Strojniški vestnik - Journal of Mechanical Engineering 66(2020)4, 266-275 © 2020 Journal of Mechanical Engineering. All rights reserved. D0l:10.5545/sv-jme.2019.6467 Original Scientific Paper Received for review: 2019-11-17 Received revised form: 2020-03-19 Accepted for publication: 2020-03-30 Pitch Stability Analysis for Mechanical-Hydraulic-Structural-Fluid Coupling System of High-Lift Hoist Vertical Shiplift Yang Zhang12 - Duanwei Shi12* - Tong Xiao2 - Ji Zhou2 - Xionghao Cheng2 Wuhan University, Key Laboratory of Hydraulic Machinery Transients, China 2Wuhan University, School of Power and Mechanical Engineering, China Pitch stability of the high-lift wire rope hoist vertical shiplift under dynamic hydraulic levelling has always been an issue of concern. It not only affects working efficiency but also brings significant challenges to operational safety. A new mechanical-hydraulic-structural-fluid (MHSF) coupling dynamics model and a developed semi-analytical method are presented for stable property analysis. The models of the hydraulic levelling subsystem, shallow water sloshing subsystem, the main hoist mechanical subsystem, and the shiplift chamber structure subsystem are built using a closed-loop transfer function, multi-modal theory, and an second-type Lagrangian equation, respectively. Then, a core twenty-one order state matrix of the MHSF coupling system is established using the state-space method. Subsequently, the Lyapunov motion stability theory and Eigen-analysis method are used in combination to judge the pitch stability and analyse the characteristics of the subsystems. Taking four typical high-lift hoist vertical shiplifts as examples, the rationality of the proposed model and method is validated. The results indicate that although the pitch stability safety factor under hydraulic dynamic levelling is reduced by about 15 % to 44 % with respect to hydraulic static levelling, hydraulic dynamic levelling still can meet stability requirements. Furthermore, for the designed 200 m level hoist vertical shiplift, the preliminary design parameters can ensure the pitch stability safety factor under dynamic hydraulic levelling of not less than 1.1. The element most prone to instability is the shallow water sloshing subsystem; increasing the synchronous shaft stiffness or the water boundary layer damping ratio can effectively enhance the pitch stability. Keywords: shiplift, pitch stability, hydraulic levelling, Eigen-analysis, dynamics Highlights • A new mechanical-hydraulic-structural-fluid (MHSF) coupling dynamics model of high-lift hoist vertical shiplift has been established. • A developed semi-analytical method has been presented to analyse the pitch stability characteristics. • Pitch stability under hydraulic dynamic levelling is reduced by 15 % to 44 % with respect to hydraulic static levelling, but hydraulic dynamic levelling will not cause instability. • For the designed 200 m level shiplift, improving the synchronous shaft stiffness by 30 % can enhance the pitch stability by about 6.2 %; doubling the water boundary layer damping ratio can raise the pitch stability about 20.57 %. 0 INTRODUCTION As a core component in a large hydropower station, the high-lift hoist vertical shiplift is pre-developed in China. It uses cables to raise the ship to overcome the water level difference between upstream and downstream. Since the lift height of the 200 m level hoist vertical shiplift (abbreviated as 200 m level shiplift) is nearly double that of the Three Gorges shiplift, the pitch stability problem becomes more complicated. For the vast majority of hoist vertical shiplifts, a hydraulic levelling subsystem has been set up, for levelling the shiplift chamber and balancing the tension of cables. However, once the hydraulic levelling subsystem is started during operation (dynamic hydraulic levelling), it will form a complex mechanical-hydraulic-structural-fluid (MHSF) coupling system. Its dynamic stability and the ability to resist the overturning are not apparent, so only starting the hydraulic levelling subsystem after stopping the shiplift (hydraulic static levelling) is allowed at present [1] and [2]. In order to ensure operational safety and avoid unnecessary downtime, pitch stability under dynamic hydraulic levelling should be studied in depth. Previous research studies on the shiplift system model are incomplete. The ship-water-chamber coupling motion equations are established to analyse the time response [3]. However, the primary hoist subsystem is ignored, which plays an essential role in ensuring pitch stability. Liao [4] and Cheng et al. [5] built a coupling dynamics model of main hoist subsystem, shiplift chamber, and sloshing, but neglected the torque counterweight in the main hoist subsystem; the latter is an essential part in the fully balanced quality system. A nine-degree-of-freedom (DOF) model of the primary hoist subsystem (including hoist, pulley, wire rope, 266 *Corr. Author's Address: School of Power and Mechanical Engineering, Wuhan University, Wuhan, China, dwshi@whu.edu.cn Strojniski vestnik - Journal of Mechanical Engineering 66(2020)4, 266-275 torque counterweight, gravity counterweight, and synchronous shaft), shiplift chamber, and shallow water sloshing was developed by Zhang et al. [6]. The pitch stability problem under hydraulic static levelling can be solved very well, but the hydraulic levelling subsystem remains ignored; it is important to balance and equalize the tension between the wire ropes and adjust the level of shiplift chamber. In the newly proposed MHSF coupling dynamics model, the main hoist subsystem, shiplift chamber subsystem, hydraulic levelling subsystem, and shallow water sloshing subsystem will be considered completely to analyse the pitch stability under hydraulic dynamic levelling. This is a critical issue for current shiplift designers and researchers. For the relatively simple mechanical [7] and [8] and mechanical-fluid [9] system, the Lyapunov function is commonly used to judge the stability. However, it is difficult to construct the Lyapunov function directly from the complex shiplift system. The convergence of the time-domain response is focused on analysing the complex hydraulic-mechanical-electrical-structural system [10]. However, for the MHSF coupling system with a large difference in response frequency, it is too easy to form ill-conditioned equations. The hydraulic levelling subsystem is treated as a first-order system, and the tension equalization characteristics and dynamic system responses are analysed [11]. It is a good attempt, but the simplification of the hydraulic levelling subsystem and the lack of shallow water sloshing subsystems are inappropriate for pitch stability analysis. The Routh-Hurwitz theory [12] and [13] and Lyapunov motion stability theory [14] and [15] have been proved to rate the stability effectively. However, its analysis of the intrinsic connection of the coupling system is powerless. Eigen-analysis and the state-space method are adopted to investigate the stability of the hydraulic-mechanical-electrical coupling mechanism of a hydropower plant [16] to [19]. In the present study, a new MHSF coupling dynamics model for high-lift hoist vertical shiplift will be established. Furthermore, a developed semi-analytical method integrating the Lyapunov motion stability theory and Eigen-analysis will be proposed to judge the pitch stability and analyse the internal connection of the coupling system under dynamic hydraulic levelling. Four typical high-lift hoist vertical shiplifts will be taken as examples to validate the proposed model and method. The subsystem stability priority will be interpreted. In addition, the influence of lift height, synchronous shaft stiffness and subsystem damping ratio on the pitch stability will be researched in detail. 1 METHODS In the process of ascension, the shiplift chamber suspended by cable is inevitably pitched due to the manufacturing error of the hoist diameter, the wire rope diameter and elastic modulus [20]. As shown in Fig. 1, the shiplift chamber is subjected to pitch motion during operation, and the clamping equipment, the safety brake, and the working brake have been released. Fig. 1. A 10D0F (çh a, Zb Z2, Z3, Z4, Z5, z22, A) MHSF coupling dynamics model of high-lift hoist vertical shiplift A new MHSF coupling system is established under the earth-fixed coordinate system (o'x'z'). In order to illustrate the mathematical model, the coupling system is decoupled as the main hoist mechanical subsystem, hydraulic levelling subsystem, rigid shiplift chamber structure subsystem and shallow water sloshing subsystem. The main hoist mechanical subsystem is a mechanical device that suspends the chamber and drives it up and down; it mainly consists of the hoist lift mechanism (Bb B2, B3, B4), synchronous shaft (C, txC), counterweight (mh m2) and wire rope (K, t4K, t2K, t3K, t5K, t6K). 1.1 Shallow Water Sloshing Subsystem Model The shiplift chamber is a typical rectangular water container with low filling depth ratio (H / L < 0.05). Pitch Stability Analysis for Mechanical-Hydraulic-Structural-Fluid Coupling System of High-Lift Hoist Vertical Shiplift 267 Strajniski vestnik - Journal of Mechanical Engineering 66(2020)4, 266-275 Since the variation range of water along the length distribution is much larger than that along the width distribution, a two-dimensional rigid shiplift chamber with a unitary width is adopted [21], as shown in Fig. 2. oxz is the shiplift chamber-fixed coordinate system located at the centre of the mean free surface (¿0). Fig. 2. Shallow water sloshing In the shiplift chamber under pitch motion Previous research [22] has shown that the peak hydrodynamic moment reaches maximum value under pure pitch motion instead of coupled vertical and pitch motions. To ensure the safety, pure pitch motion (more dangerous) will be further investigated. According to the studies of [6] and [23], the modal system of shallow water sloshing and hydrodynamic moment can be written as: S o 20 4Hg 4 H P1 + 2%wmlPl + w[pl = ——a + -a, (1) M =--— pBHLa + -2L2 pBHL3 ¡3,. (2) 2500 n* H 1.2 Hydraulic Levelling Subsystem Model The hydraulic levelling subsystem consists of levelling hydraulic cylinders and a hydraulic control system. If the situation of Fig. 1 appears, the control strategy is keeping the state of A11 point unchanged, and controlling the A21 point to approach A11 through the hydraulic system. The modelling of the hydraulic levelling subsystem mainly includes the proportional Fig. 3. Closed-loop transfer function block diagram of hydraulic levelling system amplifier, the proportional speed regulating valve, valve controlled asymmetric hydraulic cylinder, and shiplift chamber level sensor. Because of the working frequency of proportional amplifiers, the proportional speed regulating valve and shiplift chamber level sensor is much higher than that of the hydraulic cylinder, so these are linearized into proportional links. As shown in Fig. 3, according to the linear flow equation, the flow continuity equation and force balance equation [24], the transfer function of the valve controlled asymmetric hydraulic cylinder G(s) can be derived as: AK G (s ) = ■ kK„. 1 + - V 2 s + = ^h =- \2(1 + n2)j3eA,2 , k ' - MV Mt _ Kce 2 a1 2 (1 + n2 )peM, V, (3a) (3b) 1 + kV 2 (1 + n2 (3c) K A2 1 —+ V k 2 (1 + n1 )pt4 (3d) here n = A2/A b Kce=Kc+Ct and k=t4 K. According to Fig. 3, the closed-loop transfer function of the hydraulic levelling subsystem GB(s) is as follows: G (s) = AK 1 t kK„. q-KKmml s s + (l£hmh + mr )s2 + (ml + l£hmrmh )s + +(1- kKqKaKsvKf mm (4) The input of the hydraulic levelling system is the displacement deviation of the shiplift chamber (aL), and the output is the piston displacement (z22). Thus, the dynamic model can be derived based on Eq. (4). z22 + (hmh + mr) ) + ( + 2^hmrmh) z22 + ' AK ^ 1+- me -KKK mrmhZ22 - (5) 268 Zhang, Y. - Shi, D.W. - Xiao, T. - Zhou, J. - Cheng, X.H. Strojniski vestnik - Journal of Mechanical Engineering 66(2020)4, 266-275 AK 1 t kK, -KKmm\La = 0. cont.(5) 1.3 Main Hoist Mechanical Subsystem Model The kinetic energy and potential energy of the main hoist mechanical and shiplift chamber structure subsystem can be defined as: fe-V m2 m2 2 t2 KR + -- m2 fe*Z + m23 t5 KR V m2 m2 J^* 4 v m J* J6 KR Z 4 - m. + —- m = 0, (p2 =0, -ft =0. (11) (12) (13) T = 1 J1 (l + ti ) + 2 J2 (l ) +1 Ja2 + +1 miZ1 + 12 m2Z22 + 2 m2Z32 + 2 miZ42 + 12 m3Z52 , (6) V = 1K i - Z5 -1 aa + 214K(r9i - z2i - Z5 + 2aa j + 213K( + Z )2 + + 212 K (Rft + Z 2 f + 215 K (2 + Z3 f + + 216K (Rp2 + Z4 f+1C ( -ft f+ 21C ( + ti ).(7) Considering the torsional vibration damping ratio ¿j and wire rope damping ratio ¿g, the dynamic model of the primary hoist mechanical subsystem can be derived using the second type Lagrangian equation. & + l(C + t2 KR2 +13 KR2 + tiC + KR2) ( + J2) (C +12 KR2 + t3KR2 + tiC + KR2) KR Z - (Ji + J2) aKR «V -a ■ tyKR Z (Ji + J2) 5 2(Ji + J2) (Ji + J2) 1 lKR „ C Z2 - (Ji + J2) (Ji + J2) 9 + ft = 0, (8) (t6 KR2 +15 KR2 + C + tlC +14 KR2) (J, + J2) (t6 KR2 +15 KR2 + C + tlC +14 KR2) ft- (J, + J2) 9V t4 KR t4 KR Z t4aKR (J, + J2) 22 (J, + J2) 5 2(J, + J2) UKR „ t6 KR _ C ~ Z A a - (J, + J2) (J, + J2 ) (J, + J2 ) 9 = 0 z, + 2 d^Z! + h*Z, t3KR 3 -9, = 0 m. (9) (10) 1.4 Shiplift Chamber Structure Subsystem Model The rigid shiplift chamber is subjected to pitch and vertical motions during operation. Considering the pitch motion damping ratio ¿p and vertical motion damping ratio ¿v, the model of the shiplift chamber structure subsystem can also be derived by using the second type Lagrangian equation. Z5 + l(K +t4K) | (K +t4K) 7 KR 5--ft - m. Ka t4Ka a - a + 2£p KRa t4 KR t4 K ■ 4 (P2 + Z22 m3 m3 Ka2 S\ (t4 Ka2 4 ). 1 4 (14) J, -a J -a - 2J3 t4 KRa t4Ka (pl + --ft - --z,. 2 J3 2 J3 Ka t4Ka J3 z - m (15) 1.5 Eigen-analysis of MHSF Coupling System With the MHSF coupling system, it is rather difficult to judge the stability by directly solving the time domain signal. The state-space method is adopted to form a core twenty-one order state matrix. Substituting Eq. (2) into Eq. (15), the Eq. (1), Eq. (5) and Eq. (8) to Eq. (15) can be further organized as follows: a + C11a + C 2a - C 3ft + C 4ft - C 5 z22 + +q6 Z5 + + Clspl = 0, (16a) + C2 l(pl + C2 2ft — C2 3Z5 - C2 4a + C2 5Zl + +C2,6 Z2 — C2ft2 = 0, (16b) 1 "'1 Pitch Stability Analysis for Mechanical-Hydraulic-Structural-Fluid Coupling System of High-Lift Hoist Vertical Shiplift 269 2 4 2 3 + m 2 2 + + + + + Strajniski vestnik - Journal of Mechanical Engineering 66(2020)4, 266-275 (2 + C3 i(p2 + C3( - C3 3Z22 - C3,4Z5 + C3,5a + +C3,6Z3 + C3,7Z4 - C3( = 0, (16c) Zi + C4Z + C42 Zi + C43ft = 0, (16d) Z2 + C51Z2 + C5,2 Z2 + C5,3ft = 0, (16e) ¿3 + C6Z + C6:2 Z3 + C6,3ft = 0, (16f) Z 4 + C71Z4 + C7,2 Z4 + C7,3ft = 0, (16 g) Z5 + C8,1Z5 + C8,2 Z5 — C8,3^1 + C8,4a — C8,5^2 + + C86 Z22 = 0, (16h) PJ + C9J$J + C9,2P1 + C93a + C94a - C9ftJ + +C9ft2 - C9,7z22 + C9 8Z5 = 0, (16i) Z22 + C10,1Z22 + C10,2 Z22 + C10,3 Z22 — C10,4a = 0, (16j) here C¡j correspond to coefficient terms in Eq. (15), Eq. (8) to Eq. (14), Eq. (1) and Eq. (5). In order to judge the pitch stability, the Eq. (16a) to (16i) should be reduced to first-order. The state vectors are selected as: x = ft, x2 = ^i, x3 = y2, x4 = x5 = a, x6 = a, x7 = Zp x8 = Zl, x9 = Z2, x10 = ^ x11 = Z3, x12 = ^ x13 = Z3, x14 = Z4, x15 = Z5, x16 = ^ x17 = P^ x18 = Pu x19 = z22, x20 = z22 and x21 = z22. (17) Combining Eq. (17) and Eq. (16a) to (16i), the state equations of the coupling system can be described by: X1 x1 dx =A dt _ X21 _ _ X21 _ = Ax. (18) This is the central concept of this paper because the pitch stability can be analysed by investigating the eigenvalues of A. The eigenvalue equation of matrix A can be derived as follows: = o. (19) Solutions of Eq. (19) are = Re, + Imj (i = 0, 1, ..., 21), which are not only the eigenvalues of matrix A but also the main indicators of pitch stability. According to the Lyapunov motion stability theory, the necessary and sufficient condition for the zero-solution stability of a linear system with constant coefficients is that the real parts of all eigenvalues are negative (Rei < 0). Since the eigenvalues ^ are not needed to be expressed analytically, numerical calculation with Mathematica software can be adopted to judge the pitch stability. The algorithm flow is shown as: (1) For a particular shiplift, substituting structural parameters except a into matrix A, and let a = 0.01i' (i' = 0, 1, 2, ...). (2) For each starting from 0, substituting into the matrix A to find its eigenvalues. If all the eigenvalues have a negative real part, stopping the cycle and output the critical value of a = acd, where output acd is the critical distance of suspension points under hydraulic dynamic levelling, which is a key evaluation indicator of the pitch stability. Otherwise, the loop continues until acd is found. 2 RESULTS AND DISCUSSION 2.1 Validation In order to illustrate the rationality of the presented model and calculation method, four typical high-lift hoist vertical shiplifts are adopted to be analysed, as shown in Table 1, where acs is calculated by [6], nd = a/acd and ns = a/acs. The decrease of acd and acs indicates pitch stability improvement. Comparing the acd and acs in Table 1, it can be seen that the dynamic hydraulic levelling is enabled to significantly reduce the pitch stability (from 15 % to 44 %). The main reason is that the hydraulic levelling subsystem is coupled in series to the original mechanical-structural-fluid system, reducing the system's overall stiffness. The calculated acd is the key parameter that has long limited application of hydraulic dynamic levelling in shiplift technical practice. T x x' = x 21 0 Table 1. The distance of suspension points for four typical high-lift hoist vertical shiplifts Shiplift name h [m] Current status ad [m] acd [m] acs [m] nd ns (ns - nd) / ns [%] GeHeyan shiplift (the first step) 40 In use 23.6 13 8.6 1.8 2.7 34 GeHeyan shiplift (the second step) 82 In use 24.0 19.6 11.6 1.2 2.1 41 GouPitan shiplift (the second step) 127 In use 36.2 33.3 18.5 1.1 2.0 44 200 m level vertical shiplift 200 Design stage 62 56.4 47.9 1.1 1.3 15 270 Zhang, Y. - Shi, D.W. - Xiao, T. - Zhou, J. - Cheng, X.H. Strojniski vestnik - Journal of Mechanical Engineering 66(2020)4, 266-275 Analysing ad and acd in Table 1, ad can guarantee that the shiplift is in stable working condition under hydraulic dynamic leveling. Especially for the 200 m level shiplift, the design parameters can ensure that even if the hydraulic levelling subsystem is started at the most dangerous position, the shiplift chamber is still stable with nd = 1.1. 2.2 Subsystem Stability Priority In order to study the subsystem stability priority under hydraulic dynamic levelling, the vibration frequency change of each subsystem in the process of gradually reducing a is explored by taking the 200 m level shiplift as an engineering example. Table 2. Eigenvalues of 200 m level shiplift coupling system under different distance of suspension points Subsystem Distance of suspension points 62 56.4 13 Hydraulic levelling -5.33±161.80j -5.33±161.80j -5.33±161.80j Hoists -3.99±46.60j -3.99±46.59j -4.00±46.56j -3.93±44.59j -3.93±44.59j -3.93±44.59j -0.17±16.24j -0.17±16.24j -0.17±16.24j Suspended -0.17±16.23j -0.17±16.23j -0.17±16.23j wire ropes -0.47±9.25j -0.46±8.93j -0.51±7.93j -0.38±7.56j -0.38±7.56j -0.38±7.56j Chamber pitch motion -0.58±5.32j -0.54±4.98j -0.47±0.34j Chamber vertical motion -0.23±3.39j -0.23±3.39j -0.23±3.39j Shallow water sloshing -1.33x10-5 ±0.22j -8.18x10-5 ±0.22j 0.33±0.37j Here, j is an imaginary number. The negative real part of eigenvalue indicates that the subsystem is stable, otherwise unstable. The larger the absolute value of the real part, the higher the stability and instability margin. The absolute value of the imaginary part represents the vibration frequency of the subsystem [25]. In the process of reducing a from 62 m (design value) to 56.4 m (stable critical value), the eigenvalue real parts of each subsystem are negative, and the shiplift system is in stable state. However, the obvious change is that stability margin of the shallow water sloshing subsystem decreases rapidly (from -1.33*10-5 to -8.18*10-8). This means that the shallow water sloshing subsystem is the most vulnerable to instability. When a is lower than 56.4 m, the eigenvalue real part of the shallow water sloshing subsystem changes from negative to positive, entering the unstable state. Continuing to reduce a to 13 m, causes the eigenvalue real part to become larger (from -8.18*10-8 to 0.33), and the unstable vibration of shallow water sloshing subsystem becomes more serious. Meanwhile, the vibration frequency of shiplift chamber structure subsystem is rapidly decreased (from 4.98 rad/s to 0.34 rad/s), close to the unstable vibration frequency of the shallow water sloshing subsystem (from 0.22 rad/s to 0.37 rad/s). The vibration frequencies of the two subsystems are close to the resonance state. It can be inferred that the most unstable one (other than shallow water sloshing subsystem) is the shiplift chamber structure subsystem. While a is reduced from 56.4 m to 13 m, the vibration frequencies of the hydraulic levelling subsystem and the main hoist subsystem remain stable, and the eigenvalue real part of the hydraulic levelling subsystem (-5.33) is farther away from the virtual axis than the main hoist subsystem (-4.00 to -0.38). The stability margin of the hydraulic levelling subsystem is greater than the main hoist subsystem. For the 200 m level shiplift, the stability priority from high to low is hydraulic levelling subsystem, the main hoist subsystem, the shiplift chamber structure subsystem, and the shallow water sloshing subsystem. 2.3 Influence of Lift Height Taking the 200 m level shiplift (a = 62 m) under hydraulic dynamic levelling as an example, variations of acd and subsystem vibration frequencies with lift height are analysed in Fig. 4. It can be seen from Fig. 4a that with the increase of lift height (0 m to 200 m), acd becomes smaller (56.4 m to 41.2 m), which means that pitch stability is gradually enhanced. The main reason is that the length of suspended wire ropes becomes shorter with the rise of lift height; thus, the stiffness of suspended wire ropes increases rapidly, and pitch stability is enhanced. Figs. 4b to 4f show that the vibration frequencies of hydraulic levelling subsystem, main hoist subsystem, shiplift chamber subsystem, and shallow water sloshing subsystem change stably in their respective regions. For the 200 m level shiplift, pitch stability under dynamic hydraulic levelling can be guaranteed in the whole operation process. Pitch Stability Analysis for Mechanical-Hydraulic-Structural-Fluid Coupling System of High-Lift Hoist Vertical Shiplift 271 Strajniski vestnik - Journal of Mechanical Engineering 66(2020)4, 266-275 1 ^Critical distance of suspension points . 1 4) (56.4-41.2) a) 20 40 60 80 100 120 140 160 180 200 Lift height [m] jj 450 ■o ¿4 CJ |-300 g tfn c 250 2 -Q >151 b) ^-Hydraulic leveling suDsystem (161.801~412.221) 0 20 40 b) Lift height [m] e) f) Fig. 4. Variations of a) critical distance of suspension points, b) hydraulic levelling subsystem vibration frequency, c) hoists vibration frequencies, d) wire ropes vibration frequencies, e) pitch and vertical motion vibration frequencies, f) shallow water sloshing vibration frequency with lift height c) 2.4 Influence of Synchronous Shaft Stiffness The synchronous shaft system ensures the synchronization between the hoists and guarantees the shiplift smooth operation. Table 3 shows the variation of acs and acd with synchronous shaft stiffness increases. Table 3. Effect of synchronous shaft stiffness on the pitch stability of the 200 m level shiplift Critical distance Synchronous shaft stiffness increment of suspension points DeSlgn 5 % 10 % 15 % 20 % 25 % value 30 % acd [m] 56.4 55.7 55.0 54.4 53.8 53.3 52.9 acs [m] 47.9 47.2 46.6 46.1 45.5 45.1 44.7 Table 3 displays that the pitch stability is enhanced with the synchronous shaft stiffness increases, regardless of hydraulic dynamic or static levelling. For the hydraulic static levelling, the pitch stability is enhanced by 6.7 % when the synchronous shaft stiffness is increased by 30 %. For the dynamic hydraulic levelling, while the synchronous shaft stiffness raises by 30 %, pitch stability enhances about 6.2 %. It can be inferred that pitch stability of 200 m level shiplift can be effectively improved by increasing the synchronous shaft stiffness, whether it is hydraulic static or dynamic levelling. 2.5 Influence of Damping Ratio The essence of stability is that system energy decays over time, and finally reaches a state of equilibrium with minimum energy. The subsystem damping plays an essential role in energy dissipation and attenuation. Table 4 analyses the influence of the subsystem damping ratio on the pitch stability. 272 Zhang, Y. - Shi, D.W. - Xiao, T. - Zhou, J. - Cheng, X.H. Strojniski vestnik - Journal of Mechanical Engineering 66(2020)4, 266-275 Table 4. Effect of subsystem damping ratio on the pitch stability of the 200 m level shiplift Series & $h acd [m] 1 0.1 0.01 0.005 0.05 0.00024 0.03 56.4 2 0.1 0.01 0.005 0.05 2x0.00024 0.03 44.8 3 0.1 0.01 0.005 0.05 0.00024 2x0.03 56.39 where $g refers to [2], $ and $ are adopted from [4], $v comes from [6], is calculated according to [26] and ;h is obtained by Eq. (3 c). Comparing series 1 and series 2, when is doubled, pitch stability safety factor is raised from 1.1 (i.e. 62 / 56.4) to 1.4 (i.e. 62 / 44.8). That is, increasing can effectively improve the pitch stability. The main reason is that the weakest stability link in the MHSF coupling system is the shallow water sloshing subsystem, and the increase of the sloshing damping ratio can accelerate the convergence and enhance stability. When the shiplift works, because of the existence of ship in the chamber, $w is greatly increased, and sloshing is suppressed. Compared with the ship-free state, pitch stability is improved, and the safety factor in the actual operation process is higher than the calculated value 1.1. Comparing series 1 and series 3, while $h is doubled, acd is essentially unchanged. That is, $h hardly affects the pitch stability. The main reason is that the hydraulic levelling subsystem has the highest stability priority in the coupling system. Increasing is equivalent to raise the upper limit of stability but does not affect the lower limit of stability. 3 CONCLUSIONS A new MHSF coupling dynamics model and a developed semi-analytical method are presented to investigate the pitch stability under hydraulic dynamic levelling. Taking four typical high-lift hoist vertical shiplifts as examples, the reliability of the proposed model and method is illustrated. The subsystem stability priority and influence factors on the pitch stability are analysed in detail. Pitch stability of the 200 m level shiplift under dynamic hydraulic levelling is focused. The key observations are summarized as follows: 1. Based on the closed-loop transfer function, multi-modal theory and second type Lagrangian equation, the models of hydraulic levelling subsystem, shallow water sloshing subsystem, the main hoist mechanical subsystem and shiplift chamber structure subsystem are built, respectively. Subsequently, a core 21 order state matrix of the MHSF coupling system is proposed using state-space method. 2. A developed semi-analytical method integrating the Lyapunov motion stability theory and the Eigen-analysis method is proposed to judge the pitch stability and analyse subsystem characteristics. The results indicate that pitch stability under dynamic hydraulic levelling is reduced by 15 % to 44 % with respect to hydraulic static levelling, but dynamic hydraulic levelling will not cause instability. For the designed 200 m level shiplift, the critical distance of suspension points is 56.4 m, and the design parameters can guarantee the pitch stability safety factor under dynamic hydraulic levelling not less than 1.1. 3. In the MHSF coupling system, the stability priority from high to low is the hydraulic levelling subsystem, the main hoist subsystem, shiplift chamber structure subsystem, and the shallow water sloshing subsystem. 4. For the designed 200 m level shiplift, pitch stability enhances significantly (maximum 26.98 %) as the lift height increases (0 m to 200 m). Improving the synchronous shaft stiffness by 30 % can enhance the pitch stability about 6.2 %. Doubling the water boundary layer damping ratio can raise the pitch stability about 20.57 %. 4 ACKNOWLEDGMENTS This study was supported by the National Key R&D Program of China (No. 2016YFC0402002), and the authors gratefully acknowledge this support. 5 NOMENCLATURES a distance of suspension points, [m] ad distance of suspension points given by designers, [m] acd critical distance of suspension points under hydraulic dynamic levelling, [m] acs critical distance of suspension points under hydraulic static levelling, [m] A state matrix of MHSF coupling system A1 non-rod cavity area of hydraulic cylinder, [m2] A2 rod cavity area of hydraulic cylinder, [m2] b, eigenvalue equation coefficients B width of shiplift chamber, [m] B1 hoist at upstream B2 pulley at upstream B3 pulley at downstream B4 hoist at downstream Pitch Stability Analysis for Mechanical-Hydraulic-Structural-Fluid Coupling System of High-Lift Hoist Vertical Shiplift 273 Strajniski vestnik - Journal of Mechanical Engineering 66(2020)4, 266-275 c center point of shiplift chamber bottom C equivalent synchronous shaft stiffness, [N-m-rad-1] Cjj coefficient expressions Ct total leakage coefficient related to load pressure, [m^N-^s-1] g gravity acceleration, [m-s-2] G(s)transfer function of valve controlled asymmetric hydraulic cylinder GB(s) closed-loop transfer function h lift height, [m] H water depth of shiplift chamber, [m] J1 equivalent inertia moment of hoist, [kg-m2] J2 equivalent inertia moment of pulley, [kg-m2] J3 equivalent inertia moment of shiplift chamber, [kg-m2] k equivalent load wire rope stiffness, [N-m-1] K equivalent stiffness of wire ropes at upstream, [N-m-1] Ka proportional amplifier gain Ksv flow gain of proportional speed regulating valve Kf sensor gain Kq flow gain Kce total pressure flow coefficient including leakage, [m5-N-1-s-1] Kc pressure-flow factor (the change rate of the load flow rate relative to the load pressure when the valve spool displacement is constant), [mS-N-^s-1] L length of shiplift chamber, [m] m1 half weight of torque counterweight, [kg] m2 half weight of gravity counterweight, [kg] m3 total weight of shiplift chamber, [kg] M hydrodynamic moment with respect to point c, [N-m] Mt equivalent mass of piston and load wire rope, [kg] n effective area ratio of two cavities of hydraulic cylinder nd pitch stability safety factor under hydraulic dynamic levelling ns pitch stability safety factor under hydraulic static levelling R radius of hoists and pulleys, [m] t1 torsional stiffness ratio of synchronous shaft and motor drive shaft t2 influence factor on torque counterweight suspended wire rope at upstream t3 influence factor on gravity counterweight suspended wire rope at upstream t4 influence factor of elasticity modulus tolerance t5 influence factor on gravity counterweight suspended wire rope at downstream t6 influence factor on torque counterweight suspended wire rope at downstream T kinetic energy, [J] V potential energy, [J] Vt total volume of two cavities of hydraulic cylinder, [m3] x state vector matrix z22 piston displacement, [m] Z1 torque counterweight vertical displacement at upstream, [m] Z2 gravity counterweight vertical displacement at upstream, [m] Z3 gravity counterweight vertical displacement at downstream, [m] Z4 torque counterweight vertical displacement at downstream, [m] Z5 shiplift chamber vertical displacement, [m] a pitch angular displacement, [rad] P1 first order modal function of sloshing elastic modulus of oil, [Pa] first order water boundary-layer damping ratio 4 hydraulic damping ratio ; torsional vibration damping ratio ;g wire rope damping ratio ;p pitch motion damping ratio ;v vertical motion damping ratio Z0 mean free surface p water density, [kg-m-3] rotating angle displacement of hoist and pulley at upstream, [rad] rotating angle displacement of hoist and pulley at downstream, [rad] m1 first order natural frequency of fluid in the shiplift chamber, [rad-s-1] mh hydraulic natural frequency, [rad-s-1] mr turning frequency of inertial link, [rad-s-1] 6 REFERENCES [1] Yu, Q., Fang, X. 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