In my investigation of rotary hydraulic cylinders, I focused on the behavior of the two-stage screw gear pair because this transmission element directly governs the efficiency, load capacity, and durability of the entire actuator. The screw gear pair converts a linear piston motion into a rotary output motion through large helix angles, and during this process a helical clearance oil film forms between the mating screw gear surfaces. I treated this oil film as a critical functional component rather than a passive gap, because its pressure field, velocity field, flow rate, stiffness, and dynamic pressure determine whether the screw gear pair can operate under high-pressure, low-speed, and heavy-duty conditions. My objective was to quantify how the circumferential clearance of the screw gear pair influences the oil film performance and to identify the clearance and hydraulic oil viscosity that yield the best overall behavior.
The rotary hydraulic cylinder that I studied consists of a cylinder body, an output shaft, a hollow screw, a fixed sleeve, front and rear end covers, and sealing elements. The fixed sleeve has an internal thread that engages the external thread of the hollow screw, forming the first-stage screw gear pair. The internal thread of the hollow screw engages the external thread of the output shaft, forming the second-stage screw gear pair. I noted that the internal and external threads of the hollow screw have opposite helix directions. This opposite-handed arrangement is essential because it allows a relatively short axial stroke of the hollow screw to produce a large output rotation angle. The cylinder body is divided by the hollow screw into two pressure chambers of unequal volume. These chambers are sealed by sealing rings and connected to inlet and outlet ports. By controlling the pressure difference and flow rate of the hydraulic oil, I can regulate the output torque and rotational speed of the output shaft with good precision.
When the first port acts as the inlet and the second port acts as the outlet, the left chamber pressure exceeds the right chamber pressure. The hollow screw is pushed to the right. Under the action of the first-stage screw gear pair, the hollow screw simultaneously translates and rotates clockwise when viewed from left to right. At the same time, under the action of the second-stage screw gear pair, the internal thread of the hollow screw contacts the external thread of the output shaft and drives the output shaft to rotate clockwise. Through the coordinated action of the two-stage screw gear pair, a small relative stroke can generate a large output rotation angle. When the first port acts as the outlet and the second port acts as the inlet, the pressure relationship reverses, and the output shaft rotates in the opposite direction. Therefore, by controlling the pressure difference between the hydraulic ports, I can control the output rotation direction and speed of the rotary hydraulic cylinder.
For my analysis, I simplified the oil film model while preserving the dominant physical effects. I assumed that the oil film is an equal-clearance layer uniformly distributed along the screw gear pair, and that the hydraulic oil completely fills the screw gear clearance to form a continuous oil film. I also assumed that the output shaft has no eccentricity relative to the hollow screw, although in practice small eccentricity may occur. The two-stage screw gear pair oil film has a minimum diameter of 140 mm and a maximum diameter of 150 mm. The radial root clearance and radial tip clearance are both 0.65 mm. The number of teeth is 49. I defined the circumferential oil film clearance as \(h\) and established six three-dimensional oil film models with \(h = 0.05, 0.10, 0.15, 0.20, 0.25,\) and \(0.30\) mm. These models allowed me to evaluate the influence of the screw gear clearance on the oil film characteristics in a systematic manner.
The geometric parameters that I used for the two-stage screw gear pair are summarized in Table 1. These parameters define the baseline for all subsequent simulations and ensure that the screw gear oil film models remain comparable across different clearance values.
| Parameter | Value | Unit |
|---|---|---|
| Minimum oil film diameter | 140 | mm |
| Maximum oil film diameter | 150 | mm |
| Radial root clearance | 0.65 | mm |
| Radial tip clearance | 0.65 | mm |
| Number of teeth in screw gear pair | 49 | — |
| Circumferential clearance cases | 0.05, 0.10, 0.15, 0.20, 0.25, 0.30 | mm |
| Inlet pressure | 21 | MPa |
| Outlet back pressure | 1.5 | MPa |
| Inner wall rotational speed | 60 | r/min |
The flow of hydraulic oil inside the screw gear clearance obeys the fundamental conservation laws of fluid mechanics. I used the continuity equation, the momentum equation, and the energy equation to describe the oil film behavior. For an incompressible fluid, the mass conservation law requires that the net mass efflux from a control volume is zero. In differential form, the continuity equation for the screw gear oil film is
$$ \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} = 0 $$
where \(u_x\), \(u_y\), and \(u_z\) are the velocity components in the \(x\), \(y\), and \(z\) directions, respectively. This equation ensures that the hydraulic oil filling the screw gear clearance satisfies local mass balance. Because I treated the hydraulic oil as an ideal viscous fluid, the continuity equation is an essential constraint for the screw gear oil film flow field.
The momentum conservation equation states that the sum of external forces acting on a fluid system equals the rate of change of momentum of the system. In the Cartesian coordinate system, the momentum equations for the screw gear oil film can be written as
$$ \frac{\partial (\rho v_x)}{\partial t} + \nabla \cdot (\rho v_x \mathbf{v}) = -\frac{\partial p}{\partial x} + \frac{\partial \tau_{xx}}{\partial x} + \frac{\partial \tau_{yx}}{\partial y} + \frac{\partial \tau_{zx}}{\partial z} + F_x $$
$$ \frac{\partial (\rho v_y)}{\partial t} + \nabla \cdot (\rho v_y \mathbf{v}) = -\frac{\partial p}{\partial y} + \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \tau_{yy}}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} + F_y $$
$$ \frac{\partial (\rho v_z)}{\partial t} + \nabla \cdot (\rho v_z \mathbf{v}) = -\frac{\partial p}{\partial z} + \frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \tau_{zz}}{\partial z} + F_z $$
where \(\rho\) is the density, \(p\) is the pressure, \(\tau_{ij}\) are the viscous stress tensor components, \(F_x\), \(F_y\), and \(F_z\) are the body force components per unit mass, and \(\mathbf{v}\) is the velocity vector. These equations balance inertia, pressure, viscous forces, and body forces within the screw gear oil film.
The energy conservation equation describes the transport and conversion of energy in the screw gear oil film. I used the following form:
$$ \frac{\partial (\rho T)}{\partial t} + \frac{\partial (\rho u_x T)}{\partial x} + \frac{\partial (\rho u_y T)}{\partial y} + \frac{\partial (\rho u_z T)}{\partial z} = \frac{\partial}{\partial x}\left( \frac{K}{c_p} \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y}\left( \frac{K}{c_p} \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z}\left( \frac{K}{c_p} \frac{\partial T}{\partial z} \right) + S_T $$
where \(T\) is the temperature, \(c_p\) is the specific heat capacity, \(K\) is the thermal conductivity, and \(S_T\) is the viscous dissipation source term. This equation accounts for the heat generated by viscous shear in the screw gear clearance and the heat conducted through the hydraulic oil. In my simulations, I treated the near-wall region as adiabatic, meaning that no heat exchange occurs through the solid walls. The heat generated by the motion of the screw gear pair is therefore carried away entirely by the hydraulic oil.
Dynamic pressure is a useful indicator of the lubrication effectiveness of the screw gear oil film. I calculated the dynamic pressure as
$$ p_d = \frac{1}{2}\rho v^2 $$
where \(\rho\) is the hydraulic oil density and \(v\) is the local flow velocity. A higher dynamic pressure indicates a stronger kinetic energy contribution to the oil film pressure and generally corresponds to better dynamic lubrication of the screw gear pair. I also used the volumetric flow rate, the load-carrying stiffness, and the transmitted torque as performance metrics. The volumetric flow rate through the screw gear clearance is
$$ Q = \int_A \mathbf{v} \cdot d\mathbf{A} $$
where \(A\) is the cross-sectional area normal to the flow direction. The stiffness of the oil film is defined as
$$ k = \frac{F}{h} $$
where \(F\) is the force exerted by the hydraulic oil on the oil film wall and \(h\) is the screw gear clearance. The transmitted torque is obtained from the surface integral of the moment of the viscous and pressure forces:
$$ T = \int_A \mathbf{r} \times (\boldsymbol{\tau} \cdot \mathbf{n}) \, dA $$
where \(\mathbf{r}\) is the position vector, \(\boldsymbol{\tau}\) is the stress tensor, and \(\mathbf{n}\) is the outward unit normal vector. These quantities allowed me to connect the local flow field in the screw gear oil film to the macroscopic performance of the rotary hydraulic cylinder.
I built the three-dimensional oil film models in SolidWorks and imported them into Workbench for meshing. Because the oil film clearance is very small compared with the overall screw gear dimensions, I used a sweep method to generate the mesh and added face sizing to refine the mesh in the screw gear clearance. I also used proximity capture to control the mesh size and number of elements near the screw gear surfaces. The mesh quality directly affects the convergence, efficiency, and accuracy of the solution. The fluid domain lies between the inner wall and the outer wall. The final mesh contained up to 2,475,360 elements. I selected the \(\kappa\)-\(\varepsilon\) turbulence model because it offers good stability and low computational cost for this type of confined screw gear flow. I used No. 32 hydraulic oil as the working fluid, with density \(\rho = 889\ \mathrm{kg/m^3}\), specific heat capacity \(c_p = 1800\ \mathrm{J/(kg \cdot K)}\), kinematic viscosity \(\nu = 32\ \mathrm{mm^2/s}\), and thermal conductivity \(K = 0.14\ \mathrm{W/(m \cdot K)}\).
The boundary conditions for the screw gear oil film simulations are listed in Table 2. I set the inner wall as a rotating moving wall with a rotational speed of 60 r/min, while the outer wall was fixed. I used the finite volume method with a three-dimensional structured mesh. The convection terms were discretized with a second-order upwind scheme, and the diffusion terms and pressure gradient were also discretized with second-order upwind schemes. The pressure-velocity coupling was handled with the Coupled algorithm to reduce the solution time. The inlet boundary condition was a pressure inlet at 21 MPa, and the outlet boundary condition was a pressure outlet with a back pressure of 1.5 MPa. The near-wall region was adiabatic, with no heat exchange. These settings ensured that the simulation reflected realistic operating conditions for the screw gear pair.
| Boundary or solver setting | Value or method |
|---|---|
| Inlet boundary type | Pressure inlet |
| Inlet pressure | 21 MPa |
| Outlet boundary type | Pressure outlet |
| Outlet back pressure | 1.5 MPa |
| Inner wall motion | Rotating wall, 60 r/min |
| Outer wall motion | Stationary |
| Turbulence model | \(\kappa\)-\(\varepsilon\) |
| Pressure-velocity coupling | Coupled algorithm |
| Convection discretization | Second-order upwind |
| Diffusion discretization | Second-order upwind |
| Pressure gradient discretization | Second-order upwind |
| Near-wall thermal condition | Adiabatic |
The hydraulic oil properties that I used are summarized in Table 3. I selected these properties to represent a typical mineral hydraulic oil used in high-pressure rotary hydraulic cylinders. The density, specific heat capacity, kinematic viscosity, and thermal conductivity are the key inputs for the screw gear oil film simulation.
| Property | Value | Unit |
|---|---|---|
| Density \(\rho\) | 889 | kg/m\(^3\) |
| Specific heat capacity \(c_p\) | 1800 | J/(kg·K) |
| Kinematic viscosity \(\nu\) | 32 | mm\(^2\)/s |
| Thermal conductivity \(K\) | 0.14 | W/(m·K) |
After the simulation converged, I extracted the pressure field, velocity field, flow rate, stiffness, and torque for each screw gear clearance. The pressure distribution inside the screw gear oil film showed a clear gradient from the inlet to the outlet. The maximum oil film pressure occurred at the inlet and was approximately 21 MPa. The minimum oil film pressure was approximately 1.2 MPa. The shape of the screw gear clearance had only a minor influence on the pressure distribution. In particular, the oil film pressure at the tooth tip and tooth root of the screw gear pair was nearly identical. This indicates that the pressure field is primarily controlled by the inlet and outlet boundary conditions and by the overall clearance size, rather than by the local tooth profile of the screw gear pair.
I also examined the internal velocity distribution for the different screw gear clearances. The velocity vector magnitude first increased and then decreased as the screw gear clearance increased. When the screw gear clearance was large, the maximum velocity occurred near the tooth root. Within the same oil film, the velocity was lowest at the walls and highest in the central part of the clearance. This behavior is consistent with the no-slip condition at the screw gear walls and with the development of a velocity profile across the oil film thickness. The axial maximum flow velocity as a function of screw gear clearance is summarized in Table 4. The dynamic pressure reached its maximum value of 9.18 MPa when the screw gear clearance was 0.10 mm, indicating that the best dynamic lubrication condition for the screw gear pair occurred at this clearance in terms of dynamic pressure alone.
| Screw gear clearance \(h\) (mm) | Maximum pressure (MPa) | Minimum pressure (MPa) | Dynamic pressure \(p_d\) (MPa) |
|---|---|---|---|
| 0.05 | 21.0 | 1.20 | — |
| 0.10 | 21.0 | 1.56 | 9.18 |
| 0.15 | 21.0 | 1.55 | — |
| 0.20 | 21.0 | 1.56 | — |
| 0.25 | 21.0 | 1.56 | — |
| 0.30 | 21.0 | 1.56 | — |
Although the dynamic pressure was highest at 0.10 mm, I could not select the screw gear clearance based on dynamic pressure alone. I therefore evaluated the torque transmitted by the hydraulic oil, the volumetric flow rate through the screw gear clearance, and the oil film stiffness. These three quantities provide a more complete picture of the screw gear pair performance. The torque exerted by the hydraulic oil on the oil film wall increased gradually as the screw gear clearance increased. However, when the screw gear clearance reached 0.20 mm, the growth rate of the torque slowed down. Further increases in the screw gear clearance produced only minor changes in torque. At the same time, the oil film stiffness decreased as the screw gear clearance increased. When the screw gear clearance reached 0.20 mm, the rate of stiffness reduction also slowed down. Further increases in clearance did not significantly change the stiffness. The volumetric flow rate increased with the screw gear clearance. When the screw gear clearance reached 0.20 mm, the growth rate of the flow rate increased significantly, and the leakage through the screw gear clearance also rose noticeably. This reduces the volumetric efficiency of the rotary hydraulic cylinder and causes an overall efficiency drop.
Table 5 summarizes the relative trends that I observed for torque, flow rate, and stiffness as functions of the screw gear clearance. The table uses qualitative indicators to show the rate of change. The key result is that the screw gear clearance of 0.20 mm provides a balanced compromise: the torque is close to its maximum practical value, the stiffness is still acceptable, and the flow rate has not yet entered the rapid-growth regime that would cause excessive leakage.
| Screw gear clearance \(h\) (mm) | Torque trend | Flow rate trend | Stiffness trend |
|---|---|---|---|
| 0.05 | Lowest | Lowest | Highest |
| 0.10 | Increasing | Increasing slowly | Decreasing |
| 0.15 | Increasing | Increasing slowly | Decreasing |
| 0.20 | High, growth slows | Moderate, growth begins to accelerate | Moderate, reduction slows |
| 0.25 | High, nearly unchanged | High, rapid growth | Low, nearly unchanged |
| 0.30 | High, nearly unchanged | Highest, rapid growth | Lowest |
Based on these results, I selected a screw gear clearance of 0.20 mm as the optimal clearance for the two-stage screw gear pair. At this clearance, the oil film performance reaches the best overall state when torque, load-carrying stiffness, internal leakage, and flow rate are considered together. This conclusion is central to my optimization of the rotary hydraulic cylinder. The screw gear pair with a 0.20 mm clearance provides sufficient torque capacity while avoiding the severe leakage that occurs at larger clearances.
To validate my simulation results, I compared them with a published screw gear pair model that has a similar rotary hydraulic cylinder configuration. I analyzed the pressure field distribution for different oil film clearances under the same operating conditions. For clearances of 0.10, 0.15, and 0.20 mm, the maximum oil film pressures in the reference model were 20.9, 21.0, and 20.9 MPa, respectively. The maximum oil film pressures in my model were 21.0, 21.0, and 21.1 MPa, respectively. The maximum difference was 0.94%. The minimum oil film pressures in the reference model were 1.20, 1.50, and 1.47 MPa, while my model gave 1.56, 1.55, and 1.56 MPa. The maximum difference was 23%. I attribute this difference to the larger oil film diameter in the reference model, which increases the effective working area and reduces the internal pressure. Nevertheless, the trends in flow rate and stiffness with increasing screw gear clearance were consistent with the reference results. Therefore, I consider the simulation results to be valid for the purpose of screw gear clearance optimization.
| Oil film clearance (mm) | Reference maximum pressure (MPa) | My maximum pressure (MPa) | Reference minimum pressure (MPa) | My minimum pressure (MPa) |
|---|---|---|---|---|
| 0.10 | 20.9 | 21.0 | 1.20 | 1.56 |
| 0.15 | 21.0 | 21.0 | 1.50 | 1.55 |
| 0.20 | 20.9 | 21.1 | 1.47 | 1.56 |
After determining the optimal screw gear clearance, I investigated how the hydraulic oil viscosity affects the screw gear oil film performance. I fixed the screw gear clearance at 0.20 mm and used four HM hydraulic oils with different viscosities. At 40 °C, the kinematic viscosity grades correspond to HM22, HM32, HM46, and HM68. The kinematic viscosities are approximately 22, 32, 46, and 68 mm\(^2\)/s, respectively. I used the same boundary conditions and solver settings as in the previous simulations. The purpose of this part of my study was to determine whether a lower-viscosity or higher-viscosity oil is more suitable for the two-stage screw gear pair under high-pressure, low-speed, and heavy-duty conditions.
| Hydraulic oil type | HM22 | HM32 | HM46 | HM68 |
|---|---|---|---|---|
| Kinematic viscosity at 40 °C (mm\(^2\)/s) | 22 | 32 | 46 | 68 |
The internal velocity distribution of the screw gear oil film changed significantly with the hydraulic oil viscosity. As the viscosity increased, the flow velocity and volumetric flow rate decreased. The oil film stiffness first increased and then decreased. The friction loss of the hydraulic components also increased, which reduced the mechanical efficiency of the rotary hydraulic cylinder. At the same time, as the viscosity increased, the leakage from the pressure chambers decreased, which improved the volumetric efficiency and enhanced the positioning accuracy of the hydraulic system. These competing effects mean that the selection of hydraulic oil viscosity must be based on the dominant operating requirements of the screw gear pair.
Table 7 summarizes the effects of hydraulic oil viscosity on the screw gear oil film performance. The flow rate and velocity decrease monotonically with increasing viscosity. The stiffness has a non-monotonic behavior: it reaches a maximum at an intermediate viscosity and then decreases. The mechanical efficiency decreases as viscosity increases because of higher friction losses. The volumetric efficiency increases as viscosity increases because of reduced leakage. For a rotary hydraulic cylinder used in high-pressure, low-speed, and heavy-duty applications, I must prioritize mechanical transmission efficiency and load-carrying performance. Therefore, I should choose a hydraulic oil with a relatively low viscosity. However, if the viscosity is too low, the temperature rise may reduce the viscosity further and cause internal leakage that prevents the screw gear pair from sustaining the required pressure. Taking these factors into account, I selected No. 32 hydraulic oil as the optimal working fluid for the two-stage screw gear pair.
| Hydraulic oil type | Flow rate trend | Velocity trend | Stiffness trend | Friction loss | Volumetric efficiency |
|---|---|---|---|---|---|
| HM22 | Highest | Highest | Lower than maximum | Lowest | Lowest |
| HM32 | High | High | High | Low | Moderate |
| HM46 | Moderate | Moderate | Highest | Moderate | High |
| HM68 | Lowest | Lowest | Lower than maximum | Highest | Highest |
The physical reason for the viscosity trends can be explained with the Reynolds number and the viscous stress. The Reynolds number of the screw gear oil film is
$$ Re = \frac{\rho v D_h}{\mu} $$
where \(D_h\) is the hydraulic diameter of the screw gear clearance and \(\mu\) is the dynamic viscosity. As the viscosity increases, the Reynolds number decreases, which tends to stabilize the flow but also increases the viscous shear stress. The viscous shear stress is proportional to the velocity gradient across the oil film:
$$ \tau = \mu \frac{\partial u}{\partial y} $$
Higher viscosity therefore increases the shear stress and the associated friction loss. At the same time, a higher viscosity reduces the leakage flow through the screw gear clearance because the flow resistance increases. The flow rate through a narrow screw gear clearance can be approximated by a pressure-driven Poiseuille flow relation:
$$ Q \propto \frac{h^3 \Delta p}{\mu L} $$
where \(h\) is the screw gear clearance, \(\Delta p\) is the pressure difference, \(\mu\) is the dynamic viscosity, and \(L\) is the flow path length. This relation explains why the flow rate decreases as the viscosity increases and why the flow rate increases strongly as the screw gear clearance increases. The cubic dependence on \(h\) is particularly important: it explains why the leakage grows rapidly when the screw gear clearance exceeds 0.20 mm.
The load-carrying stiffness of the screw gear oil film can be related to the pressure distribution and the clearance:
$$ k = \frac{\partial F}{\partial h} \approx \frac{\Delta p A}{h} $$
where \(A\) is the effective area of the screw gear oil film. The stiffness depends on both the pressure field and the clearance. As the viscosity changes, the pressure distribution changes because the flow resistance changes. This is why the stiffness first increases and then decreases with increasing viscosity. At low viscosity, the flow rate is high and the pressure gradient is relatively small, so the stiffness is limited. At high viscosity, the flow rate is low and the viscous losses are high, but the pressure distribution may become less favorable for load support, so the stiffness decreases again. The optimum viscosity for stiffness is therefore an intermediate value. However, because the rotary hydraulic cylinder operates under high pressure and heavy load, I need a balance between stiffness and mechanical efficiency, and No. 32 hydraulic oil provides this balance.
The torque transmitted by the screw gear pair is another important performance metric. The torque is generated by the pressure difference across the screw gear threads and by the viscous shear stress acting on the screw gear surfaces. The total torque can be written as
$$ T = T_p + T_\mu $$
where \(T_p\) is the pressure-induced torque and \(T_\mu\) is the viscous torque. The pressure-induced torque is approximately
$$ T_p \approx \Delta p \, A \, r_m \tan \lambda $$
where \(r_m\) is the mean radius of the screw gear pair and \(\lambda\) is the helix angle. The viscous torque is approximately
$$ T_\mu \approx \mu \frac{v}{h} A r_m $$
These expressions show that the torque increases with pressure difference, effective area, mean radius, and helix angle. The viscous torque increases with viscosity and velocity and decreases with clearance. When I increase the screw gear clearance, the pressure-induced torque tends to increase because the effective pressure area changes, but the viscous torque decreases. The net effect observed in my simulations is that the total torque increases with clearance up to about 0.20 mm and then levels off. This is consistent with the balance between pressure-induced torque and viscous torque.
The helix angle of the screw gear pair is a critical geometric parameter. For a screw gear with lead \(L\) and mean diameter \(d_m\), the helix angle is
$$ \tan \lambda = \frac{L}{\pi d_m} $$
The output rotation angle for an axial displacement \(s\) is
$$ \theta = \frac{2\pi s}{L} $$
This relation shows that a larger lead produces a larger rotation angle for the same axial stroke. The two-stage screw gear pair uses opposite helix directions to amplify the rotation angle. The first-stage screw gear pair rotates the hollow screw, and the second-stage screw gear pair rotates the output shaft. The total output rotation is the sum of the rotations produced by the two stages. This is why the rotary hydraulic cylinder can achieve a large swing angle with a compact structure. However, the large helix angle also increases the axial force component and the sensitivity of the screw gear pair to clearance and lubrication. Therefore, the oil film clearance must be optimized to maintain efficient torque transmission and adequate load capacity.
The oil film pressure field in the screw gear clearance is governed by the Reynolds equation for thin viscous films. For an incompressible Newtonian fluid, the steady-state Reynolds equation in a helical coordinate system can be written as
$$ \frac{\partial}{\partial x}\left( h^3 \frac{\partial p}{\partial x} \right) + \frac{\partial}{\partial z}\left( h^3 \frac{\partial p}{\partial z} \right) = 6\mu U \frac{\partial h}{\partial x} + 12\mu \frac{\partial h}{\partial t} $$
where \(x\) is the direction of relative motion, \(z\) is the transverse direction, \(U\) is the sliding velocity, and \(h\) is the film thickness. In my simulations, I solved the full three-dimensional Navier-Stokes equations with the \(\kappa\)-\(\varepsilon\) turbulence model rather than the Reynolds equation alone, because the screw gear clearance flow can exhibit inertial and turbulent effects at high pressure and high flow rate. The Reynolds equation nevertheless provides a useful analytical framework for understanding the influence of clearance, viscosity, and sliding velocity on the pressure field.
The flow regime in the screw gear oil film can be characterized by the Reynolds number and the clearance ratio. The clearance ratio is
$$ \varepsilon = \frac{h}{R} $$
where \(R\) is the characteristic radius of the screw gear pair. For my model, \(R\) is on the order of 70 to 75 mm, and \(h\) varies from 0.05 to 0.30 mm. Therefore, \(\varepsilon\) is on the order of \(10^{-3}\) to \(4 \times 10^{-3}\). This thin-film geometry justifies the use of a fine mesh across the clearance and supports the assumption that the pressure is nearly constant across the film thickness. The flow is primarily driven by the pressure difference between the inlet and outlet, with an additional contribution from the rotation of the inner wall. The rotational speed of 60 r/min corresponds to a surface velocity of
$$ U = \omega R = \frac{2\pi n}{60} R $$
where \(n\) is the rotational speed in r/min. For \(n = 60\) r/min and \(R = 75\) mm, the surface velocity is approximately \(U = 0.471\) m/s. This velocity is moderate, but it is sufficient to generate a measurable dynamic pressure contribution in the screw gear oil film.
The dynamic pressure that I calculated is \(p_d = \rho v^2 / 2\). For the maximum dynamic pressure of 9.18 MPa at \(h = 0.10\) mm, the corresponding maximum velocity is
$$ v_{\max} = \sqrt{\frac{2 p_d}{\rho}} = \sqrt{\frac{2 \times 9.18 \times 10^6}{889}} \approx 143.7\ \mathrm{m/s} $$
This high local velocity occurs in the central region of the screw gear clearance where the flow area is smallest. It indicates that the screw gear oil film can experience strong dynamic effects even at a moderate rotational speed, because the clearance is small and the pressure difference is large. This is one reason why the oil film clearance must be carefully controlled: a very small clearance can produce high velocity and high dynamic pressure, but it also increases viscous losses and the risk of thermal degradation. A very large clearance reduces viscous losses but increases leakage and reduces stiffness. The optimal clearance of 0.20 mm balances these competing effects.
The volumetric flow rate through the screw gear clearance can be used to estimate the leakage and the volumetric efficiency. If the ideal flow rate required by the output motion is \(Q_{\text{ideal}}\), the volumetric efficiency is
$$ \eta_v = \frac{Q_{\text{ideal}}}{Q_{\text{ideal}} + Q_{\text{leak}}} $$
where \(Q_{\text{leak}}\) is the leakage flow through the screw gear clearance. My simulations showed that \(Q_{\text{leak}}\) grows rapidly when \(h > 0.20\) mm. Therefore, the volumetric efficiency drops rapidly for larger clearances. For \(h \leq 0.20\) mm, the leakage is moderate and the volumetric efficiency remains acceptable. This is another reason why I selected 0.20 mm as the optimal screw gear clearance.
The mechanical efficiency of the screw gear pair is affected by the viscous friction in the oil film. The friction torque can be estimated as
$$ T_f = \int_A \tau \, r \, dA \approx \mu \frac{U}{h} A r_m $$
The mechanical efficiency is then
$$ \eta_m = \frac{T_{\text{output}}}{T_{\text{output}} + T_f} $$
As the viscosity increases, \(T_f\) increases, so \(\eta_m\) decreases. As the clearance increases, \(T_f\) decreases, so \(\eta_m\) increases. However, increasing the clearance also increases leakage and reduces stiffness. The optimal design must therefore balance mechanical efficiency and volumetric efficiency. My simulations indicate that \(h = 0.20\) mm and No. 32 hydraulic oil provide a good compromise for high-pressure, low-speed, and heavy-duty operation.
To further interpret my results, I computed the pressure gradient along the screw gear flow path. The pressure gradient is approximately
$$ \frac{\Delta p}{L} = \frac{21\ \mathrm{MPa} – 1.5\ \mathrm{MPa}}{L} $$
where \(L\) is the effective flow path length along the screw gear helix. For a representative helix length of the screw gear pair, this pressure gradient drives the leakage flow and contributes to the load-carrying pressure field. The pressure gradient is highest near the inlet and decreases toward the outlet. This is consistent with the pressure distribution that I observed in the simulations. The pressure field is relatively uniform at the tooth tip and tooth root, which means that the screw gear tooth profile does not create strong local pressure concentrations. Instead, the overall clearance and the inlet-outlet pressure difference dominate the pressure field.
The velocity field in the screw gear clearance is influenced by both the pressure-driven flow and the Couette flow induced by the rotating inner wall. The combined velocity profile can be approximated as
$$ u(y) = \frac{1}{2\mu} \frac{dp}{dx} (y^2 – h y) + U \frac{y}{h} $$
where \(y\) is the coordinate across the film thickness, \(h\) is the clearance, and \(U\) is the wall velocity. The first term represents the pressure-driven Poiseuille flow, and the second term represents the shear-driven Couette flow. This equation shows that the velocity profile is a superposition of a parabolic profile and a linear profile. The maximum velocity occurs at a location that depends on the relative magnitudes of the pressure gradient and the wall velocity. In my simulations, the pressure-driven component was dominant because the inlet-outlet pressure difference was large. However, the wall velocity still contributed to the velocity field, especially near the rotating inner wall.
The load-carrying capacity of the screw gear oil film depends on the pressure distribution and the effective area. The total normal force is
$$ F = \int_A p \, dA $$
The stiffness is the derivative of the force with respect to the clearance:
$$ k = -\frac{\partial F}{\partial h} $$
In my simulations, the force increased with clearance up to a point, but the rate of increase slowed down at \(h = 0.20\) mm. The stiffness decreased with clearance, but the rate of decrease also slowed down at \(h = 0.20\) mm. This behavior is consistent with the transition from a pressure-dominated regime to a leakage-dominated regime. At small clearances, the pressure field is strong and the stiffness is high, but the flow rate is low and viscous losses are high. At large clearances, the pressure field is weaker, the stiffness is lower, and the leakage is high. The optimal clearance is the point where the marginal gain in torque and flow rate is balanced by the marginal loss in stiffness and volumetric efficiency.
My optimization procedure can be summarized as follows. I first defined the geometric parameters of the two-stage screw gear pair. I then created six oil film models with different circumferential clearances. I solved the three-dimensional turbulent flow field in each model using the \(\kappa\)-\(\varepsilon\) model and the Coupled algorithm. I extracted the pressure field, velocity field, dynamic pressure, torque, flow rate, and stiffness. I then compared these quantities across the six clearance values. I found that the dynamic pressure was highest at \(h = 0.10\) mm, but the torque, flow rate, and stiffness trends indicated that \(h = 0.20\) mm provided the best overall performance. I validated the simulation results against a published screw gear pair model and found good agreement in the maximum pressure and consistent trends in flow rate and stiffness. I then fixed the clearance at 0.20 mm and varied the hydraulic oil viscosity. I found that as the viscosity increased, the flow velocity and flow rate decreased, while the stiffness first increased and then decreased. Considering the high-pressure, low-speed, and heavy-duty operating conditions, I selected No. 32 hydraulic oil as the optimal working fluid.
The practical implications of my study are significant for the design of rotary hydraulic cylinders with two-stage screw gear pairs. A screw gear clearance of 0.20 mm provides a good balance between torque capacity, stiffness, and leakage. A smaller clearance may increase stiffness but can also increase viscous heating and reduce flow rate, which may lead to inadequate lubrication. A larger clearance may reduce viscous losses but can cause excessive leakage and loss of stiffness, which degrades the positioning accuracy and load capacity. The use of No. 32 hydraulic oil ensures that the screw gear pair operates with low friction and sufficient load-carrying capacity under high pressure. These recommendations can help improve the mechanical efficiency, volumetric efficiency, and service life of the screw gear pair and the rotary hydraulic cylinder as a whole.
In addition to the clearance and viscosity effects, I also considered the influence of the screw gear tooth profile. The pressure distribution at the tooth tip and tooth root was nearly uniform, which suggests that the tooth profile has a secondary effect compared with the overall clearance. However, the tooth profile still affects the local film thickness and the contact pattern. In future work, I would like to incorporate more detailed tooth surface roughness and thermal effects into the screw gear oil film model. I would also like to study transient operating conditions, such as rapid reversals and shock loads, because these conditions can cause sudden changes in the screw gear oil film pressure and velocity fields. Such studies would further improve the understanding of screw gear lubrication and support the development of more efficient and durable rotary hydraulic cylinders.
Overall, my analysis demonstrates that the oil film in the two-stage screw gear pair is a critical determinant of rotary hydraulic cylinder performance. By combining three-dimensional flow simulation with systematic clearance and viscosity studies, I identified \(h = 0.20\) mm as the optimal circumferential clearance and No. 32 hydraulic oil as the optimal working fluid. These results provide a quantitative basis for the design and optimization of screw gear pairs in high-pressure, low-speed, and heavy-duty hydraulic actuators. The methodology that I used can be extended to other screw gear geometries and operating conditions, and it offers a reliable approach for improving the efficiency, load capacity, and durability of rotary hydraulic cylinders.
| Optimization variable | Range considered | Best value | Primary reason |
|---|---|---|---|
| Screw gear clearance \(h\) | 0.05–0.30 mm | 0.20 mm | Balances torque, stiffness, flow rate, and leakage |
| Hydraulic oil viscosity | HM22, HM32, HM46, HM68 | No. 32 | Balances mechanical efficiency, load capacity, and leakage |
In conclusion, the two-stage screw gear pair in a rotary hydraulic cylinder forms a helical clearance oil film that controls the transmission efficiency and load-carrying behavior. My simulations showed that the pressure field is dominated by the inlet-outlet pressure difference and that the maximum pressure is approximately 21 MPa at the inlet. The velocity field and dynamic pressure depend strongly on the screw gear clearance. The torque increases with clearance but levels off at 0.20 mm, while the stiffness decreases with clearance but also levels off at 0.20 mm. The flow rate increases with clearance, and the growth becomes rapid above 0.20 mm. Therefore, I selected 0.20 mm as the optimal screw gear clearance. For hydraulic oil viscosity, increasing viscosity reduces velocity and flow rate, while stiffness first increases and then decreases. Considering the high-pressure, low-speed, and heavy-duty application, I selected No. 32 hydraulic oil. These findings provide a practical basis for optimizing screw gear pairs in rotary hydraulic cylinders and for improving their mechanical efficiency, volumetric efficiency, and service life.