Sommerfeld effect in a two-disk rotor dynamic system ...

This is a preprint. The original article is published in Meccanica, Springer. Please cite this article as: Bisoi, A., Samantaray, A.K. & Bhattacharyya, R. Meccanica (2017). https://doi.org/10.1007/s11012-017-0757-3

Sommerfeld effect in a two-disk rotor dynamic system at various unbalance conditions Alfa Bisoi · A.K. Samantaray 1 · Ranjan Bhattacharyya

Abstract Near resonance behavior of a shaft-rotor system driven by a motor with limited power gives rise to the Sommerfeld effect wherein the rotor speed gets stuck till a specific level of excitation power is reached. In this paper, a semi-analytical method is used to observe the Sommerfeld effect of a two-disk rotor system driven through a direct current (DC) motor. The effect of relative unbalance position on vibration amplitude of a two disk rotor-shaft system is studied in detail. We observe the change in critical power input for passage through resonance by changing the unbalance positions on the disks. We also investigate the case where two close resonance frequencies interfere with the jump phenomenon associated with the Sommerfeld effect. Prediction of critical power input in complex rotor system is highly essential. Else the system may be destroyed due to large vibration amplitudes generated during operations. In this article, drive and rotor dynamics are modeled together by using multi-energy domain bond graph approach and theoretically obtained steady-state characteristics are validated through transient simulations. Keywords Two-disc rotor · Rotor dynamics · Sommerfeld effect · Bond graph.

1 Introduction The rising need for rotating equipment demands a full phase analyses of all type of rotor systems with various unbalance conditions. Unbalance in rotor system exists due to various well-known reasons. Some of these are blow holes in castings, uneven number and position of bolt holes, parts fitted off-center, machined diameters eccentric to the bearing locations, etc. When the system becomes complex (e.g. multi-disk systems), the chance of residual unbalance is high and also some different unbalance conditions arise. A small unbalance can cause large vibration and sometimes it is enough to destroy the whole system [1]. In rotating systems, more power consumption occurs at resonance zone [2, 3]. This leads the vibration amplitudes to a very high value. The vibration amplitudes and power consumption drop down to a lower value after the rotation speed makes a sudden increase beyond the resonance speed. This jump phenomena is known as the Sommerfeld effect [4]. In a recent article, these authors considered a continuous rotor system driven by a non-ideal drive and studied the importance of the Sommerfeld effect emphasizing actuator sizing [5]. Non-ideal sources are affected by system response [6]. Hence, the consideration of drive dynamics is highly essential in the study of motor driven rotor systems. In some articles, non-ideal drives are used as a power source and drive-rotor interactions are explored in them [7]–[13]. In a multi-degree of freedom system excited by non-ideal drive, undesirable chaotic regimes may exist under certain conditions [14, 15], especially when the system is lightly damped and there is some rotor unbalance. Also, non-ideal systems can exhibit chaos in presence of strong non-linearity such as stick-slip friction, cubic springs, and parametric excitation [16]. Noise in parameters, most notably those in mechanical system which change the system natural frequencies or in the drive system which change the mechanical power output at the resonance, can also lead to chaotic response in the regime of resonance. The synchronization phenomenon is well-known in the scientific community as a spontaneous move towards order and uniformity in a system. In a mysterious fashion, synchronization permeates most of nature’s dynamics and it is explained as a countervailing force to the thermodynamic law of entropy or general disorderness of the universe. Synchronization of physical devices was first described by Dutch physicist Christian 1

Corresponding author. Tel: 91 3222 282998, email: [email protected] Systems, Dynamics and Control Laboratory, Department of Mechanical Engineering, Indian Institute of Technology - Kharagpur, 721302 India.

Huygens for two mechanical pendulum clocks hanging from the same wall, which reportedly went in and out of sync periodically [17]. Thereafter, more such synchronized mechanical systems were studied – such as clocks, metronomes and engineering based rotating systems [18]. When there are two or more non-ideal drives interacting with a common driven or excited system, there is a possibility of synchronization between the nonideal drives even without any direct kinematic coupling between the drives [19−21]. However, ideal drives cannot synchronize because they do not follow the nature’s law. Sources are also vastly affected by residual unbalance and source-system coupling of unbalanced rotor systems is difficult to model. It is found that the bond graph approach is suitable for modeling the interaction dynamics of a non-ideal drive and flexible shaft-rotor system with both internal and external damping in a convenient way [5]. Most of the rotating equipment are complex and they contain multiple disks, however, a few researchers have paid attention towards two-disk unbalanced rotor systems. Critical speeds and periodic response of discrete two-disk rotor system are numerically obtained in [23, 24]. Some researchers have studied the Sommerfeld effect of a DC motor driven two-disk rotor system but ignored the effect of gyroscopic coupling in the system [11]. Coupling behavior between shaft torsion and blade bending of a two disk system is studied in [25]. Numerical results of this work show that material damping plays a major role in the stability of the system. Internal damping parameter estimation of a two-disk rotor system on journal bearing is reported in [26]. In a recent article, a mathematical model of rotor bearing system with two disks is developed [27] in which the simulations of the model are performed by considering lumped mass and the sliding bearing is modeled by a non-linear oil-film force. The results of this article revealed that the vibration amplitude is affected by the angular position of the unbalance forces in the disks and it has considerable impact on oil-film instability of the system. However, the system used an ideal drive and the Sommerfeld effect (drive system coupling) is not considered. The authors believe that this effect may significantly influence the overall performance of this system. In this article, therefore, we develop both discrete and continuous mathematical models of a motor- driven two-disk rotor-shaft system with gyroscopic coupling and unbalance on the disks considering the motor to be a non-ideal drive. The aim of this paper is to predict the jump voltage so as to overcome the critical speed zones for different unbalance conditions obtained by changing the phase difference between the unbalance forces. A numerical approach followed by a bond graph model is used for modeling and simulation of the discrete model of the system. For continuous system, a finite element model is developed. Modal analyses of this system are done by FE analysis. The transient simulations of the continuous system are done by using Bond graph approach for most critical unbalance conditions in the two disks. We begin the article with the description of the model.

2 Discrete two-disk rotor dynamic system A shaft having torsional stiffness and damping in addition to transverse ones with two eccentric disks is attached to a non-ideal source. Two ideal bearings are attached at two ends of the shaft. Disks are asymmetrically placed on shaft, i.e. disk1 is attached at one third length of shaft and disk 2 is located at 0.78th length of shaft length from left end of the shaft. The two disks are equal in size and have the same mass. The translational and rotational dampings are present on the disks and their values are assumed equal in both disks. Before mounting on the rotor shaft, the disks have been balanced individually and the residual unbalance in them is identical. The disks are attached perpendicular to the shaft so that there is no moment unbalance.

y

DC motor

𝜙𝑦

𝜙𝑥 x 𝜃

𝑒1 z

𝑒2

Fig. 1 Two-disk rotor system driven by a DC motor. In the following analysis, unless otherwise specified, we shall neglect the shaft mass. Internal or material damping is present in the shaft, and bearing damping value is also taken into account during modeling of the

system. A DC motor is attached to the system to drive the shaft through a coupling whose effect on the dynamics is not considered.

2.1 Mathematical model Stiffness of the shaft is calculated numerically by influence coefficient method. At static condition, forces and moments are assumed at each disk position and deflection of shaft is calculated at respective points. Let Fx1 , Fy1 ,

M x1 and M y1 be the forces and moments in x, y co-ordinate directions at disk 1 position. Similarly Fx 2 , Fy 2 , M x 2 and M y 2 are the forces and moments at disk 2 position. Deflections and rotations at disk 1 position are denoted with subscript 1 and are expressed as

uφ l11 Fy1 − uφ11 M x1 + uφ l 21 Fy 2 − uφ 21 M x 2 , x1 = ul11 Fx1 + ulφ11 M y1 + ul 21 Fx 2 + ulφ 21 M y 2 , −φ= x1 y1 = ul11 Fy1 − ulφ11 M x1 + ul 21 Fy 2 − ulφ 21 M x 2 and φ y1 = uφ l11 Fx1 + uφ11 M y1 + uφ l 21 Fx 2 + uφ 21 M y 2 .

(1)

Similarly, deflections and rotations at disk 2 location are denoted with subscript 2 and are expressed as

x2 = ul12 Fx1 + ulφ12 M y1 + ul 22 Fx 2 + ulφ 22 M y 2 , −φ= uφ l12 Fy1 − uφ12 M x1 + uφ l 22 Fy 2 − uφ 22 M x 2 , x2 y2 = ul12 Fy1 − ulφ12 M x1 + ul 22 Fy 2 − ulφ 22 M x 2 and φ y 2 = uφ l12 Fx1 + uφ12 M y1 + uφ l 22 Fx 2 + uφ 22 M y 2 .

(2)

In the above expressions, the last two subscripts of influence coefficient u, i.e. subscripts u.ij, indicate influence coefficient at disk j due to force or moment at disk i and i, j may be 1 or 2. In the initial subscripts of u, l and φ indicate direct linear and rotational terms, and lφ and φ l indicate cross-coupled terms. These influence coefficients are given in Appendix-1. The negative signs appear in the equations due to the sign convention used to define the bending moments [28]. Equations (1−2) can be rearranged in matrix form as

(x

1

φ y1 y1 −φx1 x2 φ y= y2 −φx 2 ) U8×8 ( Fx1 M y1 Fy1 − M x1 Fx 2 M y 2 Fy 2 − M x 2 ) , 2 T

T

(3)

where U is the matrix of influence coefficients. The stiffness matrix K d is calculated by inverting the influence coefficient matrix such that = Fk

(F

x1

M y1 Fy1 − M x1 Fx 2 M y 2 Fy 2 −= M x2 ) K d8×8 ( x1 φ y1 y1 −φx1 x2 φ y 2 y2 −φx 2 ) T

T

(4)

in which, the translational, rotational and cross coupled stiffness are represented in the matrix form K l 21 K lφ 21 0 0 0 0   K l1 K lφ 1 K 0 0 Kφ l 21 Kφ 21 0 0   φ l 1 Kφ 1  0 K l1 K lφ 1 K l 21 K lφ 21  0 0 0   Kφ l 1 Kφ 1 0 0 0 0 Kφ l 21 Kφ 21  K d =  K K lφ 12 K l 2 K lφ 2 0 0 0 0   l12  Kφ l 2 Kφ 2 0 0 0 0   K lφ 12 Kφ 12  0 K l12 K lφ 12 K l 2 K lφ 2  0 0 0   Kφ l 2 Kφ 2  0 Kφ l12 Kφ 12 0 0  0

(5)

The elements denoted by sub-scripts l , φ , lφ and φ l for translational, rotational and cross coupled stiffness parameters, respectively. The internal damping forces of shaft are calculated in a rotating frame whose z-axis is aligned with the zaxis of the inertial reference frame. The rotating frame variables are given with subscript r and subscript d indicates the damping forces/moments. The material/internal damping forces, in terms of the proportional damping coefficient denoted by λi , are given as

= Fi

(F

x1r

(

T M y1r Fy1r − M x1r Fx 2 r M y 2 r Fy 2 r = λi K d8×8 x1r φy1r y1r −φx1r x2 r φy 2 r y 2 r −φx 2 r −M x 2r ) d

)

(6) The values of proportional damping coefficient are experimentally obtained and are listed in various handbooks [29]. Note that when the equations are written in the inertial reference frame, due to the rotating material damping force in Eq. (6), the so-called circulatory forces arise. The velocities in rotating reference frame can be written as 𝐕𝐫 = 𝐕 − 𝛚 × 𝐫

(7)

where 𝐕 is velocity vector in inertial reference frame, 𝛚 is the angular velocity of the rotating frame and 𝐫 is the  ˆ . Thus, the material damping forces are displacement vector in the fixed frame. Here, ω = θk

Fi = λi K d Vr = λi K d V − λi K dθkˆ × r

(8)

where the term λi K d V is equivalent to the direct external damping in the fixed reference frame and the term

−λi K dθ × r is equivalent to the stiffness in fixed reference frame. The terms −λi K dθ × r form an antisymmetric matrix and are the cause of non-conservative circulatory forces. In addition to the internal forces due to shaft stiffness and material/internal damping, external damping forces are assumed to act on each disk. External damping occurs due to the presence of the medium in which the rotor operates, such as due to air drag. The forces and moments due to external damping are expressed in the inertial reference frame as T T = − M x 2e ) Fe ( Fx1e M y1e Fy1e − M x1e Fx 2 e M y 2 e Fy 2 e = R e8×8 ( x1 φy1 y1 −φx1 x2 φy 2 y 2 −φx 2 ) d (9) where, assuming same external damping on both the disks,  Re Rlφ 0 0 0 0 0 0  R R 0 0 0 0 0 0  eφ  lφ  0 0 Re Rlφ 0 0 0 0    0 0 Rlφ Reφ 0 0 0 0  R e =  0 0 0 0 Re Rlφ 0 0     0 0 0 0 Rlφ Reφ 0 0   0 0 0 0 0 0 R R  e lφ    0 0 0 0 0 0 Rlφ Reφ  (10) and subscript 𝑒, 𝑒𝜙 and 𝑙𝜙 indicate direct damping on translation, direct damping on rotation, and cross-coupled damping between translation and rotation, respectively. It is assumed here, as in most literature, that there is no cross-coupling in external damping between x and y direction vibrations. The total restoring force on the disks becomes sum of the forces appearing in Eqs. (4), (6) and (9). 𝑦

M e

O

G

𝛿𝑖

M – Mass center G – Geometric center O – Origin of reference

θ 𝑥

Fig. 2 Shaft cross section showing mass center and geometric center of ith disk, i = 1, 2.

T

To describe the unbalance forces on the rotor disks, consider the schematic of the disk shown in Fig. 2 where the geometric center is at G and the mass center is at a small distance (eccentricity) e, which has been exaggerated in the representation. The eccentricities in the rotor disks and their phases (as shown in Fig. 2) are denoted by subscripts indicating the disk number. The positions of the mass center of ith disk is (𝑥𝑚𝑖 , 𝑦𝑚𝑖 ) are defined as

xm1 = x1 + e1 cos(ωt + δ1 ) , ym1 = y1 + e1 sin(ωt + δ1 ) ,

xm 2 = x2 + e2 cos(ωt + δ 2 ) , ym 2 = y2 + e2 sin(ωt + δ 2 ) ,

(11)

where the positions of the geometric center is (𝑥𝑖 , 𝑦𝑖 ) and 𝜃 is the angle between the x-axis and the line passing from the geometric centre through the mass centre of the rotor. The phase angle 𝛿𝑖 is arbitrary for the disks. Taking two successive time derivative of Eq. (11),

xm1 = x1 − e1 sin(θ + δ1 )θ , y m1 = y1 + e1 cos(θ + δ1 )θ ,

xm 2 = x2 − e2 sin(θ + δ 2 )θ , y m 2 = y 2 + e2 cos(θ + δ 2 )θ ,

(12)

and

y1 e1 sin(θ + δ1 )θ 2 + e1 cos(θ + δ1 )θ ,   xm1 =− x1 e1 cos(θ + δ1 )θ 2 − e1 sin(θ + δ1 )θ , ym1 =−

  y2 − e2 sin(θ + δ 2 )θ 2 + e2 cos(θ + δ 2 )θ , xm 2 =− x2 e2 cos(θ + δ 2 )θ 2 − e2 sin(θ + δ 2 )θ , ym 2 =

(13)

The inertial forces are obtained by multiplying the mass center accelerations with corresponding inertia parameters. Here 𝑚𝑖 , 𝐼𝑑𝑖 and 𝐼𝑝𝑖 , 𝑖 = 1,2 denote the mass, diametral and polar moments of inertia of i-th disk. Since the disks are situated perpendicular to the shaft, moment unbalances are absent. In addition to the inertial forces, we also consider the gyroscopic coupling in the disks. Some authors have considered gravity in the analysis of non-ideal systems [15] whereas most have neglected it [21]. Gravity is neglected in this formulation under the assumption that the torque due to gravity is small with respect to other load torques [18, 22], at least in the neighborhood of the first critical speed. Gravity can be neglected in the case of vertical rotors but may have significant influence on the dynamics of horizontal rotors with large eccentricity and/or slow rotational speed. The resulting equations of motion of shaft rotor system may then be written as

 K y − θλ  K φ m1  x1 + ( Re + λi K l1 ) x1 + ( Rlφ + λi K lφ1 )φy1 + λi K l 21 x2 + λi K lφ 21φy 2 + θλ i l1 1 i lφ 1 x1  K y − θλ  K φ + K x + K φ + K x + K φ= m e cos(θ + δ )θ 2 + me sin(θ + δ )θ +θλ i l 21 2 i lφ 21 x 2 l1 1 lφ 1 y1 l 21 2 lφ 21 y 2 1 1 1 1 1

 K y I d 1φy1 + ( Reφ + λi Kφ1 )φy1 − θ I p1φx1 + ( Rlφ + λi K lφ1 ) x1 + λi Kφ 21φy 2 + λi K lφ 21 x2 + θλ i lφ 1 1  K φ + θλ  K y − θλ  K φ +K x +K φ +K x +K φ = −θλ 0, i φ 1 x1 i lφ 21 2 i φ 21 x 2 lφ 1 1 φ 1 y1 lφ 21 2 φ 21 y 2  K x − θλ  K φ m1  y1 + ( Re + λi K l1 ) y1 − ( Rlφ + λi K lφ 1 )φx1 + λi K l 21 y 2 − λi K lφ 21φx 2 − θλ i l1 1 i lφ 1 y1  K x − θλ  K φ + K y − K φ + K y − K φ= m e sin(θ + δ )θ 2 − m e cos(θ + δ )θ , −θλ i l 21 2 i lφ 21 y 2 l1 1 lφ 1 x1 l 21 2 lφ 21 x 2 1 1 1 1 1 1

 K x I d 1φx1 + ( Reφ + λi Kφ 1 )φx1 + θ I p1φy1 − ( Rlφ + λi K lφ 1 ) y1 + λi Kφ 21φx 2 − λi K lφ 21 y 2 + θλ i lφ 1 1  K φ + θλ  K x + θλ  K φ −K y +K φ −K y +K φ = +θλ 0, i φ 1 y1 i lφ 21 2 i φ 21 y 2 lφ 1 1 φ 1 x1 lφ 21 2 φ 21 x 2  K y − θλ  K φ m2  x2 + ( Re + λi K l 2 ) x2 + ( Rlφ + λi K lφ 2 )φy 2 + λi K l12 x1 + λi K lφ12φy1 + θλ i l2 2 i lφ 2 x 2  K y − θλ  K φ + K x + K φ + K x + K φ= m e cos(θ + δ )θ 2 + m e sin(θ + δ )θ , +θλ i l12 1 i lφ 12 x1 l12 1 lφ 12 y1 l2 2 lφ 2 y 2 2 2 2 2 2 2  K y I d 2φy 2 + ( Reφ + λi Kφ 2 )φy 2 − θ I p 2φx 2 + ( Rlφ + λi K lφ 2 ) x2 + λi Kφ 2φy 2 + λi K lφ 2 x2 + θλ i lφ 12 1  K φ + θλ  K y − θλ  K φ +K x +K φ +K x +K φ = −θλ 0, i φ 12 x1 i lφ 2 2 i φ 2 x2 lφ 12 1 φ 12 y1 lφ 2 2 φ 2 y2

(14)

(15)

(16)

(17)

(18)

(19)

 K x − θλ  K φ m2  y2 + ( Re + λi K l 2 ) y 2 − ( Rlφ + λi K lφ 2 )φx 2 + λi K l12 y1 − λi K lφ 12φx1 − θλ i l12 1 i lφ 12 y1  K x − θλ  K φ + K y − K φ + K y − K φ= m e sin(θ + δ )θ 2 − m e cos(θ + δ )θ , −θλ i l2 2 i lφ 2 y 2 l12 1 lφ 12 x1 l2 2 lφ 2 x 2 2 2 2 2 2 2

(20)

and

 K x I d 2φx 2 + ( Reφ + λi Kφ 2 )φx 2 + θ I p 2φy 2 − ( Rlφ + λi K lφ 2 ) y 2 + λi Kφ 2φx 2 − λi K lφ 2 y 2 + θλ i lφ 12 1

(21)

 K φ + θλ  K x + θλ  K φ −K y +K φ −K y +K φ = +θλ 0. i φ 12 y1 i lφ 2 2 i φ 2 y2 lφ 12 1 φ 12 x1 lφ 2 2 φ 2 x2

In the left hand side of Eq. (14), the first is the inertial term, the second and third terms are the direct and cross coupled external damping and the material damping part appearing as external damping as per Eq. (8), the fourth and fifth terms are also due to material damping terms appearing as external damping, the sixth to ninth terms are due to circulatory terms as per Eq. (8), and the rest are due to the shaft stiffness. In addition to similar terms appearing in Eq. (15), a new term due to gyroscopic coupling appears in the third place. All other equations can be likewise explained. Note that the same set of equations can be derived from other approaches such as extended Hamilton’s principle [30], Umbra-Lagrangian [31] and Umbra-Hamiltonian formulations [32] which can take care of non-conservative circulatory forces.

2.2 The Campbell diagram For steady-state analysis, we will consider the case of constant rotor speed 𝜃̇ = 𝜔 and 𝜃̈ = 0. Equations (17) and (21) are multiplied by −1 and then Eqs. (14−21) are recast in state-space form

= x A16×16 x + Bz ,

(22)

[ x1 φy1 y1 −φx1 x2 φy 2 y 2 −φx 2 x1 φ y1 y1 −φx1 x2 φ y 2 y2 −φx 2 ]T , input vector where state vector x = T

 m1e1ω 2 cos(ωt + δ1 ) m1e1ω 2 sin(ωt + δ1 ) m2 e2ω 2 cos(ωt + δ 2 ) m2 e2ω 2 sin(ωt + δ 2 )  , and matrices z= T

1 0 0 0 0 0 0 0  0 0 1 0 0 0 0 0   B1   ,B = B =   with B1 = 2 0 0 0 0 1 0 0 0 B 2    0 0 0 0 0 0 1 0

[0]8×4 , and

[ A1 ]8×8 [ A 2 ]8×8  Α= . [ A 3 ]8×8 [ A 4 ]8×8 

The sub-matrices of A are detailed in Appendix-2. The eigenvalues of the state matrix A appearing in (22) give the stability threshold of the rotor system and its critical speeds. The coefficients of matrix A are dependent on the rotor speed θ = ω and hence the natural frequencies of the system change with the rotor speed. At critical speed, one of the natural frequencies becomes equal to the rotor speed. Sommerfeld effect happens when the rotor speed is near the critical speeds. Therefore, a Campbell diagram of the rotor system is used to determine the critical speeds numerically. The model parameters used for numerical analysis are listed in Table 1. Note that we assume both the rotor disks to be identical, i.e., 𝑚1 = 𝑚2 = 𝑚, 𝐼𝑑1 = 𝐼𝑑2 = 𝐼𝑑 , 𝐼𝑝1 = 𝐼𝑝2 = 𝐼𝑝 and 𝑒1 = 𝑒2 = 𝑒, except the phases 𝛿1 and 𝛿2 . Using the values mentioned in Table 1, and the influence coefficient expressions given in Appendix 1, the numeric values of the coefficients of stiffness matrix shown in Table 2 are calculated. The imaginary and real parts of eigenvalues are then used to obtain the Campbell diagram of the system by plotting these with rotor speed ( ω = θ ) separately in Figs. 3 and 4, respectively. Figure 3 is used to determine the critical speeds and Fig. 4 gives the stability threshold of the rotor system. In these figures, FW and BW indicate forward and backward whirls, respectively. The first two critical speeds obtained from the intersection of 1x line with the 1st and 2nd forward whirl frequencies in Fig. 3 are 142.1 rad/s and 436.9 rad/s, respectively. Note that backward whirl frequencies are not excited unless there is asymmetry in the system, such as in bearings, and hence intersections of 1x line with those are not considered. Theoretically, there are two stability threshold speeds for 1FW and 2FW obtained at zero decay (real part of eigenvalue) in Fig. 4 as 532.5 rad/s and 538.8 rad/s, respectively. Note that the system is

marginally stable when the real parts of all of its eigenvalues are negative and when the real part of any one of its eigenvalues is zero then the system is marginally stable. Here, the 1st forward whirl becomes unstable before the second. Therefore, according to [12, 13], the shaft-rotor system as a whole becomes unstable when the rotor spin exceeds 532.5 rad/s. Table 1. The system parameters, their descriptions and values. Parameter 𝜆𝑖 𝐸 𝑑 𝜌 𝑉𝑠 𝑅𝑚 𝜇𝑚 𝑅𝑏 𝐿

𝑎 𝑏 𝑚 𝐼𝑝 𝐼𝑑 𝑒

𝑅𝑒

𝑅𝑒𝜙

𝑅𝑙𝜙

Description Material damping co-efficient Young’s modulus of shaft Shaft diameter Density of shaft material DC motor supply voltage DC motor armature resistance DC motor characteristic constant Bearing rotational damping coefficient Shaft length

Value 2 × 10−4 Ns/m 2× 1011 N/m2 0.02 m 7800 kg/m3 Variable 5Ω 0.5Nm/A

5 × 10−4 Nms/rad

Length from motor end Length from motor end Mass of the disk Polar moment of inertia of disk Diametral moment of inertia of disk Eccentricity in rotor disk External translational damping coefficient on disks External rotational damping coefficient on disks External cross-coupled damping coefficient on disks

0.9m Disk1 0.3𝑚

Disk2

5 kg 0.15 kgm2 0.075 kgm2 0.001m

0.7m 5 kg 0.15 kgm2 0.075 kgm2 0.001m

1 Nms/rad

1 Nms/rad

40 Ns/m

40 Ns/m

−0.5 Ns/rad

−0.5 Ns/rad

Table 2. Stiffness coefficient values. Coefficient 𝐾𝑙1 𝐾𝑙𝜙1 = 𝐾𝜙𝑙1 𝐾𝜙1 𝐾𝑙21 = 𝐾𝑙12 𝐾𝑙𝜙21 = 𝐾𝑙𝜙12 𝐾𝜙𝑙21 = 𝐾𝜙𝑙12 𝐾𝜙12 = 𝐾𝜙21 𝐾𝑙2 𝐾𝑙𝜙2 = 𝐾𝜙𝑙2 𝐾𝜙2

Value 469057.23 N/m 6544.98 N/rad 31415.92 Nm/rad −294524.31 N/m 58904.86 N/rad −58904.86 N/rad 7853.98 Nm/rad 883572.93 N/m 58904.86 N/rad 39269.90 Nm/rad

Real part of eigenvalue, rad/s

Frequency, rad/s

500 2FW

400 300

1x

2BW

200 1FW

100 1BW

0

0

100

200 300 ω, rad/s

400

500

Fig. 3 The Campbell diagram showing natural frequency vs. shaft speed.

20 10 0

1FW

-10

1BW 2FW

-20

10−3

1FW

0

-30

2FW

2BW

-40 0

−10−3 530

534

538

100 200 300 400 500 600 700 ω, rad/s

Fig. 4 Campbell diagram showing real part of eigenvalues vs. shaft speed. The zero-crossing is zoomed in the inset figure.

2.3 Steady state whirl amplitudes To find the steady state whirl amplitudes from the equations of motion we substitute the assumed solutions − B1 (ω ) sin (ωt= + ξ1 ) , φ y1 B1 (ω ) cos (ωt + ξ1 ) , = x1 A1 ( ω ) cos ( ωt= + ζ 1 ) , y1 A1 ( ω ) sin ( ωt + ζ 1 ) , φx1 =

= x2 A2 ( ω ) cos ( ωt = + ζ 2 ) , y2

B2 (ω ) cos (ωt + ξ 2 )

− B2 (ω ) sin (ωt += ξ 2 ) and φ y 2 A2 ( ω ) sin ( ωt + ζ 2 ) , φx 2 =

in Eqs. (14)−(21) . Thereafter, separating cos (ωt ) and sin (ωt ) parts, 16 equations are obtained containing eight unknowns A1 , B1 , ζ 1 , ξ1 , A2 , B2 , ζ 2 and ξ 2 (See the Appendix-3 for the related equations). Indeed, there are 8 independent equations and others are linear combinations of those. These algebraic equations are then solved to obtain the unknown values at various values of rotor speed. Results for the amplitudes versus rotor speed are shown in Fig. 5 for various values of the orientation angle δ2 and for δ1 = 0 (taken as reference). For this rotationally symmetric system belonging to the special orthogonal group SO(2), the absolute values of δ1 and δ2 are not important in this study, but their difference (δ1−δ2) or phase matters. The variables used to plot the results are steady state normalized transverse amplitudes 𝛼1 = 2𝜔12 𝐴1 (𝜔)⁄(𝜋 2 𝑔) and 𝛼2 = 2𝜔12 𝐴2 (𝜔)⁄(𝜋 2 𝑔), normalized rotary displacement amplitudes 𝛽1 = 1.5𝜋 2 𝐵1 (𝜔) and 𝛽2 = 1.5𝜋 2 𝐵2 (𝜔) and normalized shaft spin speed 𝛾 = 𝜔/𝜔1 , where 𝜔1 is the first critical speed of the rotor-shaft system and 𝑔 is the acceleration due to gravity. The constant multipliers are suitably chosen scales and have no physical meaning. We observe from these figures that the steady state vibration amplitudes are large, in general, in critical speed regions and the phase variation between disks leads to variation in vibration amplitudes, as well. It turns out that the vibration amplitudes gradually decrease and then increase in the 1st critical zone and the trend is reversed in the 2nd critical region as δ2 is increased from zero to 𝜋. Also, the amplitudes do not depend on the 12𝜋 ≡ −𝜋/2 in Fig. 5). The phases ζ 1 , ξ1 , ζ 2 and sign of the phase difference (See results for δ2= 𝜋/2 and δ2=

ξ 2 are not plotted here, but are required for the next step.

10 0 20 10 0 20

𝛼1

10 0 20

0 2

δ2 =𝜋⁄8

1

δ2 =𝜋⁄2

1

δ2 =6𝜋⁄8

1

0 2 0 2

0 20 0 20

0 2

0 20 10 0

1

δ2 =14𝜋⁄8 2

0 2

𝛾

0 2 1 4

0

20 0 40 20 0 40 20

20 0 40

0

1

δ2 =14𝜋⁄8 2

20 0 40 20 3

4

0

0 1

δ2 =𝜋⁄8

0.5

δ2 =𝜋⁄2

0.5

δ2 =6𝜋⁄8

0.5

0

𝛾

0.5

δ2 =𝜋

0 1

0.5

𝛾

δ2 =6𝜋⁄8

0 1

δ2 =12𝜋⁄8

2

δ2 =𝜋⁄2

0 1

0.5

1

δ2 =𝜋⁄8

0 1

δ2 =10𝜋⁄8

δ2 =14𝜋⁄8

δ2 =0

0.5

δ2 =𝜋

0 40

δ2 =12𝜋⁄8

1

3

δ2 =𝜋⁄2

0 40

δ2 =10𝜋⁄8

1

δ2 =12𝜋⁄8

10

20

δ2 =𝜋

1

δ2 =10𝜋⁄8

10

0 40

δ2 =𝜋⁄8

(d) 1

δ2 =0

20

δ2 =6𝜋⁄8

0 2

δ2 =𝜋

10

δ2 =0

1

𝛽1

0 20

(c) 40

𝛼2

δ2 =0

10

0

(b) 2

𝛽2

(a) 20

8

δ2 =10𝜋⁄8

0 1

δ2 =12𝜋⁄8

0 1 0.5 3

4

0

0

1

δ2 =14𝜋⁄8 2

3

4

𝛾

Fig. 5 Steady state normalized transverse vibration amplitudes (a) 𝛼1 , (b) 𝛽1 , (c) 𝛼2 and (d) 𝛽2 versus normalized shaft speed (𝛾).

2.4 Sommerfeld effect characterization of the system We will use the ideal system’s steady-state whirl amplitudes and phases to characterize the Sommerfeld effect in the non-ideal system. Here, the DC motor acts as a source of limited energy. The control variable to change the DC motor mechanical power output is the supply voltage. The voltage at which the amplitude and motor speed

suddenly jump to lesser and higher values, respectively, is called the jump voltage. In the present case, the jump voltage of the system is predicted from power balance between the rotor system and the mechanical power output from the motor. In steady-state synchronous whirl with θ = ω , the power dissipated from the rotor system due to external damping on the disk and the bearing damping is given as C 0 0 0  0 C 0 0  q + R ω 2 , Pd q T  = b 0 0 C 0    0 0 0 C (23) R R   lφ  x1 φ y1 y1 −φx1 x2 φ y 2 y2 −φx 2  and C =  e where qT =  is a damping matrix. Note that R R  lφ eφ  material/internal damping does not dissipate any power during synchronous rotor whirl and hence it is not present in Eq. (23). The mechanical power output from the motor is given as = Pm τ= µm i= µmω (Vs − µmω ) Rm , mω mω

(24)

where, τ m is the torque of motor, im is armature current, µm is motor characteristic constant, Vs is DC motor supply voltage, Rm is the armature resistance, τ m = µ m im , back electromotive force Vb = µ mω and i= m

(Vs −Vb )

Rm . Finally, equating dissipative and motor powers, we obtain a relation between the motor supply

voltage and the steady angular speed (θ = ω ) as

µmω (Vs − µmω ) Rm = A12ω 2 Re + B12ω 2 Reφ + 2 A1 B1ω 2 Rlφ cos (ζ 1 − ξ1 ) + A2 2ω 2 Re + B2 2ω 2 Reφ + 2 A2 B2ω 2 Rlφ cos (ζ 2 − ξ 2 ) + Rbω 2

4

UNSTABLE REGION 𝛿2 = 𝜋⁄8 𝑉𝑠 = 365.7 V

3.5 3

𝛾

2.5 2

1.5

Jump

𝛿2 =14 𝜋⁄8 𝛿2 = 𝜋⁄2 𝑉𝑠 = 607.5 V 𝑉𝑠 = 1398 V

𝛿2 =0 𝑉𝑠 = 281.012 V

1.1

Jump

0.5 0

𝛿2 =𝜋 𝑉𝑠 = 2515 V Jump

1 0.9

1

0

(25)

0.8 0.7 0

1000

𝑉𝑠 , V

50

ω=142.3 rad/s 𝑉𝑠 = 133.012 V

2000

100

150

3000

Fig. 6 Steady state normalized shaft speed (𝛾) vs. voltage (Vs) for e=0.001m. Characterization of Sommerfeld effect is done from (25) and the rotor speeds are numerically obtained for various values of input voltages using the parameters in Table 1 and the amplitude responses at various speeds given in Fig. 5. From the computational viewpoint, it is easier to evaluate values for Vs at different values of ω . This result is shown in Fig. 6. The jump phenomena during rotor coast up are observed at 1st and 2nd critical speed zones. Such double resonance capture (double Sommerfeld effect) has been also shown in [33] for an eccentric motor mounted on flexible foundation with different stiffness in two directions. From Fig. 6, it is seen that the rotor speed linearly increases with increase in supply voltage when the rotor speed is away from resonance speeds. But in 1st and 2nd critical speed zones, the rotor speed gets almost stuck for a certain range of voltage. After a critical supply voltage, the rotor speed jumps to higher values. These jumps for the most adverse cases are shown by dashed vertical lines. The jump voltages at 1st and 2nd critical speed zones are 133.012 V and 281.012 V, respectively, for 𝛿1 = 𝛿2 = 0. The jumps during rotor coast down at 1st and 2nd critical speeds occur at 75 V and 195 V, respectively. Keeping 𝛿1 = 0 and increasing 𝛿2 from 0 to 𝜋 results increase in jump voltages at 2nd critical speed zone and decrease in jump voltage in 1st critical speed zone. The jump voltage near 2nd critical

speed zone becomes 2515V when the phase difference between eccentric masses of both disks is 𝜋 whereas the Sommerfeld effect vanishes in the 1st critical speed zone. Note that if gravity were considered in the model then the maximum load torque due to gravity on the motor is less than 0.1 Nm whereas the motor mechanical torque output at the first critical speed is above 6 Nm. This justifies our assumption to neglect gravity from the model. It is also found from Fig. 6 that at critical speed zones, a single input of supply voltage can give three possible values of rotor speeds. If there is only one unique value of rotor speed near the resonance regime for each supply voltage then the Sommerfeld effect is absent at that critical speed. In effect, Sommerfeld effect is present when there are multiple possible rotor speeds for a single input voltage. If there are three possible rotor speed for a single input voltage then out of the three, the two stable solutions satisfy [34–36]

d dω >0 ( Pm − Pd ) < 0 or dω dVs

(26)

The unstable branch of solutions are, however, not achievable either during coast up or during coast down operations. As a consequence, there is a range of missing speeds just above the resonance frequencies, which is a classic characteristic of Sommerfeld effect. For design purpose, we need to consider the worst case scenario. For the chosen data, the voltage requirement in the second critical speed range is impractical. If the rotor has to be operated at high speed then it has to be balanced more precisely. We have initially considered eccentricity 𝑒 = 0.001m so that the phase influences can be clearly demonstrated. For high speed operation in the second critical speed region, we would henceforth consider 𝑒 = 0.0001m. At this smaller eccentricity, the Sommerfeld effect is characterized as shown in Fig. 7.

4

UNSTABLE REGION 1.1

3.5 3

𝛾

2.5 2

1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 0.9 55 56 57 58 59 60 61 62 63 64

1.5 1

𝛿2 =0

𝛿2 =𝜋

ω=436.83 rad/s 𝑉𝑠 = 203.8 V

0.5 0

0

30

60

90 120 𝑉𝑠 , V

150

180

210

Fig. 7 Steady state normalized shaft speed (𝛾) vs. voltage (Vs) for 𝑒 = 0.0001m.

In Fig. 7, only the results for two critical bounds, i.e., with reference 𝛿1 = 0 and 𝛿2 =0 and 𝜋 are shown. From a design view point, it is impossible to know the eccentricity phases a priori and hence the two extreme cases are considered. Here, 𝛿2 =0 is the extreme case for first critical speed and 𝛿2 = 𝜋 is that for the second critical speed. At the lower eccentricity, it is found from the zoomed part in Fig. 7 that Sommerfeld effect is absent in the 𝑑𝜔 > 0 for all corresponding input voltages and there is always a single solution first critical speed region, i.e., 𝑑𝑉𝑠

for rotor speed for every input voltage. In the second critical speed region, jumps occur at 203.8 V during rotor coast up when 𝛿2 = 𝜋. However, for much smaller values of 𝛿2 (e.g., 𝛿2 = 0), Sommerfeld effect may be absent in the second critical speed region as well. Since the unbalance phase difference is usually unknown and can be any value, 𝛿2 = 𝜋 should be considered during a safe design process. The rotor system parameters are known to affect the severity of Sommerfeld effect [36]. Variation of rotor mass, rotary inertia or shaft stiffness can change the critical speeds. When the critical speed increases, more motor mechanical output power is required for passage through the resonance. Likewise, lesser external damping or larger unbalance leads to larger vibration amplitudes and consequently more power is needed to pass through the resonance. For sufficiently large external damping, the Sommerfeld effect may altogether vanish [36]. The motor parameters such as characteristic constant and armature resistance also change the mechanical power output and have significant influence on the passage through resonance. The motor parameters may be adjusted for sizing the

motor for specific rotor dynamic applications in order to promote smoother transition through the critical speeds with lesser vibration amplitudes [5].

2.5 Bond graph model of two-disk discrete rotor system In order to proceed with transient analysis of the system, we construct the bond graph model of the system by representing the kinematic and force balance relations in terms of bond graph junction structures and constraints. A bond graph model, which is especially suitable for multi-energy domain system dynamics study, easily represents the dynamic coupling between the motor and rotor [37–40]. With change of order of states, the stiffness matrix can be written as

(F

x1

M y1 Fx 2 M y 2 Fy1 −= M x1 Fy 2 − M x 2 ) K d8×8 ( x1 φ y1 x2 φ y 2 y1 −φx1 y2 −φx 2 ) T

T

(27)

where Kd

 K l1 K lφ 1 K l 21 K lφ 21  K   K 4× 4 0 4× 4   φ l1 Kφ 1 Kφ l 21 Kφ 21  . = and symmetric matrix K 0   K l12 K lφ 12 K l 2 K lφ 2   4× 4 K 4× 4     K lφ 12 Kφ l 2 Kφ l 2 Kφ 2 

 R i 4× 4 0 4× 4  and material damping= matrix is given as Rd =  with R i λi K.  0 4× 4 R i 4× 4 

The material damping is modelled in a frame rotating with the shaft. Note that material damping has influence only when the rotor shaft has asynchronous whirl. In synchronous whirl, there is no strain rate in the shaft as seen from the rotating frame and hence material damping has no influence. In this article, rotating frames are denoted by a subscript r post-fixed to the other subscripts. The origin of the rotating frame is fixed at the origin of the inertial reference frame and its z-axis is fixed along that of the inertial reference frame. The linear velocities in the rotating frame are given as

 x   cos θ =     y  r  − sin θ

sin θ   x    − sin θ   +θ  cos θ   y   − cos θ

cos θ   x    − sin θ   y 

(28)

and the rotational velocities in the rotating frame are given as

 φx   cos θ =       φ y  r  − sin θ

sin θ   φx    − sin θ   +θ  cos θ   φy   − cos θ

cos θ   φx    − sin θ   φy 

(29)

Figure 8 shows the bond graph for the considered non-ideal motor-shaft-rotor system. In this simpler model, the mass of the shaft is neglected in comparison with that of the disks. Thus, the resulting model is that of a discrete rotor system. Torsional vibrations of the rotor shaft have no influence on the Sommerfeld effect [41] and hence, are neglected in the model. In the bond graph model, four I elements model the inertia of disks ( m ) and are connected to 1x1 , 1y1 , 1x 2 , 1y 2 junctions. The diametral rotary inertia of both disks are modeled with another four I elements connected to 1−φx1 ,

1φ y1 , 1−φ x 2 , 1φ y 2 junctions. The gyroscopic coupling is modeled using two separate gyrator elements whose gyrator modulus is I pθ . The variable transformers (TF elements) in the model synthesize the kinematic relations given in Eq. (2) to model the eccentricities in the two disks which setup the energetic coupling between the shaft spin and the flexural/structural vibrations. The modulation of the gyrators and transformers by shaft spin speed is not shown in the figure. The two C-fields in the model represent the stiffness matrix Kd. The CTF blocks [5, 41] represent the coordinate transformation from fixed to rotating frame and two R-fields denoted as Ri model the material damping. Two more R-fields denoted as Re model the external damping on the rotor. Two transformers with modulus −1 are used to transform the angular velocity −φxi to φxi (i=1,2 indicating the disk number) before coordinate transformation as per Eq. (29) and the same transformers perform a reverse transformation of bending moments.

Fig. 8 Bond graph model of discrete shaft-rotor disk and DC motor with annotations indicated by block arrows.

R :𝑅𝑚

1

Se :𝑉𝑠

C :Kt

1

I :𝐽𝑚

Disk 1 CG y translation

R :𝑅𝑏

R :Rt

1

0

I :𝐽𝑒

I:m

I:m

Shaft coupling

𝜇𝑚 .. GY

DC Motor

Disk 1 spin and eccentricity

0

TF:

1𝜃̇

TF:

0

CTF

−𝑒sin 𝜃 + 𝛿1 𝑒 cos 𝜃 + 𝛿1

Disk 1 CG x translation

TF: -1

1𝜙̇𝑦1𝑟 1𝜙̇𝑥1𝑟

Shaft stiffness and external damping (bending about x-axis)

Disks 1 and 2 rotations about inertial x axis

I : 𝐼𝑑

Inertial and moving frame coordinate transformation

1𝑦̇ 1𝑟

1𝑥̇ 1𝑟

I : 𝐼𝑑

Disks 1 and 2 rotations about inertial y axis

1𝑦1̇

1−𝜙̇𝑥1

TF :-1

CTF

1𝜙̇𝑦1

1𝑥1̇

TF :-1

CTF

1𝜙̇𝑦2

Re

C

1𝑦2̇

Disks 1 and 2 center x translation

TF:-1

1𝑦̇ 2𝑟

1𝑥̇ 2𝑟

Disks 1 and 2 center y translation

I : 𝐼𝑑

Inertial and moving frame coordinate transformation

1𝜙̇𝑥2𝑟

1𝜙̇𝑦2𝑟

I : 𝐼𝑑

CTF

Shaft stiffness and external damping (bending about y-axis)

1−𝜙̇𝑥2

Material damping (bending in rotating frame x-axis)

Ri

Gyroscopic couplings

GY :𝜇𝜙𝑥𝑦1 GY :𝜇𝜙𝑥𝑦2

Ri

Material damping (bending in rotating frame y-axis)

Re

C

1𝑥2̇

0

TF:

1𝜃̇

TF:

0

Disk 2 CG y translation

I:m

Disk 2 spin and eccentricity

I:m

Disk 2 CG x translation

−𝑒sin 𝜃 + 𝛿1 𝑒 cos 𝜃 + 𝛿1

For the DC motor, the voltage supply is modeled by a source of effort element, i.e. Se : Vs , armature resistance by R element as R: 𝑅𝑚 and motor characteristic constant µ m by a Gyrator element. The total polar moment of inertia 𝐽𝑒 = 2𝐼𝑝 of the rotor system is modeled at 1𝜃̇ junction by an I element and the total bearing resistance by R: 𝑅𝑏 at the same junction. The torsional stiffness K t and torsional damping Rt of the shaft coupling are modeled by C and R elements at 1𝜃̇ junction of the bond graph model.

2.6 Transient analysis of non-ideal discrete rotor system

To validate the bond graph model, the electrical motor part in the bond graph model (Fig. 11) is replaced by a constant source of flow representing the motor/rotor speed. The steady state vibration amplitudes are then obtained at various spin speeds and they match perfectly with the values given in Fig. 5. Maximum amplitudes are also obtained at the predicted forward critical speeds and the system becomes unstable at the predicted stability threshold. Thereafter, the full system model with electrical motor is simulated to validate the Sommerfeld effect characterization done earlier in Fig. 6. Non-dimensional variables used to plot the transient response are defined 2 2 2 2 as 𝛼1∗ = 2𝜔12 �𝑥12 + 𝑦12 �(𝜋 2 𝑔), 𝛼2∗ = 2𝜔12 �𝑥22 + 𝑦22 �(𝜋 2 𝑔), 𝛽1∗ = 1.5𝜋 2 �𝜙𝑥1 + 𝜙𝑦1 , 𝛽2∗ = 1.5𝜋 2 �𝜙𝑥2 + 𝜙𝑦2

and 𝛾 ∗ = 𝜃̇ /𝜔1 . In steady state, 𝛼1∗ ≈ 𝛼1 , 𝛼2∗ ≈ 𝛼2 , 𝛽1∗ ≈ 𝛽1 , 𝛽2∗ ≈ 𝛽2 and 𝛾 ∗ ≈ 𝛾. Transient simulation results of the model gives the vibration amplitudes and rotational speeds for various values of voltage input, as shown in δ= 0 . From these Figs. 9 to 12 which depict jump phenomenon at the 1st and 2nd critical speed zones for δ= 1 2 results, the first and second mode jump voltages are obtained as 133.01 V and 281.01 V, respectively, which are found earlier in Fig. 6. In Figs. 9 and 11 the rotor speed gets stuck at 1st and 2nd critical speeds, i.e., γ * = 1 and 3.075, respectively. Figure 10 shows that at supply voltage of Vs = 133.01 V, the rotor speed jumps from γ * = 1 to γ * = 2.2 (i.e. ω = 312.62 rad/s) and the vibration amplitudes reduce accordingly (See Fig. 5).

8

7

7

𝛾∗

5

4

4

Caught at 1st critical speed

𝛼2∗

𝛼1∗

𝛼1∗

𝛼2∗

6

5 3

3

2

2

1

1

0 0

Passage through 1st critical speed

𝛽1∗

𝛽2∗

6

𝛾∗

𝛽2∗

𝛽1∗

8

50 100 150 200 250 300 350 400 Time, s

Fig. 9 Transient response during coasting up before the jump for Table 1 data at 𝑉𝑠 =133V and 𝛿1 =𝛿2 =0.

0

0

100

200 Time, s

300

400

Fig. 10 Transient response during coasting up after the jump for Table 1 data at 𝑉𝑠 =133.01V and 𝛿1 =𝛿2 =0.

Similarly, in Fig. 12, there is a jump from 2nd critical speed to higher value. However, the rotor speed cannot exceed the stability threshold speed (See Fig. 6) and it gets stuck near the stability threshold value ( γ * = 3.75 ). Such capture of rotor speed at stability threshold is another kind of Sommerfeld effect which has been established previously by various researchers [12, 13, 42]. The rotor speed does not remain constant when the system is caught near the stability threshold speed; it approximately contains two frequencies where one is asynchronous whirl at the natural frequency of the system and the other is the synchronous whirl at the rotor speed. Consequently, the vibration amplitudes are also not constant (See [12, 13] for details). The transient simulation results presented here agree very well with the steady-state analysis. However, in some other cases with low rotor disk polar moment of inertia, there can be premature jumps [5] and quasi-static voltage increment may be necessary to validate steady state results. The transient simulations validate the jump voltages predicted through power balance approach. For other phase shift conditions, i.e. 𝛿2 ≠0, the jump voltages can be likewise validated. Since the most critical case for the passage through second critical speed is at 𝛿2 = 𝜋, we specifically consider that case here. However, considering

the practical range of voltage values and the amount of allowable unbalance at high rotor speed, we use eccentricity 𝑒 = 0.0001m and the corresponding Sommerfeld effect characterization given in Fig. 7.

7

Caught at critical speed

𝛽2∗

5

𝛾∗

𝛾

∗

𝛽2∗

𝛽1∗

6

𝛼2∗

𝛼2∗

𝛼1∗

3

𝛼1∗

4 Passage through 1st critical speed

2 1 0 0

18 Passage Passage through 16 through 1st 2nd critical speed 14 critical speed 12 Caught at stability 10 threshold speed 8 6 4 2 0 0 40 80 120 160 200 240 280 Time, s

𝛽1∗

2nd

20

40

60 80 Time, s

100

120

Fig. 11 Transient response during coasting up before the jump for Table 1 data at 𝑉𝑠 =281 V and 𝛿1 =𝛿2 =0.

Fig. 12 Transient response during coasting up after the jump for Table 1 data at 𝑉𝑠 =281.01V and 𝛿1 =𝛿2 =0.

The transient simulation results reveal that with 𝑒 = 0.0001m and 𝛿2 = 𝜋, the rotor speed gets stuck at the second critical speed ( γ * = 3.075 ) at supply voltage 𝑉𝑠 = 203.7V, but escapes it (See Fig. 13) at 𝑉𝑠 = 203.8V. Also, in Fig. 13, no symptom of Sommerfeld effect is seen while the system transits through the first critical speed.

3.5 3 2.5

𝛾∗

𝛽2∗

𝛽1∗

Steady state speed below stability threshold speed

𝛼2∗

𝛼1∗

2

1.5

Smooth passage through 1st critical speed

1 0.5

Passage through 2nd critical speed

0 0

40

80 120 Time, s

160

200

Fig. 13 Transient response during coasting up after the jump for Table 1 data at 𝑉𝑠 =203.7 V, 𝛿1 =0 and 𝛿2 = 𝜋.

The results presented in this section use a discrete rotor model. However, rotor shaft mass can change the critical speeds of the rotor system and thus, change the Sommerfeld effect characteristics. In the next section, we consider the mass of the shaft in a distributed rotor system model.

3 Distributed rotor-shaft system model with two disks The two disk shaft-rotor system is modeled as a continuous system in which mass of the shaft is included. For the purpose of modal analysis of the purely mechanical rotor dynamic system, finite element (FE) method is used whereas for the simulation of the electro-mechanical rotor-motor system, bond graph model is used.

3.1 Modal Analysis The two disk shaft rotor system is modeled in ANSYS. Each shaft element is modeled by 18 numbers of BEAM188 elements, disks of the system are modeled by two MASS21 elements, external damping on each disk are modeled by MATRIX27 elements and BETAD command is used for assigning internal damping in the

70

Frequency, Hz

2FW

56 2BW

42 1x

28 1FW

14 1BW

0 0

90

180 270 ω, rad/s

360

450

Fig. 14 The Campbell diagram of distributed rotor shaft system showing natural frequency vs. shaft speed.

Real part of eigenvalue, Hz

model. Bearing and other constraints are modeled by giving boundary conditions to the model. Modal analysis of the continuous system is a complicated and time consuming task with Bond graph approach [5] but using FE analysis in ANSYS, modal analysis is easily performed and the Campbell diagram can be obtained directly. The Q-R damped solver is used for modal analysis of the system. For detail description of codes and commands, one can refer ANSYS 14.5 help manual. The 1D beam model in ANSYS has 18 numbers of 1D beam elements and 19 numbers of nodes. Each element has equal length. The disks are positioned at 7th and 15th nodes of the FE model and bearings are modeled at the end nodes. The shaft material density and diameter parameter values given in Table 1 are used. The first and second critical speeds and stability threshold of the system are obtained from Campbell diagrams, shown in Figs. 14 and 15. From the Campbell diagrams, the forward critical speed values are found to be 129.296 rad/s and 414.195 rad/s, respectively, and the stability threshold of the system is 495.3 rad/s. Note that the corresponding discrete rotor model gave first and second forward critical speeds at 142.1 rad/s and 436.9 rad/s, respectively, and the stability threshold speed at 532.5 rad/s. Therefore, the continuous or distributed parameter model of the system, which is much more accurate representation of the system dynamics, gives lower critical speeds and stability threshold values. As a consequence, we expect that motor power (here supply voltage) requirement to pass through the resonance (critical speeds) would be less in the distributed parameter rotor system in comparison to the corresponding discrete parameter rotor system. In the following sections, we verify this hypothesis.

0.002 Zero decay line

0.0012 0.0004

1FW 2FW

-0.0004 -0.0012 -0.002 490

494

498 502 ω, rad/s

506

510

Fig. 15 The Campbell diagram distributed rotor shaft system showing real part of eigenvalue vs. shaft speed.

4.2 Bond graph model of the two-disk distributed rotor system A FE bond graph model of the two disk rotor system is used here with Rayleigh beam elements. Such bond graph model is already available in literature [5, 32, 41] and hence, we do not elaborate on it here. The bond graph model employs 18 number of shaft elements and 19 number of node elements. Out of the 19 numbers of node elements, the left and the right most elements are called bearing elements which implement the geometric boundary conditions, 15 numbers are called hub elements which model the rotor shaft mass, rotary inertia and gyroscopic couplings, and two are disk elements which model disk mass, rotary inertia, gyroscopic coupling and external damping. Each shaft element includes model of shaft bending stiffness and material/internal damping in a rotating frame. As per causality, each shaft element receives nodal velocities (linear and rotational velocities) from the two nodes at its ends and outputs the shear forces and bending moments to be applied on those nodes. The bond graph model developed using the aforementioned sub-models is shown in Fig. 16. The supports are ideal (pin-pin support) and modeled at two ends of shaft with zero sources of flow, i.e. null transverse velocities. Half of the rotary inertia of shaft element is taken at both bearing ends. Each bearing damping is indicated by 𝑅: 𝑅𝑏 /2. The voltage supply to the motor is represented with 𝑆𝑒: 𝑉𝑠 element. This source is linked to a 1-junction where the armature resistance is modeled as 𝑅: 𝑅𝑚 element and the DC motor characteristic constant 𝜇𝑚 is modeled with GY element. The total polar moment of inertia of the rotor shaft is modeled by 𝐼: 𝐽𝑒 element i.e. 1 = Je J ps + I m , where I m is the inertia of rotating part of motor. In Fig. 16, H blocks with subscripts indicate 2 the hub model where H1 is the second node; the first node is the one where left bearing boundary conditions are implemented. D6 and D15 are the disks of the system positioned at the 7th and 15th nodes of the model. Shaft submodels are denoted by S1 to S18 blocks. Each gyrator element with modulus 𝐽𝑝𝑠 𝜃̇/2 models gyroscopic coupling and the unitary gyrator is used for causality matching at sub-system interface [5].

𝟏𝒙̇

SF: 0 I :𝑱𝒅𝒔 /𝟐

I :𝑱𝒅𝒔 /𝟐

SF: 0

𝟏𝒙̇

𝟏𝝓̇𝒚

𝟏𝝓̇𝒚

GY :𝑱𝒑𝒔 𝜽̇/𝟐

S1

H1

… D6

…… D15

……H18

I :𝑱𝒆

𝟏𝝓̇𝒙 𝟏𝒚̇

Motor

I :𝑱𝒅𝒔 /𝟐

SF:0

GY :1

𝟏𝜽̇

R :𝑹𝒃 /2

𝟏𝒊

R :𝑹𝒎

GY:𝝁𝒎 SE :𝑽𝒔

I :𝑱𝒅𝒔 /𝟐

GY :𝑱𝒑𝒔 𝜽̇/𝟐

S18

𝟏𝝓̇𝒙 𝟏𝒚̇

SF:0

R :𝑹𝒃 /2

1

Fig. 16 FE bond graph model of the distributed parameter rotor-disk system with DC motor drive.

4.3 Transient analysis of the two-disk distributed rotor system The bond graph model was first validated with ideal drive simulations where the motor model in Fig. 16 was replaced by a constant source of flow representing rotor spin. Peak vibration amplitudes are obtained exactly at the critical speeds obtained from ANSYS model and the rotor system becomes unstable at the predicted stability threshold speed. In addition, when the density of the shaft material is reduced to almost zero, it is found that all the corresponding simulation results from the distributed parameter model approach those from the discrete rotor model presented in Figs. 9−13. The transient simulations of the validated bond graph model are performed to observe the jump voltages at st δ= 0 and other values in Table 1 reveal that the 1 and 2nd critical speed regions. The simulation results for δ= 1 2 motor/rotor speed gets stuck at the first critical speed for supply voltage 𝑉𝑠 =114.3V and is able to escape the resonance at first critical speed for 𝑉𝑠 =114.4V (See Fig. 17). When the rotor speed escapes a critical speed, the rotor speed jumps to a higher value and all the vibration amplitudes reduce noticeably. Note that discrete system model has predicted the voltage requirement for accelerating through the first critical speed to be 133.01V. Likewise, the critical voltage to escape through the second critical speed is found to be 253.3V (See Fig. 18), which is well below the corresponding limit of 281.01V obtained from the discrete system model. In Fig. 18, the motor/rotor speed is captured at the stability threshold after escaping through the second critical speed.

6

14

10

Passage through 2nd critical speed

𝛾

∗

Caught at stability threshold speed

8

3

𝛼2∗

𝛼1∗

𝛼1∗

𝛼2∗

𝛽1∗

𝛽1∗

𝛽2∗

𝛾∗

4

12

𝛽2∗

Passage through 1st critical speed

5

Passage through 1st critical speed

2

6 4

1

2

0 0

50

100 150 200 250 300 Time, s

Fig. 17 Transient response during coasting up of distributed two-disk rotor-shaft system for Table 1 data at 𝑉𝑠 =114.4V and 𝛿1 =𝛿2 =0.

0

0

50

100 150 200 250 300 Time, s

Fig. 18 Transient response during coasting up of distributed two-disk rotor-shaft system for Table 1 data at 𝑉𝑠 =253.3V and 𝛿1 =𝛿2 =0.

As discussed earlier, the result in Fig. 17 corresponds to the most critical combination of eccentricity phases at the first critical speed. However, the result in Fig. 18 is for the least critical combination of eccentricity phases at the second critical speed. The most critical phase combination at the second critical speed is obtained when the phase difference is 𝜋 radians. However, the voltage requirement to transit through the second critical speed in this case becomes impractical (See Fig. 6) and also, the residual unbalance 𝑒 = 0.001m is unacceptable for such a heavy rotor operating at high speed. As a consequence, the maximum eccentricity is limited here to 𝑒 = 0.0001m. For such a case, the Sommerfeld effect characterization at the most critical phase combination (𝛿1 = 0 and 𝛿2 = 𝜋) with assumption of discrete rotor model is given in Fig. 7. The transient simulation of the distributed parameter two-disk shaft-rotor system model shows that the rotor speed remains caught at the second critical speed for supply voltage 𝑉𝑠 =192.9V and escapes the resonance at second critical speed for 𝑉𝑠 =193V (See Fig. 19) whereas the corresponding discrete rotor model predicted a voltage requirement of 203.7V.

3.5 3

𝛾∗

𝛽2∗

𝛽1∗

Steady state speed below stability threshold speed

2.5 2

Passage through 2nd critical speed

𝛼2∗

𝛼1∗

1.5

Smooth passage through 1st critical speed

1 0.5 0

0

20

40

60 80 Time, s

100

120

Fig. 19 Transient response during coasting up of distributed two-disk rotor-shaft system for 𝑉𝑠 =193V, 𝑒 = 0.0001m, 𝛿1 = 0 and 𝛿2 = 𝜋.

The continuous rotor system with distributed shaft mass has lower critical speeds and the maximum value of steady state vibration amplitude at resonance is lower than that of the discrete system. Lower critical speed and smaller amplitude leads to less dissipated power (See Eq. (25). As the dissipated power of the continuous system is lower than that of the discrete system, then there is high possibility of crossing the resonance with smaller jump voltage for continuous system. Thus, the discrete system model overestimates the motor power requirement for passage through the critical speeds.

4 Conclusions We developed a discrete mathematical model of an unbalanced two-disk rotor system which is driven by a nonideal, i.e. limited power, DC motor drive. We successfully predicted the steady-state speed-voltage curves for different unbalance conditions. Bond graph tool has been used effectively to implement drive dynamics in the discrete as well as continuous rotor models. Numerical and semi-analytical approaches are used for the simulations of discrete and continuous systems. Major changes in the minimum voltage/power requirements to accelerate through critical speeds are observed with the change in the phase difference between the unbalance positions in the disks. Unbalance is the primary contributor to Sommerfeld effect in rotor dynamic systems. In multi-disk rotor, the phase of the unbalances in various disks influences the Sommerfeld effect. It is shown that under certain conditions, Sommerfeld effect may vanish at certain critical speeds and may become severe at some other critical speeds. From a design perspective, the most severe condition of Sommerfeld effect needs to be accounted for. In the considered two disk system, it is seen that the most severe cases of Sommerfeld effect at first and second critical speeds occur for the in phase and out of phase unbalance positions, respectively. The phases can change for different rotor disk masses and locations. Thus, the most critical phase combination has to be determined for each critical speed and the corresponding motor power requirement to operate at a super-critical speed needs to be determined accordingly. Unless the motor is properly sized (minimum power, current, voltage etc. ratings), it may not be possible to operate the rotor at the rated speed. An oversized motor, on the other hand, adds to the manufacturing cost, energy consumption, weight, etc. and is not optimal for the system. Thus, the possibilities of Sommerfeld effect at various critical speeds have to be considered during the design stage itself. It is found that the discrete model of the rotor system marginally overestimates the minimum required voltage (power) input whereas the distributed parameter model is more accurate. However, it is much easier to

handle the discrete rotor model and estimate the Sommerfeld effect characteristics using power balance principle. The distributed parameter model is complex and needs several simulations to estimate the Sommerfeld effect characteristics. Therefore, assuming a small factor of safety and using engineering approximations, the discrete rotor model is proposed as the design tool for electrical motor sizing in multi-disk rotor systems. This article considers the bearings to be ideal. The method proposed in this article can be extended to multidisc rotor systems with non-ideal bearing supports. In such a case, additional rigid-body modes such as cylindrical and conical modes would appear leading to possibility of more critical speeds. In addition, if the supports provide asymmetric stiffness then there is possibility of backward whirls getting excited and Sommerfeld effect can exist for the rigid body modes and backward whirl modes. Bearings can also affect the stability of the rotor dynamics systems. For example, journal bearings provide non-linear support stiffness and damping and exhibit the typical half-frequency whirl behavior which loads the drive system and may show Sommerfeld effect during asynchronous half-frequency whirl. These are some of the open problems which can be taken up as future research.

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Appendix-1 Fx1

Fx2

My1 a1

b2 L

Mx2

a1

b1 a2

Fy2

Fy1 Mx1

My2

b1 b2

a2 L

Fig. A1. Load and deflection location definitions for computation of influence coefficients.

The disc locations are specified as a1 = 0.3 m , b1 = 0.6 m , a2 = 0.7 m , b2 = 0.2 m , with a1 + b1 = a2 + b2 = L . The expressions for influence coefficients in Eqs. (1, 2) are given as follows: = ul11

)

(

a1b1 2 a1 b1 , ulφ 11 L − b12 − a12 = (6a1 L − 3a12 − 2 L2 − a12 ) , = uφ l11 ( L2 − b12 − 3a12 ) , 6 EIL 6 EIL 6 EIL

uφ 11 =

(

− 6a1 L − 3a12 − 2 L2 − 3a12 6 EIL

) ,= u l 22

(

)

a2 b2 2 a2 , ulφ 22 L − b22 − a22 = (6a2 L − 3a22 − 2 L2 − a22 ) 6 EIL 6 EIL

(

)

− 6a2 L − 3a22 − 2 L2 − 3a22 b1  L b2 3 2 2 3 2 2 2 , , ul12 = = uφ l 22 ( L − b2 − 3a2 ) uφ 22 =  ( a2 − a1 ) + a2 L − b1 − a2  6 EIL 6 EIL 6 EIL  b1  ulφ 12 = uφ 12 = uφ l 21 =

(a

3 2

− 3a22 L + 3a2 a12 − 3a12 L + 2a2 L2

φ l12

6 EIL

(

− 6a2 L − 3a12 − 2 L2 − 3a22 6 EIL

), u

) ,= u l 21

(

=

(3L(a − a ) 2

(

2

1

(

)

+ b1 L2 − b12 − 3b1a22 6 EIL

)

),

b2 a1 2 −a1 , ulφ 21 L − b22 − a12 = (6a2 L − 3a22 − 2 L2 − a12 ) 6 EIL 6 EIL

(

)

)

− 6a2 L − 3a22 − 2 L2 − 3a12 b2 . ( L2 − b22 − 3a12 ) , uφ 21 = 6 EIL 6 EIL

Appendix-2 The sub-matrices of [ A ]16×16 in Eq. (22) are detailed as follows:

 −( Re + λi K l1 ) −( Rlφ + λi K lφ1 ) 0 0  m1 m1   −( R + λ K ) −( R + λ K ) −θ I p1 lφ i φ l1 eφ i φ1  0  I d1 I d1 I d1  −( Re + λi K l1 ) −( Rlφ + λi K lφ1 )  0 0  m1 m1   θ I p1 −( Rlφ + λi Kφ l1 ) −( Reφ + λi Kφ 1 )  0 I I d1 I d1  d1 [ A1 ] =  −λi K lφ 12 −λi K l12  0 0  m2 m2   −λi Kφ l12 −λi Kφ 12 0 0  Id 2 Id 2   −λi K lφ 12 −λi K l12  0 0  m2 m2  −λi Kφ l12 −λi Kφ 12  0 0  Id 2 Id 2 

− K l1 m1

− K lφ1

− Kφ l 1

− Kφ 1

I d1

I d1

m1

 K  K θλ θλ i lφ 1 i l1 m1

m1

 K  θλ i φ l1 θλi Kφ 1 I d1

I d1

− K l12 m2 − Kφ l12 Id 2  θλ K

φ 12

l12

m2  θλ K

φ l12

m2  θλ K

Id 2

Id 2

i

I d1 − K l1 m1 − Kφ l 1 I d1

 K − K lφ 12 −θλ i l12 m2 m2  K −K −θλ Id 2  θλ K

i

 K  K −θλ −θλ i lφ 1 i l1 m1 m1  K  K −θλ −θλ i φ l1 i φ1

i

i

lφ 12

φ 12

i

φ l12

Id 2 − K l12 m2 − Kφ l12 Id 2

     I d1   − K lφ 1   m1  − Kφ 1   I d1   K  −θλ i lφ 12   m2   K  −θλ i φ 12  Id 2  − K lφ 12   m2  − Kφ 12   Id 2 

 −λi K lφ 21 −λi K l 21 0 0  m1 m1   −λ K −λi Kφ 21 i φ l 21  0 0  I d1 I d1  −λi K lφ 21 −λi K l 21  0 0  m1 m1   λi Kφ 21 −λi Kφ l 21  0 0 I d1 I d1  [A2 ] =   −( Re + λi K l 2 ) −( Reφ + λi K lφ 2 ) 0 0  m2 m2   −( Rlφ + λi Kφ l 2 ) −( Reφ + λi Kφ 2 ) θ I p 2 0  Id 2 Id 2 Id 2   −( Re + λi K l 2 ) −( Rlφ + λi K lφ 2 )  0 0  m2 m2  θ I p 2 −( Rlφ + λi Kφ l 2 ) −( Reφ + λi Kφ 2 )  0  Id 2 Id 2 Id 2 

[ A 3 ] = [I ]8×8

and [ A 4 ] = [ 0]8×8

− K l 21 m1 − Kφ l 21 I d1  K θλ i

l 21

m1  K θλ i φ l 21

I d1 − Kl 2 m2 − Kφ l 2

Id 2  K θλ i l2 m2  K θλ i φl 2 Id 2

 K   K −θλ − K lφ 21 −θλ i lφ 21 i l 21  m1 m1 m1    − Kφ 21 −θλi Kφ l 21 −θλi Kφ 21   I d1 I d1 I d1    K θλ − K lφ 21  − K l 21 i lφ 21  m1 m1 m1   K  K K θλ − − i φ 21 φ l 21 φ 21  I d1 I d1 I d1   K   K −θλ − K lφ 2 −θλ i lφ 2  i l2  m2 m2 m2   K   − Kφ 2 −θλ i φ l 2 −θλi Kφ 2  Id 2 Id 2 Id 2   K θλ − K lφ 2  − Kl 2 i lφ 2  m2 m2 m2   K θλ − Kφ l 2 − Kφ 2  i φ2  Id 2 Id 2 Id 2 

.

Appendix-3 The 16 numbers of equations after separating the cos (ωt ) and sin (ωt ) parts as detailed in Section 2.3 are −m1 A1ω 2 cos ζ 1 − Re A1ω sin ζ 1 − Rlφ B1ω sin ξ1 + K l1 A1 cos ζ 1 + K lφ1 B1 cos ξ1 + K l 21 A2 cos ζ 2 + K lφ 21 B2 cos ξ 2 = me1ω 2 cos (δ1 ) m1 A1ω 2 sin ζ 1 − Re A1ω cos ζ 1 − Rlφ B1ω cos ξ1 − K l1 A1 sin ζ 1 − K lφ1 B1 sin ξ1 − K l 21 A2 sin ζ 2 − K lφ 21 B2 sin ξ 2 = −me1ω 2 sin (δ1 ) − I d 1 B1ω 2 cos ξ1 − B1 Reφ ω sin ξ1 + ω 2 I p B1 cos ξ1 − Rlφ A1ω sin ζ 1 + Kφ l1 A1 cos ζ 1 + Kφ1 B1 cos ξ1 + Kφ l 21 A2 cos ζ 2 + Kφ 21 B2 cos ξ 2 = 0 I d 1 B1ω 2 sin ξ1 − B1 Reφ ω cos ξ1 − ω 2 I p B1 sin ξ1 − Rlφ A1ω cos ζ 1 − Kφ l1 A1 sin ζ 1 − Kφ 1 B1 sin ξ1 − Kφ l 21 A2 sin ζ 2 − Kφ 21 B2 sin ξ 2 = 0 −m1 A1ω 2 sin ζ 1 + Re A1ω cos ζ 1 + Rlφ B1ω cos ξ1 + K l1 A1 sin ζ 1 + K lφ1 B1 sin ξ1 + K l 21 A2 sin ζ 2 + K lφ 21 B2 sin ξ 2 = me1ω 2 sin (δ1 ) −m1 A1ω 2 cos ζ 1 − Re A1ω sin ζ 1 − Rlφ B1ω sin ξ1 + K l1 A1 cos ζ 1 + K lφ1 B1 cos ξ1 + K l 21 A2 cos ζ 2 + K lφ 21 B2 cos ξ 2 = me1ω 2 cos (δ1 ) − I d 1 B1ω 2 sin ξ1 + B1 Reφ ω cos ξ1 + ω 2 I p B1 sin ξ1 + Rlφ A1ω cos ζ 1 + Kφ l1 A1 sin ζ 1 + Kφ1 B1 sin ξ1 + Kφ l 21 A2 sin ζ 2 + Kφ 21 B2 sin ξ 2 = 0 − I d 1 B1ω 2 cos ξ1 − B1 Reφ ω sin ξ1 + ω 2 I p B1 cos ξ1 − Rlφ A1ω sin ζ 1 + Kφ l1 A1 cos ζ 1 + Kφ1 B1 cos ξ1 + Kφ l 21 A2 cos ζ 2 + Kφ 21 B2 cos ξ 2 = 0

−m2 A2ω 2 cos ζ 2 − Re A2ω sin ζ 2 − Rlφ B2ω sin ξ 2 + K l12 A1 cos ζ 1 + K lφ12 B1 cos ξ1 + K l 2 A2 cos ζ 2 + K lφ 2 B2 cos ξ 2 = me2ω 2 sin (δ 2 ) m2 A2ω 2 sin ζ 2 − Re A2ω cos ζ 2 − Rlφ B2ω cos ξ 2 − K l12 A1 sin ζ 1 − K lφ 12 B1 sin ξ1 − K l 2 A2 sin ζ 2 me2ω 2 cos (δ 2 ) − K lφ 2 B2 sin ξ 2 = − I d 2 B2ω 2 cos ξ 2 − B2 Reφ ω sin ξ1 + ω 2 I p B2 cos ξ 2 − Rlφ A2ω sin ζ 2 + Kφ l12 A1 cos ζ 1 + Kφ12 B1 cos ξ1 0 + Kφ l 2 A2 cos ζ 2 + Kφ 2 B2 cos ξ 2 = I d 2 B2ω 2 sin ξ 2 − B2 Reφ ω cos ξ 2 − ω 2 I p B2 sin ξ 2 − Rlφ A2ω cos ζ 2 − Kφ l12 A1 sin ζ 1 − Kφ12 B1 sin ξ1 − Kφ l 2 A2 sin ζ 2 − Kφ 2 B2 sin ξ 2 = 0 −m2 A2ω 2 sin ζ 2 + Re A2ω cos ζ 2 + Rlφ B2ω cos ξ 2 + K l12 A1 sin ζ 1 + K lφ 12 B1 sin ξ1 + K l 2 A2 sin ζ 2 + K lφ 2 B2 sin ξ 2 = me2ω 2 sin (δ 2 ) −m2 A2ω 2 cos ζ 2 − Re A2ω sin ζ 2 − Rlφ B2ω sin ξ 2 + K l12 A1 cos ζ 1 + K lφ12 B1 cos ξ1 + K l 2 A2 cos ζ 2 me2ω 2 cos (δ 2 ) + K lφ 2 B2 cos ξ 2 = − I d 2 B2ω 2 sin ξ 2 + B2 Reφ ω cos ξ 2 + ω 2 I p B2 sin ξ 2 + Rlφ A2ω cos ζ 2 + Kφ l12 A1 sin ζ 1 + Kφ 12 B1 sin ξ1 0 + Kφ l 2 A2 sin ζ 2 + Kφ 2 B2 sin ξ 2 = − I d 2 B2ω 2 cos ξ 2 − B2 Reφ ω sin ξ1 + ω 2 I p B2 cos ξ 2 − Rlφ A2ω sin ζ 2 + Kφ l12 A1 cos ζ 1 + Kφ12 B1 cos ξ1 + Kφ l 2 A2 cos ζ 2 + Kφ 2 B2 cos ξ 2 = 0