2.2.1. Model of a Proportional Pressure-Reducing Valve
The proportional pressure-reducing valve is the core control element of the free-fall hook pressure control system. Its control signal can continuously and proportionally control the output pressure, which is approximately linear, and can realize high-precision conversion between electro-hydraulics.
By disassembling and mapping the proportional pressure reducer, as shown in
Figure 2, it can be determined that the effective displacement of the spool is 0.5 mm, and the maximum displacement of the spool is 1.6 mm. To calculate the micro-element area of the spool of the pressure-reducing valve under the valve opening x, the formula for calculating the spool overflow area is as follows.
When 1.1 < x < 1.6:
where
is the area of a single orifice,
is the number of orifices, and
is the diameter of the orifice of the pressure relief valve.
The change curve of spool overflow area with spool displacement is shown in
Figure 3.
As can be seen in
Figure 3, with the displacement of the spool, the A-T port overflow area gradually decreases. At the spool movement displacement of 0.84 mm, the A-T port is completely closed. At the displacement of 1.1 mm, the P-A port begins to open, and at the displacement of 1.6 mm, the P-A port overflow area is the largest and the P-A port is completely open. At this time, the proportional pressure-reducing valve outputs the maximum pressure.
- 2.
Spool force analysis of proportional pressure-reducing valves
The equation for the output force of solenoid is:
where
is the current value,
is the electromagnet current–force gain coefficient,
is the electromagnet displacement–force gain coefficient, and
y is the armature displacement.
The equation for the spool force balance is:
where
is the pressure-reducing valve orifice pressure,
is the pressure-reducing valve spool displacement,
is the transient hydrodynamic damping coefficient,
is the reset spring stiffness of the pressure-reducing valve,
is the feedback area of the pressure-reducing valve,
is the mass of the pressure-reducing valve spool,
is the viscous damping coefficient of the spool, and
is the steady state hydrodynamic stiffness of the pressure-reducing valve.
2.2.2. Modeling of Free-Fall Hook Wet Clutches
In the first stage of the braking process (
t1–
t2), the dynamic and static friction plates are not in direct contact, the gap between the friction plates is filled with a film of oil, and there is only the phenomenon of band displacement torque. According to Newton’s law of internal friction, the formula for calculating the slipping torque of a wet clutch is [
27,
28]:
where
is the relative speed of the dynamic and static friction discs,
is number of contact surfaces on the dynamic and static friction discs,
is the dynamic viscosity of the lubricant,
is the outer diameter of the friction disc,
is the inner diameter of the friction disc, and
is the dynamic and static friction disc gap.
In the free-fall hook system, brake cavity 5 is continuously supplied with oil, the pressure in brake cavity 5 increases, the pressure difference between brake cavity 5 and cavity 6 decreases, and the spring force overcomes the combined force between the two chambers, causing the piston to move. The equation for the instantaneous force balance of the piston is:
where
,
ps is the pressure in cavity 6,
is the pressure in brake cavity 5,
is the stiffness coefficient of the reset spring, and
x0 is the initial compression of the reset spring.
From this, the pressure required in brake cavity 5 at
t1 of the first stage of braking is:
At the oil-filling stage (
t0–
t1), the flow rate of brake cavity 5 is
, and the oil pressure in brake cavity 5 rises rapidly to
, which pushes the piston to move to eliminate the gap between the movable and static friction discs. At this time, the pressure in the brake cavity 5 is expressed:
where
is the clutch piston displacement. Neglecting oil leakage, the flow balance equation is:
where
is the volume of brake cavity 5 and
is the volumetric modulus of elasticity of the fluid. Substituting Equation (6) into Equation (7) obtains:
Assuming that the piston moves at a constant speed in the first stage, the flow in brake cavity 5 is determined by the speed of piston movement and is a certain value.
To avoid flow fluctuations, let
, then the time to eliminate the gap is
, and integrate over
:
At the end of the first stage (
t =
t2), the pressure in braking cavity 5 reaches:
In the second stage of braking, also called the slippery wear stage (
t2–
t3), the movable and static friction plates come into contact and produce relative sliding. With the increase in pressure in brake cavity 5, the lubricant film between the movable and static friction plates is completely destroyed, in which the rough friction torque plays a major role, and the torque generated by this process makes the load brake. Ignoring the band-rolling torque, the torque calculation equation of friction vice [
29,
30] is:
where
u is the coefficient of kinetic friction of the friction disc and
.
The acceleration of the load during braking is calculated as:
where
is the tension on the wire rope,
is the total mass of the load,
is the radius of the winch, and
is the transmission ratio between the winch and the output shaft of the clutch.
The pressure in brake cavity 5 at the completion of braking should reach
:
In the third stage of braking, also called the full rough friction stage (
t3–
t4), the friction plate is affixed and pressed together, the movable and static friction plates are relatively static, and the torque generated by the clutch achieves full braking of the load. At this time, the friction type is static friction, and the torque calculation equation is:
where
is the coefficient of static friction of the friction disc. The clutch brake cavity 5 pressure at this stage is:
2.2.3. Transfer Function of a Free-Fall Hook System
The first stage of clutch braking has little effect on braking, and this article focuses on the second and third stages of the braking process.
To model the second and third stages of clutch braking, the force equations for the spool of a proportional pressure-reducing valve are:
The linear flow equation for a proportional pressure-reducing valve is:
When the load flow rate is zero:
carry out the Laplace transform:
The resulting open-loop transfer function of the system is obtained as:
From the known parameters, the characteristic equation of the system is derived as:
A stability analysis of the system is carried out. Based on the characteristic equations, a table of the Rouse criterion is presented as shown below:
According to Equation (24), the Rouse criterion table shows that the elements of the first column are all positive and the coefficients of the characteristic equations of the system are all positive, so the system is determined to be stable.