## 1 Introduction  
The double tripod industrial driveshafts (DTID) are used in the cement industry plant, food industry, and agriculture industry to link the power supplier with the industrial mill. In most cases, the power supplier is the asynchronous electric motors, and the link to a moderate heavy-duty mill is DTID, the industrial system presented in Figure 1. Generally, in heavy-duty use, a drivetrain line from the electric motor to the mill is covered by a double cardan driveshaft, see Figure 2, but this type of driveshaft, having two Hooke's joints, induces a nonuniformity that must be avoided [1] (pp. 5-9). For a moderate heavy-duty industrial milling system, to avoid the non-uniformity of Hooke's joints, a double tripod industrial driveshaft (DTID) is used in the driveline transmission because this type of driveshaft has two constant velocity joints (CV) [1] (pp. 52-79). The DTIDs are, in fact, quasi-homokinetic driveshafts, as proven by [PERSON] et al. [1] (p. 78), in [2], and mentioned in [3]. This is the type of DTID that the present work deals with, and the first effect considered is the kinematic nonuniformity already studied in detail by the papers [2,3]. [PERSON] et al. developed [4] a double roller tripod joint to obtain a better constant velocity. The excitation torque load, an input of the asynchronous electric motor, can be considered harmonic type, as mentioned by [PERSON] in the paper [5]. The present work investigates nonstationary forced torsional vibrations (NFTV) of DTID using designed methods that certify the possible chaotic deterministic or ergodic dynamic behavior. In [6], [PERSON] et al. developed a torsional model with two degrees of freedom (2 DOF) to investigate the damping of torsional vibrations in variable speed drive, particularly the torsional mode excitations. [PERSON] et al. studied [7] the amplitude-frequency characteristics for torsional vibration in an automotive powertrain using a model with 29 DOF. In his doctoral thesis [8] (pp. 6-100), [PERSON] investigated the modeling of nonlinear driveshaft systems with double U-joints, observing the passage effect and the Sommerfeld effect for both 2 DOF and 3 DOF models. Based on findings in the literature [6, 7, 8], it is evident, in opposition to the models in [9], that the modified physically consistent model (MPCM) for FTV of DTID is a better approach to industrial realities. The next paragraph will discuss the MPCM for FTV of DTID that considers the following aspects: double tripod of the joints for ID, geometric and kinematic nonuniformity of CVJs, and harmonic excitation of torsional torque load due to the use of an asynchronous electric motor for the moderate heavy-duty industrial milling system as well as the reactive harmonic torque induced by the mill. In their paper, [PERSON] and [PERSON] remarked for the first time using numerical data [10] that the most critical resonance for a driveshaft with a universal joint is the principal parametric resonance. [PERSON] et al. in the article [11], investigated the parametric instability of driveshafts with U-joints using the monodromy matrix method. Another study [12] investigated the dynamic instability of a double cardan joint using the FEM (finite element method) coupled with the monodromy matrix method. As suggested in the literature [10, 11, 12], the most crucial resonant case of a driveshaft is the principal parametric resonance; therefore, the present work is dedicated to investigating the nonstationary dynamic instability of FTV for DTID in the principal parametric resonance region (PPRR). [PERSON] et al. [13] investigated the values of tangential stress for a pumping unit shaft in the resonant case, confirming the need for avoiding such a torsional resonance region. The paper [14] studied the designing needs of models for torsional vibrations of shafts of mechanical systems to control the dynamic behavior in torsion, by vibration dampers and equivalent stiffness of double-mass oscillating systems. [PERSON] et al.