## 1 Introduction  
The present work presents a perturbation approach, specifically the asymptotic method approach (AMA) [1], to investigate the principal parametric resonance (PPR) for the forced bending vibrating (FBV) movements of the elements of an automotive driveshaft (AD), with the AMA being a powerful tool for the investigation of vibrations induced by shocks, as mentioned by [PERSON] in the literature [2]. To analyze the PPR region, a model for FBV movements of the AD elements was designed, with a specific multibody dynamic structure considering the following aspects: the geometric nonuniformity of the AD elements and the shock excitation induced by the road geometry. The excitations of the FBV movements of the AD elements are due to the shock excitations acting on the automotive wheel, being an excitation induced by the road geometry [3]. The geometric nonuniformities of the AD elements are undoubtedly the leading cause of nonlinear parametric vibrations of the AD in the range of 0.1-15 kHz, as established by experimental data in the literature [4] (pp. 98-123). The first researchers who considered the specific geometry of the AD elements were [PERSON] and [PERSON]. They first analyzed the nonlinear dynamic behavior of the AD elements in the PPR region [5]. The lateral vibration of the AD was analyzed by [PERSON] and [PERSON] for superharmonic resonance in [6]. In [7], [PERSON] et al., studied the bending vibrations of a shaft for 4 WD (four-wheel drive) drivetrains to design the control methods at low frequencies. Analysis of the parametric bending vibrations for a cardan shaft was performed by [PERSON], and the data are presented in [8]. The NVH phenomena related to the vehicle driveline architecture were investigated by [PERSON] et al. in [9] within different frequency ranges. At the same time, in [10], the same authors realized the driveline integration process. In [11], [PERSON] et al., performed dynamic analysis and vibration tests on the carbon-fiber-reinforced plastics drive-line transmission for machine tools. [PERSON] studied the vibration control of driveshafts considering the Sommerfeld effect on multiple linked shafts, and the data are presented in [12]. [PERSON] et al. [13] conducted an extended analysis of the vibrations for a driveshaft with a crack using experimental modal analysis and finite element analysis (FEA) for isotropic materials such as structural steel. The increased interest in achieving a high standard of comfort in the automotive industry led to developing research on dynamic vibration absorbers for the bending vibration of the vehicle propeller shaft [14]. In the same area, [PERSON] et al. investigated the use of photonic crystals in the vibration dampers of the AD [15]. In recent years, composite materials gained advancements in industrial utilization. Therefore, analysis of their dynamic behavior was needed, so [PERSON] and [PERSON] estimated the deflection and the natural frequencies of an AD made of such materials using the FEA [16]. The authors of [17] analyzed an AD carrying torque from the principal car differential to the rear wheel differential using the FEA for lateral and torsional vibration, considering the conventional and composite materials. [PERSON] et al. in [18] performed a detailed investigation of the vibration characteristics of the composite AD using the finite element method (FEM). This paper aims to investigate the dynamic behavior of the AD elements for the FBV movements in the PPR region. This investigation supposes the design of a model for FBV movements for an AD's elements, the computation of the stationary and nonstationary amplitude spectrum in the PPR region, and the analysis of the dynamic instability frontiers in the same area. The novelty for this investigation is the design of the model of FBV movements of the AD elements and the use of a perturbation approach (AMA) to compute the amplitude, the phase angle, and the dynamic instability frontiers for both stationary and nonstationary motions.