration control strategies, utilizing force actuators and sophisticated control algorithms, offer a promising solution for mitigating unwanted vibrations in various engineering applications. Unlike passive methods that rely on inherent material properties, active control systems actively counteract vibrations by applying opposing forces, achieving superior vibration suppression and adaptability across a wider frequency range. A typical active vibration control system consists of sensors, electronic circuits, and actuators working in unison. Sensors detect vibrations and relay feedback signals to the electronic circuit, which processes the information and determines the appropriate actuation force. Actuators, in turn, receive commands from the electronic circuit and apply the necessary counteracting force to the system [11,12]. Researchers have explored various control strategies for active vibration absorption, including the following. Linear velocity and acceleration feedback control: These methods adjust the actuation force proportionally to the measured velocity or acceleration of the vibrating structure. Cubic velocity feedback control: This approach introduces nonlinearity into the control law, potentially improving performance for certain types of vibrations. Nonlinear saturation controller: NSCs limit the maximum actuation force, preventing actuator saturation and ensuring stability. Positive position feedback controller: PPF controllers utilize a positive feedback loop to target specific vibration modes, effectively suppressing resonant vibrations [13, 14, 15, 16]. The Multiple Time Scales Method has proven to be a valuable tool for analyzing the complex dynamic behavior of systems with multiple resonances, such as those encountered in series hybrid electric vehicles. By decomposing the system's motion into multiple time scales, MTSM enables the study of slow and fast dynamics, providing insights into phenomena like ultra-harmonic resonances and steady-state responses. For instance, MTSM has been successfully employed to investigate the nonlinear characteristics of a cantilever beam subjected to external forcing and self-excitation. The study revealed that external stimulation resonance can induce harmonic motion at the beam's support, highlighting the importance of considering nonlinear effects in vibration analysis [17; 18; 19; 20]. Researchers have employed various control strategies and analytical techniques to mitigate vibrations in nonlinear systems subjected to harmonic excitations. One such approach involves using a positive position feedback controller to dampen vibrations in a harmonically excited nonlinear beam. The Multiple Time Scales Method has been instrumental in deriving analytical solutions for this system, providing valuable insights into its dynamic behavior [21]. In another study, a negative linear velocity feedback controller was implemented to regulate a dangling cable system subjected to mixed excitation forces, characterized by cubic and quadratic nonlinearities. A hybrid control system, termed NNPDCVF, is implemented to mitigate nonlinear vibrations in a dynamic beam system. This approach integrates cubic velocity feedback with a negative nonlinear proportional derivative component to counteract the system's inherent nonlinearities [22]. Further research has delved into the nonlinear vibration patterns of a cantilever beam system, designated as the primary system, interacting with a secondary system through a nonlinear absorber.