## Abstract

Many driveline systems are designed to accommodate angular misalignment by the use of flexible couplings or Universal Joints (U-Joints) which link individual shaft segments. The Sommerfeld effect is a nonlinear phenomenon observed in some rotor systems being driven through a critical speed when there is not enough power to accelerate the rotor through resonance. Previous studies have shown that rotor speed can become captured when transitioning through natural frequencies due to nonlinear interactions between a non-ideal driving input and rotor imbalance. This paper, for the first time, shows that this type of rotor speed capture phenomena can also be induced by driveline misalignment. During rotor spinup under constant motor torque, it is found that misalignment-induced rotor speed capture phenomena can occur as the shaft speed approaches ½ the first elastic torsional natural frequency. Depending on misalignment level and motor torque, the shaft speed will either dwell near this speed and then pass through, or the speed will become trapped. Here, a nonlinear rotordynamics model of a segmented driveshaft connected by two U-joints including effects of angular misalignment and load torque is developed for the study. This analysis also determines the minimum driveline misalignment angle for which the shaft speed capture phenomena will occur for a given motor torque and load torque condition.

## 1 Introduction

One of the key parts of this model will be to accurately capture the drivesystem rotordynamics which will be especially important at higher rotational speeds and torque levels. To accommodate angular misalignment and center offsets, most driveshafts are equipped with some type of flexible couplings or universal joints (i.e., U-joint or Hooke’s joint) [1], which link individual shaft segments. U-joints can also be a significant source of vibration due to their kinematics which, depending on the level of angular misalignment, can result in shaft speed fluctuations about the nominal input speed. Asokanthan and Wang [1] and Asokanthan and Meehan [2] explored the torsional vibration of a simplified drivesystem consisting of two shafts connected by a single U-joint. It was found that sufficient levels of angular misalignment resulted in parametric instabilities for shaft operating speeds in the vicinity of the principle and sum-type torsional natural frequency combinations. Bulut and Parlar further explored the stability of two massless torsional flexible shafts connected by one U-joint using the Monodromy matrix method [3,4]. Furthermore, Iwatsubo and Saigo [5] analyzed the transverse vibration of a single U-joint/shaft system under the effect of a follower torque load. Here, it was found that the follower torque produced both flutter and parametric instabilities. Xu and Marangoni [6,7] found that angular misalignment resulted in 2/rev excitations which produced significant lateral vibrations for shaft speeds operating near 1/2 the critical speeds. Kato et al. [8] showed that coupled torsion and lateral vibrations occur for shaft speeds near ½ the sum of the lateral and torsional natural frequencies. In addition, both Kato and Ota [9] and Saigo et al. [10] considered effects of viscous and Coulomb friction between the cross-pin and yokes within the U-Joint coupling. Mazzei et al. [11] explored the dynamic stability of a misaligned single U-joint shaft system supported by a flexible bearing and subjected to a constant follower torque. DeSmidt et al. [12] found that misalignment induced dynamic coupling between torsion and lateral vibrations and that this interaction can cause parametric instability near sum-type combinations of torsional–lateral natural frequencies. Crolla [13] and Sheu et al. [14] showed that the viscous joint damping had a strong effect on shaft vibration amplitudes near critical speed. DeSmidt et al. [15] showed that load torque induced bending–bending combination parametric instability regions, while misalignment resulted in both bending–bending and torsion–bending combination parametric instabilities. Mazzei [16] studied the effect of angular acceleration rate on shaft vibration amplitudes in a single U-joint/shaft system. It was shown that no sweep rate was able to avoid the 2/rev forced motion resonance induced by the angular misalignment. Finally, most recently, Browne and Palazzolo [17] explored the nonlinear vibration of a single U-joint/driveshaft including a tuned damper. Their results show significant vibration jump phenomena measured during shaft acceleration and run-down. These vibration levels were proportional to misalignment and load rotational inertia.

The nonlinear Sommerfeld effect was first introduced by Sommerfeld [18] and was also described by Kononenko [19], who conducted experiments consisting of an imbalanced rotor driven by a small motor with the bounded energy source mounted on the end of a cantilever beam structure. Here, the speed jump phenomenon was detected in the motor as the rotational speed was increased through the beam first bending natural frequency. As motor driving power was increased, rotational speed remained constant and then suddenly jumped to a higher value above the system natural frequency. It was concluded that rotor speed was affected by the lateral vibration of the support and that below as certain threshold the additional power was sent into increasing lateral vibrations rather than increasing rotor speed. Krasnopolskaya and Shvets [20] investigated nonlinear coupled torsion/lateral vibrations of an imbalanced shaft. They determined the parametric conditions for the rotor speed capture and pass through and also found two chaotic regions that were close to the fundamental frequencies. Dimentberg et al. [21] and Belato et al. [22] both further investigated the unbalanced shaft driven with limited power. Dimentberg et al. applied Krylov–Bogoliubov averaging-over-the-period to obtain a torque–speed curve. They experimentally demonstrated that adjusting the lateral support stiffness during passage is an effective method to avoid Sommerfeld rotor speed capture phenomena. Tsuchida et al. [23] explored this effect on a more complicated support structure involving two degrees-of-freedom (DOFs) plus the rotor. They demonstrated the occurrence of Sommerfeld rotor speed jump behavior in the vicinity of both modes. Furthermore, Bolla et al. [24] considered effects of nonlinear support stiffness on the coupled support/rotor system, and a frequency response curve was obtained to determine the bifurcation phenomenon. In addition, Felix and Balthazar [25] explored the use of an electromechanical vibration absorber on the same system to suppress the vibration amplitudes. In another study, Gonçalves et al. [26] analytically and experimentally determined the minimum motor power required for traversing coupled nonlinear torsion–lateral resonances in a motor/rotor/beam system. For the case of more complex rotordynamic systems, Samantaray et al. [27] explored Sommerfeld behavior in a flexible shaft/disk rotor driven by a non-ideal motor including effects of external and internal damping and gyroscopic forces. They concluded that the torsional material damping had a significant effect on suppressing the Sommerfeld effect in this system. Karthikeyan et al. [28] studied the Sommerfeld effect of rotor-motor using commercial software ansys. The regression equations of the steady-state transverse and rotating displacement were developed to fit the data point. Verichev [29] studied pure torsional vibration in the resonance zone of a shaft driven by a limited power source. In this case, Sommerfeld behavior was observed near the pure torsional resonance zones as well. Felix et al. [30] describe the energy harvesting involved with the portal frame type vibration under an excitation with time-variable frequency. The nonlinear energy sink (NES) absorbers are presented for reducing the vibration. Moreover, the microelectromechanical systems (MEMS) topics are investigated [31]. The wavelet transform has been used for the analysis of the Sommerfeld effect with only time response and control design proposed by Varanis et al. [32,33].

Despite the large body of existing research on dynamics of U-joint drivelines [117], Sommerfeld rotor speed capture and jump phenomena have not been formally observed or characterized in these systems. This is partially due to the fact that, in these prior investigations, the input shaft speed has been treated as a prescribed model parameter rather than being considered as degrees-of-freedom. Also, many prior studies only considered the case of constant input shaft speed and small misalignments and hence only employed linearized equations-of-motion. This study considers the full nonlinear torsion and lateral equations-of-motion of a representative double U-joint driveshaft system shown in Fig. 1.

Fig. 1
Fig. 1
Close modal

In this system, the input and output shafts are maintained in a state of parallel offset misalignment in a single plane by the suspension linkages as shown. In this condition, the input and output shafts remain parallel with equal but opposite misalignment angles at the U-joints. Furthermore, as is common, the driveline is assembled with the two U-joints having a 90 deg relative phase about the shaft rotation axis (90 deg indexing) which guarantees that the input and output shaft speeds remain identical (i.e., acts as a constant velocity joint). This ideal case can be essentially perturbed in two main ways: (1) if the input-side and output-side misalignment angles become different or (2) if the 90 deg phase condition becomes altered due to shaft torsional windup under load. This investigation focuses on the later scenario. To the authors’ knowledge, this situation has not been studied in the prior literature.

## 2 Double U-Joint Driveline Model

The general double U-joint driveline system is shown in Fig. 2.

Fig. 2
Fig. 2
Close modal
The system kinematics are formulated in terms of the input shaft rotation angle, ϕ(t), U-joint A yoke rotation angles, α1(t), β1(t), U-joint B yoke rotation angles, α2(t), β2(t), and the shaft elastic twist angle φ(t) which are all degrees-of-freedom with respect to time t. The coordinate frames shown in Fig. 2 are linked by the following Euler rotation sequence, which starts from the stationary frame ${n}=[n^1n^2n^3]$ and proceeds to the final body-fixed output shaft frame ${c}=[c^1c^2c^3]$
${n}→ϕn^1{a}→α1a^2{a′}→β1a^′3{b}⏟U−jointA→(π2+φ)b^1⏟90degindexing+shaftelastictwist{b′}→α2b^′2{c′}→β2c^′3{c}⏟U−jointB$
(1)
Here, the unit vectors of each successive coordinate frame $n^i$, $a^i$, $a^i′$, $b^i$, $b′^i$, $c^i$, and $c^i′$ for directions $[i=1,2,3]$ are related by the following coordinate transformations:
$[a^1a^2a^3]=Tna[n^1n^2n^3],[a^1′a^2′a^3′]=Taa′[a^1a^2a^3],[b^1b^2b^3]=Ta′b[a^1′a^2′a^3′][b^1′b^2′b^3′]=Tbb′[b^1b^2b^3],[c^1′c^2′c^3′]=Tb′c′[b^1′b^2′b^3′],and[c^1c^2c^3]=Tc′c[c^1′c^2′c^3′]$
(2)
with rotational transformation matrices Tna, $Taa′$, $Ta′b$, $Tbb′$$Tb′c′$ and $Tc′c$ given as
$Tna=[1000cosϕsinϕ0−sinϕcosϕ],Taa′=[cosα10sinα1010−sinα10cosα1]Ta′b=[cosβ1sinβ10−sinβ1cosβ10001],Tbb′=[1000cos(π/2+φ)sin(π/2+φ)0−sin(π/2+φ)cos(π/2+φ)]Tb′c′=[cosα20sinα2010−sinα20cosα2],andTc′c=[cosβ2sinβ20−sinβ2cosβ20001]$
(3)
Here, the 90 deg relative phasing of the U-joints is explicitly accounted for by including the angle π/2 inside the $Tbb′$ rotation matrix. Furthermore, based on the Euler rotation sequences in Eqs. (1)(3), the various coordinate frame angular velocities are
$ωa=ϕ˙n^1,ωb=ωb1b^1+ωb2b^2+ωb3b^3ωc=ϕ˙n^1+α˙1a^2+β˙1a^3′+φ˙b^1+α˙2b^2′+β˙2c^3′$
(4)
where $ωa$, $ωb$, and $ωc$ are the angular velocities of coordinate frames ${a}$, ${b}$, and ${c}$, respectively, and $d/dt=(∙)$ indicates differentiation with respect to time t. Next, since the drivesystem being considering in this investigation (see Fig. 1) is in a state of parallel offset misalignment, we have the following kinematic constraint:
$c^1=n^1$
(5)
This constraint means that the U-joint angles α1, β1, α2, and β2 are not all independent. Based on the Euler rotation sequence (1), we can also express the constraint in Eq. (5) as
$c^1=a^1$
(6)
Furthermore, from Eqs. (2) and (3), we have
$[c^1c^2c^3]=Tc′cTb′c′Tbb′Ta′bTaa′[a^1a^2a^3]$
(7)
which yields
$c^1=c11a^1+c12a^2+c13a^3$
(8a)
with coefficients
$c11=cosα1cosα2cosβ1cosβ2+sinφ(cosβ2sinα1sinα2+cosα1sinβ1sinβ2)⋯+cosφ(sinα1sinβ2−cosα1cosβ2sinα2sinβ1)$
(8b)
$c12=cosα2cosβ2sinβ1−sinφcosβ1sinβ2+cosφ(cosβ1cosβ2sinα2)$
(8c)
and
$c13=−cosα2cosβ1cosβ2sinα1+sinφ(cosα1cosβ2sinα2−sinα1sinβ1sinβ2)⋯+cosφ(cosβ2sinα1sinα2sinβ1+cosα1sinβ2)$
(8d)
In order to enforce the paralleled offset misaligned constraint, we require
$c11=1,c12=0,c13=0$
(9)
which results in the following conditions on the Euler angles of U-joint B:
$tanα2=sinφtanα1cosβ1−cosφtanβ1$
(10a)
and
$tanβ2=cosα1sinβ1sinφ+cosφsinα11−(cosα1sinβ1sinφ+cosφsinα1)2$
(10b)
Next, as depicted in Fig. 3, the orientation of the ${b}=[b^1b^2b^3]$ frame attached to the intermediate driveshaft is expressed in terms of two projected angle coordinates, δ(t) and γ(t), in the $n^1−n^2$ and $n^1−n^3$ planes, respectively. In particular, the $b^1$ unit vector which is aligned with the shaft rotation axis, is
$b^1=cosδcosγ1−sin2δsin2γn^1+sinδcosγ1−sin2δsin2γn^2+cosδsinγ1−sin2δsin2γn^3$
(11)
Fig. 3
Fig. 3
Close modal
Since the motion of the double U-joint drivesystem is confined to a single plane by the suspension system (see Fig. 1), the model is developed for case with
$γ(t)=0$
(12)
having only misalignment, δ(t), in the $n^1−n^2$ plane. Using the coordinate transformation relations in Eqs. (2) and (3), the ${b}$ frame unit vectors can also be expressed in terms of the shaft rotation ϕ and Euler angles α1 and β1 via
$[b^1b^2b^3]=Ta′b(β1)Taa′(α1)Tna(φ)[n^1n^2n^3]$
(13)
which yields
$b^1=cosα1cosβ1n^1+(cosϕsinβ1+sinϕcosβ1sinα1)n^2⋯+(sinβ1sinϕ−cosβ1sinα1cosϕ)n^3$
(14)
Finally, by equating Eqs. (10) and (14) and using constraint (12), we obtain
$tanα1=sinϕtanδandtanβ1=cosϕsinδ1−cos2ϕsin2δ$
(15)
Furthermore, expressions for the Euler angle rates can be obtained via differentiation of Eq. (15) as
$α˙1=sinϕ1−cos2ϕsin2δδ˙+cosϕcosδsinδ1−cos2ϕsin2δϕ˙$
(16a)
$β˙1=cosϕcosδ1−cos2ϕsin2δδ˙−sinϕsinδ1−cos2ϕsin2δϕ˙$
(16b)
Finally, after combining Eqs. (4), (10), (15), and (16), the ${b}$ frame angular velocity becomes
$ωb=δ˙sinϕcosϕsinδ+ϕ˙cosδ1−cos2ϕsin2δb^1+δ˙sinϕ1−cos2ϕsin2δb^2+δ˙cosϕcosδ1−cos2ϕsin2δb^3$
(17)
Furthermore, the ${c}$ frame angular velocity is
$ωc=ωLc^1=ωLn^1$
(18)
With rotational speed ωL expressed as the rational trigonometric function
$ωL=ϕ˙Δ+1+cos2δ−sin2δcos2ϕ2Δcosδφ˙…+tanδΔ(sin2φ4cosδ(sin2δ−(1+cos2δ)cos2ϕ)+sin2φsin2ϕ)δ˙$
(19a)
with denominator term
$Δ=1+tan2δ2(sin2φ(sin2δ−(1+cos2δ)cos2ϕ)−cosδsin2φsin2ϕ)$
(19b)

Here, as shown in Eq. (18), $ωc$ indicates rotation about the fixed $n^1$ axis which is a result of the parallel offset misalignment constraint (5).

Next, a rotordynamics model of the suspended double U-joint driveshaft system, which is depicted in Fig. 1, is developed. Figure 4 shows a schematic of this model which obeys the kinematic relations derived in Eqs. (1)(19).

Fig. 4
Fig. 4
Close modal
This model has 3DOF which are the input shaft rotation angle, ϕ(t), the single plane parallel offset misalignment angle, δ(t), and the shaft elastic twist angle, φ(t), each as shown in Fig. 4. Here, φ(t) is due to shaft torsional flexibility, and the shaft misalignment angle
$δ(t)=δ0+δ^(t)$
(20)
is due to both the nominal design misalignment level, δ0, plus an additional perturbation angle, $δ^(t)$, due to elastic deformation of the suspension (see Figs. 1 and 4). In order to track driveline lateral positions, we introduce the following position vector:
$RAB=Lb^1=L[cosδn^1+sinδn^2]$
(21)
which locates the center position of U-joint B relative to the fixed center point of U-joint A. Here, L is the length of the intermediate shaft (Fig. 4). Furthermore, the position vector of the output-side mass center, GL, is
$RAGL=RAB+dc^1$
(22)
where length d is the axial distance from point B to GL. From Eq. (21), the velocity of point B is determined as
$R˙AB≡vB=Lδ˙[−sinδn^1+cosδn^2]$
(23)
Furthermore, the velocity of the mass center GL is
$vGL=vB$
(24)
which is a due of the parallel axis constraint (5) enforced by the suspension linkages allowing only curvilinear translation motion of the output shaft.
The driveline system equations-of-motion are formulated using standard variational mechanics and energy methods approach. Based on the kinematics, the driveline kinetic energy is
$T=12Jmφ˙2+12JLωL(φ,ϕ,δ,φ˙,ϕ˙,δ˙)2+12mLL2δ˙2$
(25)
where Jm represents rotational inertia of the input side such as a motor connected to the input shaft, JL is the total effective rotational inertia of the driven load attached to the output shaft, and mL is the total suspended mass of the load end. Here, note that the output shaft rotational speed, ΩL, is a highly nonlinear function of ϕ, δ, and φ as given in Eq. (19). In this model, the intermediate shaft is considered massless since its inertia contributions are typically much smaller compared with Jm, JL, and mL. Next, the system strain energy is expressed as
$V=12kL2[sinδ−sinδ0]2+12ksφ2$
(26)
where k is the suspension stiffness coefficient and ks is the intermediate shaft torsional stiffness
$ks=GsLπ2(ro4−ri4)$
(27)
calculated in terms of the shaft inner and outer radii, ri and ro, and the shaft material shear modulus, Gs. Note, the suspension stiffness, k, only reacts to the component of deflection in the $n^2$ vertical direction about the nominal misalignment δ0. Axial travel along in the $n^1$ direction is not reacted by the spring as shown schematically in Fig. 4 by the roller interface between the suspension stiffness and the output shaft. Another way this could be done would be to employ a telescoping driveshaft. These effects could be ignored if δ were considered as a small angle; however, no such restriction is imposed in this model. Next, in order to model damping effects, the following Rayleigh dissipation function is utilized
$D=12cL2cosδ2δ˙2+12ξsksφ˙2$
(28)
where ξs is the viscous damping parameter of the shaft material which accounts for shaft internal damping and c is the suspension damping coefficient (see Fig. 4). Similar to the suspension stiffness, since δ is not restricted to a small angle, the suspension damping only reacts to the velocity component in the $n^2$ direction giving rise to the Lcosδ term in Eq. (28). Since δ0 is constant and considering Eq. (20), the model DOF vector is defined as
$q(t)=[φ(t)ϕ(t)δ^(t)]T$
(29)
Finally, the motor driving torque applied to input shaft, Tm, and the load torque, TL, on the output shaft are accounted for through the variation work, ΔW
$ΔW=TmΔφ+TL∂ΩL∂q˙⋅Δq=QTΔq$
(30)
where Δ is the variational operator and Q is the associated generalized force vector. Finally, the equations-of-motion of the suspended double U-joint driveline system (Fig. 4) are obtained via Euler–Lagrange equations as
$ddt[∂T∂q˙]−∂T∂q+∂V∂q+∂D∂q˙=Q$
(31)
All numerical simulations presented in Sec. 3 of this paper are based on the full nonlinear driveline system model as obtained directly from Eq. (31). The equations-of-motion in raw form are prohibitively long and cannot be practically shown in full form. However, upon assuming small elastic deformations, i.e., ϕ(t) and $δ^(t)$ having O(ɛ), a simplified model which captures many of the essential features of Eq. (31) is also obtained. This results in the following nonlinear equations which can expressed in quasi-linear form as
$M(q)q¨+Cq˙+K(q)q=N(q,q˙)+F(q)$
(32)
where M is the nonlinear inertia matrix
$M=M0+Ms2sin2φ+Mc2cos2φ+Ms4sin4φ+Mc4cos4φ$
(33a)
with matrix components
$M0=[Jm+JLJL4cosδ0(3+cos2δ0)0JL4cosδ0(3+cos2δ0−32δ^sin2δ0)JL64cos2δ0(41+20cos2δ0+3cos4δ0)0JL2ϕsinδ0tan2δ00mLL2]$
(33b)
$Ms2=JLϕtanδ0[2sinδ000tanδ04(3+cos2δ0)00000]$
(33c)
$Mc2=−JLtanδ02[0sinδ00sinδ0+32δ^cosδ0tanδ02(3+cos2δ0)0ϕ(3+cos2δ0)2cosδ000]$
(33d)
and
$Ms4=[000−JL4φsin2δ0tan2δ000000]andMc4=[0000JL8sin2δ0tan2δ00000]$
(33e)
Furthermore, the nonlinear system stiffness matrix is
$K=[0−TLsinδ0tanδ0sin2ϕ00ksTLsinδ0(1+cos2ϕ)0TLsinδ0(1+sec2δ02cos2ϕ−tan2δ02)kL2cosδ0]$
(34)
which is due to the shaft torsion stiffness, ks, and suspension stiffness, k, as well as skew symmetric-like terms related to the load torque, TL, and nominal misalignment, δ0. Furthermore, the system damping matrix due to shaft structural damping and suspension damping is
$C=[0000ξsks000cL2cos2δ0]$
(35)
Finally, the right-hand side of Eq. (32) contains fully nonlinear terms due to the load inertia JL
$N=−JLtan2δ0(ϕ˙φ˙sin2ϕ+ϕ˙2φcos2ϕ)[2cosδ012(3+cos2δ0)0]+JLtan2δ0ϕ˙φ˙2sin4ϕ[0sin2δ00]$
(36)
as well as a forcing vector due to the system torques, TL and Tm
$F=[Tm+TLTL3+cos2δ04cosδ0−TLtanδ0sinδ02cos2ϕ0]$
(37)

Note, the stiffness matrix K in Eq. (34) is not positive definite as can be observed by the zero in the first diagonal position, i.e., [K]11 = 0. This is due to the fact that the input shaft rotation coordinate ϕ(t) is an unconstrained DOF meaning that the system will have one rigid-body mode and two higher flexible modes.

To gain more understating of the driveline dynamics, the equations-of-motion of the aligned driveline (i.e., δ0 = 0) are obtained by substituting δ0 = 0 into Eqs. (32)(37) which yields the following linear system:
$M¯q¨+C¯q˙+K¯q=F¯$
(38a)
with mass, damping, and stiffness matrices
$M¯=[Jm+JLJL0JLJL000mLL2],C¯=[0000ξsks000cL2],andK¯=[0000ks000kL2]$
(38b)
and force vector due to the applied motor torque and load torque
$F¯=[Tm+TLTL0]$
(38c)
Based on Eq. (38), the undamped natural frequencies of the aligned driveline are
$[ωrigid=0,ωtorsion=ksJm+JLJmJL,andωlateral=kmL]$
(39)

Here, ωrigid = 0 is the rigid-body rotation mode associated with ϕ(t), ωtorsion is the elastic twisting mode (out-of-phase vibration of motor and load inertias) associated with φ(t), and ωlateral is the lateral vibration mode due to the suspension support compliance associated with dynamic misalignments, $δ^(t)$. The baseline system natural frequencies (39) provide reference points for studying the full nonlinear system response in Sec. 3.

## 3 Driveline Spinup Response

To investigate the possibility of Sommerfeld rotor speed capture phenomena, the response of the driveline during angular acceleration (i.e., spinup response) is numerically evaluated using Eq. (32). In each simulation, a constant value of input torque, Tm, is applied to the drivesystem which is initially at rest and possesses constant nominal misalignment δ0. The value of applied input torque used in the simulations is determined via
$Tm=(Jm+JL)α0$
(40)
where α0 is some specified shaft angular acceleration rate of the aligned (δ0 = 0 deg), fully unloaded (TL = 0) drive system. Furthermore, the load torque, TL, used in the simulations is modeled as a viscous drag torque of the form
$TL=−DgωL$
(41)
Where the load torque drag coefficient, Dg, is determined based on
$PL=DgΩL2$
(42)
where PL is a specified load power dissipation level occurring at specified operating speed ΩL. The driveline model parameters used in the simulations are summarized in Table 1.
Table 1

Driveline model parameters

Supercritical case
ωtorsion > ωlateral
Subcritical case
ωtorsion < ωlateral
ParameterValueValue
Shaft length, L3.335 m3.335 m
Shaft outer radius, ro0.0571 m0.0571 m
Shaft inner radius, ri0 m (solid shaft)0 m (solid shaft)
Shaft material shear modulus, Gs27 GPa10 GPa
Shaft proportional damping parameter, ξs2 × 10−4 s2 × 10−4 s
Motor rotational inertia, Jm0.313 kg m20.313 kg m2
Load rotational inertia, JL15.5838 kg m215.5838 kg m2
Load viscous drag coefficient, Dg0.0181 N m s0.0181 N m s
Suspended mass, mL210.8 kg210.8 kg
Suspension stiffness, k1 × 107 N/m20 × 107 N/m
Suspension damping, c1.0 & 1 × 104 (N s)/m20 & 2 × 105 (N s)/m
Torsional natural frequency, ωtorsion105.64 Hz64.3 Hz
Lateral natural frequency, ωlateral34.67 Hz155.02 Hz
Supercritical case
ωtorsion > ωlateral
Subcritical case
ωtorsion < ωlateral
ParameterValueValue
Shaft length, L3.335 m3.335 m
Shaft outer radius, ro0.0571 m0.0571 m
Shaft inner radius, ri0 m (solid shaft)0 m (solid shaft)
Shaft material shear modulus, Gs27 GPa10 GPa
Shaft proportional damping parameter, ξs2 × 10−4 s2 × 10−4 s
Motor rotational inertia, Jm0.313 kg m20.313 kg m2
Load rotational inertia, JL15.5838 kg m215.5838 kg m2
Load viscous drag coefficient, Dg0.0181 N m s0.0181 N m s
Suspended mass, mL210.8 kg210.8 kg
Suspension stiffness, k1 × 107 N/m20 × 107 N/m
Suspension damping, c1.0 & 1 × 104 (N s)/m20 & 2 × 105 (N s)/m
Torsional natural frequency, ωtorsion105.64 Hz64.3 Hz
Lateral natural frequency, ωlateral34.67 Hz155.02 Hz
Considering the baseline system (39), two different model parameter sets are considered which lead to two fundamentally difference cases. These two cases are summarized as
$CaseI,(Supercritical)ωtorsion>ωlateralCaseII,(Subcritical)ωtorsion<ωlateral$
(43)

In this paper, the supercritical case is defined as the condition when the elastic torsional natural frequency is higher that the lateral natural frequency, and the subcritical case has the opposite situation (43). The first group of simulation results shown in Figs. 57 are for the supercritical case. Figures 5(a)5(c), respectively, show the dynamic misalignment, $δ^(t)$, elastic twist, φ(t), and shaft speed, ωL, responses during spinup for the case with δ0 = 11 deg. Also, as a baseline for comparison, the spinup response for the perfectly aligned driveline (δ0 = 0 deg) is shown on the same figure.

Fig. 5
Fig. 5
Close modal
Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal

By comparing the output shaft speed responses, ωL, in Fig. 5(c), it is observed that the angular acceleration rate is initially unaffected by the presence of δ0 since the values ωL rise identically. However, once ωL begins to cross through the vicinity around 3000 rpm (revolutions per minute), the misalignment, δ0 = 11 deg, causes the angular acceleration rate to become reduced. During this same time interval, it is observed that the shaft torsional vibration amplitude in Fig. 5(b) is increasing significantly. Once ωL passes beyond a certain rpm, the acceleration rate returns to the baseline rate and the torsional vibration reduces. Next, Fig. 6 shows the spinup response for the case with nominal misalignment level increased to δ0 = 25 deg. Here, as shown in Fig. 6(c), ωL initially accelerates and then becomes trapped at a certain shaft speed while the case without misalignment continues to accelerate due the applied input torque Tm.

As shown in Figs. 6(a) and 6(b), due to the nonlinear coupled torsion and lateral dynamics of the double U-joint driveline system, once the shaft speed becomes trapped, the torsional and lateral vibration responses enter steady-state limit-cycles which are powered from input energy delivered by the input torque. Next, Fig. 7 shows a compilation of shaft speed spinup responses for five different nominal misalignment levels for the supercritical case.

This figure shows that for misalignments greater than about δ0 > 11 deg, the driveshaft rotational speed, ωL, becomes captured. This rotor speed capture phenomenon is the hallmark of the so-called Sommerfeld effect which has been found in other nonlinear rotor systems [1829]. The key finding in this paper is that the rotor speed capture is induced by driveline misalignment. In all prior studies on Sommerfeld rotor speed capture, this phenomenon has only been shown to be induced by rotor imbalance. This figure also shows that the value of the captured shaft speed is inversely related to the misalignment level. In particular, for the case with δ0 = 15 deg, the maximum speed attained is ωL ≈ 3000 rpm; however, when δ0 = 35 deg, the maximum attained shaft speed is only ωL ≈ 2340 rpm. Note, in each of these simulations, the same value of applied input torque is used.

The next group of simulation results in Figs. 810 are for the subcritical case, see Eq. (43). As seen in Figs. 8(c) and 9(c), shaft speed dwell and capture phenomena similar to that found in the supercritical case also occur in the subcritical case, although at different shaft speed values. Furthermore, the shaft twist limit-cycle vibration behavior shown in Figs. 8(b) and 9(b) is also similar to that of the supercritical case. The main difference between sub and supercritical cases is observed in the lateral vibration responses as shown in Figs. 5(a) and 6(a) (supercritical) and Figs. 8(a) and 9(a) (subcritical). In the supercritical case, ωtorsion > ωlateral, hence lateral vibration becomes excited well before the rotor speed becomes captured.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal
Furthermore, Fig. 10 shows a compilation of shaft speed spinup responses for five different nominal misalignment levels for the subcritical case. The results are all qualitatively similar to the supercritical case shown in Fig. 7. One observation which can be made by the comparisons between the sub and supercritical cases is that the shaft speed capture phenomena are not sensitive to the proximity of the driveline lateral natural frequency in relation to the elastic torsional mode. Even though all modes are coupled in a nonlinear sense, the misalignment-induced speed capture phenomena are primarily related to the elastic torsional mode. Moreover, Figs. 7 and 10 also illustrate that the capture speed value is gradually decreasing when the misalignment angle increases. Finally, it is also observed that the shaft speed capture occurs when ωL is in the vicinity of ½ the first elastic torsional natural frequency of the nominal system. Hence, the approximate misalignment-induced shaft capture speed conditions can be stated as
Misalignment-inducedshaftcapturespeedωL≈ωtorsion2
(44)

Furthermore, Figs. 11 and 12 show the influence of suspension damping on the constant torque spinup response for the supercritical and subcritical cases, respectively. Figures 11 and 12 show that when the suspension damping coefficient c is increased, it primarily affects the lateral response amplitudes as seen Figs. 11(a) and 12(a). However, the suspension damping has little effect on the misaligned-induced speed capture phenomena shown in Figs. 11(c) and 12(c). This indicates that the misalignment-induced speed capture is most strongly related to the double U-joint driveline torsional dynamics.

Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal

Next, Figs. 13 and 14 both show the effect of increasing the value of Tm on the driveshaft spinup response. Here, as shown, larger values of acceleration torque are able to accelerate the rotor speed through the critical zone while the cases with less acceleration torque are more prone to being trapped.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

The next two figures (Figs. 15 and 16) illustrate an approximate procedure to estimate the minimum value of δ0 such that the shaft speed capture will occur.

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal

In this method, the spinup simulation is started with the misalignment level initially set to a value known to cause shaft speed capture based on prior simulation results. Once ωL becomes captured and after the system enters into a steady-state limit-cycle, the value of δ0 is then slowly ramped down to zero in a quasi-steady sense. As δ0 is decreased, it eventually falls below the minimum threshold level required to maintain the captured speed and the shaft speed then begins to accelerate. The value of δ0 where the shaft speed breakaway begins is then considered to be the critical misalignment value, δ0crit, below which ωL speed capture will not occur.

Figure 17 shows the same data displayed in Figs. 15 and 16 and plots the maximum shaft speed as a function of misalignment angle for both the sub and supercritical cases. As the nominal misalignment angle of the driveline is decreased below a specific threshold, the driveshaft speed escapes capture and rises to the maximum speed dictated by the load torque level. Based on the simulation results in Figs. 1517, Table 2 lists the approximate maximum allowable values of driveline misalignment, δ0crit, necessary to avoid shaft speed capture. Qualitatively, these results show that for a fixed value of available input torque, larger values of load torque (i.e., larger Dg values) decrease δ0crit making the driveshaft more prone to nonlinear speed capture.

Fig. 17
Fig. 17
Close modal
Table 2

Maximum allowable driveline misalignment angles to avoid shaft speed capture

Input torque, Tm corresponding to α0 = 2 rad/s2 and load torque, TL, viscous drag coefficient, DgCritical misalignment, δ0crit
Supercritical case
ωtorsion > ωlateral
Subcritical case
ωtorsion < ωlateral
Dg = 0.0181 N m sδ0crit = 11.32 degδ0crit = 13.05 deg
Dg = 0.0363 N m sδ0crit = 10.48 degδ0crit = 12.51 deg
Dg = 0.0543 N m sδ0crit = 9.07 degδ0crit = 11.99 deg
Input torque, Tm corresponding to α0 = 2 rad/s2 and load torque, TL, viscous drag coefficient, DgCritical misalignment, δ0crit
Supercritical case
ωtorsion > ωlateral
Subcritical case
ωtorsion < ωlateral
Dg = 0.0181 N m sδ0crit = 11.32 degδ0crit = 13.05 deg
Dg = 0.0363 N m sδ0crit = 10.48 degδ0crit = 12.51 deg
Dg = 0.0543 N m sδ0crit = 9.07 degδ0crit = 11.99 deg

Finally, for more insight into the dynamics of the misaligned double U-joint driveline spinup response, a time-frequency spectrogram analysis plot showing both the lateral and torsional vibration amplitudes is given in Fig. 18. This figure is computed by performing a series of short time fast Fourier transforms (FFT) on windowed segments of the spinup time history data.

Fig. 18
Fig. 18
Close modal

The time-frequency analysis shows that multiple harmonics at even integer multiples of shaft speed (2ωL, 4ωL, and 6ωL) are involved in the response. It can also be observed that wideband responses are transiently excited during the onset of rotor speed capture. Finally, Fig. 18(b) shows that the relation (44) holds true since the rotor speed becomes captured as the 2ωL harmonic approaches the vicinity the 1st elastic torsion torsional natural frequency (ωtorsion ≈ 64 Hz for the baseline system).

## 4 Numerical Validation in matlab®simscape™ Environment

To further validate these findings and understand the relative importance of torsional and lateral dynamics, a model of the misaligned double U-joint driveline with flexible intermediate shaft considering only torsional dynamics is evaluated within the matlab®simscape™ simulation environment. Figure 19 shows the block diagram of this model. Here again, the driveline is driven by a constant input torque source, and the load torque is a viscous drag torque. Furthermore, within the U-joint blocks, the initial phase angles of U-joints A and B are specified as 0 deg and 90 deg, respectively, which accounts for the proper indexing of the two U-joints. The instantaneous phasing of the U-joints is due to both this initial 90 deg phase plus the angular twist of the intermediate shaft resulting in the dynamic phasing effect discussed in Secs. 2 and 3. Figures 20 and 21 show a comparison between the derived model simulation results from Secs. 2 and 3 and the matlab®simscape™ model. Figure 20 shows the output speed spinup response for several misalignment levels. Here, misalignment-induced shaft speed capture phenomena are observed in both models. In each case, based on the model parameters taken from Table 1, the shaft speed becomes fully captured for misalignment levels greater about δ0 = 14 deg.

Fig. 19
Fig. 19
Close modal
Fig. 20
Fig. 20
Close modal
Fig. 21
Fig. 21
Close modal

Furthermore, Figs. 21(a) and 21(b) show the shaft twist angle responses calculated from the derived simulation model and simscape™ model, respectively. Also, Fig. 21(c) shows the corresponding driveshaft speed response for each case. When δ0 = 15 deg, the derived simulation model predicts that the shaft speed becomes captured at about ωL = 1850 rpm while the simscape™ model predicts speed capture around ωL = 2000 rpm. This difference could be due to the effect of the lateral dynamic misalignment $δ^(t)$ degree-of-freedom, which is included in the derived simulation model but is not accounted for in the simscape™ torsional dynamics model. Moreover, there is the limitation of numerical solvers that are available for the simulation of the simscapeTM model.

Due to the overall similarity between the two models regarding the misalignment-induced speed capture, it is clear that this effect is primarily dominated by the torsion dynamic behavior. For a sufficient level of parallel offset misalignment, driveshaft speed capture can result for shaft speeds in the vicinity of ½ the first out-of-phase torsional mode. The shaft dynamic twist response alters the ideal 90 deg U-joint phasing condition resulting in shaft speed oscillations. Depending on the input torque level, the speed can become trapped rather than accelerating though. It can also be concluded that the lateral compliance makes the system somewhat more prone to speed capture as additional input energy can be absorbed into the lateral motion.

## 5 Conclusions

This work explores the coupled torsion and lateral dynamic behavior of a double U-joint driveline including the effects of angular misalignment, load torque, lateral suspension stiffness, and shaft torsional flexibility. This model utilizes the full kinematics of the double U-joint system including the effects of relative phasing between U-joints. In the parallel offset misalignment condition, the double U-joint configuration with 90 deg relative phasing is commonly used to achieve perfect constant velocity speed transmission. The model and analysis developed in this study accounts for the situation when the 90 deg U-joint phasing becomes disrupted due to shaft elastic twist deformation. This, in fact, is the basic mechanism by which all of the nonlinear torsion and lateral vibration behavior originate in this drivesystem. Without windup, all nonlinear time-varying terms in the system vanish. Another key aspect of this model is, rather than just having a prescribed input shaft speed, the input shaft rotation is considered as a degree-of-freedom. This enables the effect of limited inputs to be evaluated. In this case, the power source is modeled by a rotational inertia term and a constant driving torque. The Sommerfeld behavior is a nonlinear phenomenon observed in some rotor systems being driven through a critical speed when there is not enough power to accelerate the rotor through resonance. Previous studies have shown that rotor speed can become captured when transitioning through natural frequencies due to nonlinear interactions between a non-ideal driving input and rotor imbalance. This paper, for the first time in the literature, demonstrates that this type of rotor speed capture phenomena can also be induced by driveline misalignment. During spinup under constant driving torque, it is found that misalignment-induced shaft speed capture phenomena can occur as the shaft speed approaches ½ the first elastic torsional natural frequency. Depending on the misalignment level, the shaft speed will either dwell near this speed and then pass through, or the speed will become trapped. When the rotor speed becomes trapped the energy being delivered by the motor input torque is sent into sustaining torsional and lateral limit-cycle vibrations rather than increasing the shaft speed. It is found that increasing motor driving torque or reducing load torque can allow the shaft to escape and continue accelerating to a higher speed. The results presented in this paper are particularly important for lightweight high-performance vehicle design where it is more critical to understand the effects of stringent limits on prime mover torque due to size and weight limits. This work can also help enable further reductions in driveline vibration by increasing the understating of how misalignment induces both torsion and lateral vibrations.

## Acknowledgment

This work was supported by the National Science Foundation under Dynamical Systems Program (Grant No. CMMI-0748022).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

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