*Article* **Modeling and Analysis of FBV Movements for Automotive Driveshafts in the PPR Region**

**Mihai Bugaru \* and Ovidiu Vasile**

Department of Mechanics, University Politehnica of Bucharest, 060042 Bucures,ti, Romania; ovidiu\_vasile2002@yahoo.co.uk

**\*** Correspondence: skmbugaru@yahoo.com; Tel.: +40-749-222333

**Featured Application: The asymptotic method approach (AMA) is used to compute the amplitudes of forced bending vibrating (FBV) movements versus the excitation frequency in the region of principal parametric resonance (PPR) for multibody mechanical systems. An FBV movements model was designed for an automotive driveshaft to realize a powerful design tool for FBV movement control of the automotive transmission.**

**Abstract:** This research's goal is to model and analyze the forced bending vibrating (FBV) movements for the elements of an automotive driveshaft using a perturbation technique, the asymptotic method approach (AMA), in the region of principal parametric resonance (PPR). The PPR region was chosen because the principal parametric resonance region is one of the essential resonance regions. The model of FBV movements for the automotive driveshaft (AD) considers the aspects of the following phenomena: geometric nonuniformity of the AD elements and shock excitation due to the road. To overcome the equations for the FBV movements of the AD elements, all inertia characteristics were reduced to the longitudinal ax of the midshaft using the variation of the geometric moments of inertia with the concurrent axis and Stener's theorem. The midshaft of the AD was considered a Timoshenko simply supported beam with a concentrated mass at both ends and springs and dampers for linear and rotational movements at both ends. To determine the equations describing the FBV movements of the AD elements, Hamilton's principle was used. After establishing the equations of motion for each AD element coupled with the specific boundary conditions, the amplitude and the phase angle were computed for stationary and nonstationary motion in the PPR region using the first order of the AMA, and the dynamic instability frontiers were determined based on the same equations. The dynamic behavior of the AD was investigated concerning the variation of the damping ratio and the variation of the parametric excitation coefficient. The AMA coupled with the model of FBV movements for the AD exhibits the future research directions for analyzing FBV movements for the AD in the regions of superharmonic resonances, subharmonic resonances, combination resonances, internal resonances, and simultaneous resonances. Additionally, the AMA can predict the endurance of the AD and design control of car damping systems.

**Keywords:** multibody dynamics; automotive driveshafts; forced bending vibrations; asymptotic method; principal parametric resonance; dynamic stability frontiers
