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Hydraulic Cylinder Design Manufacturing-Technical Foundation
1.2.3 Hydraulic Cylinder Parameter Calculation Example
The previous article in this series discussed the Hydraulic Cylinder Main Parameters and Hydraulic Cylinder Parameter Calculation Formula; today this article will list Hydraulic Cylinder Parameter Calculation Example, this help it easier to better understand the hydraulic cylinder through specific calculation cases.
(1) Theoretical Output Force Calculation Example
After selecting the hydraulic cylinder nominal pressure, cylinder barrel tube inner diameter, and piston rod outer diameter, the theoretical output push force and pull force of a double-acting single-piston rod hydraulic cylinder at the nominal pressure value can be calculated according to formulas (1-1) and (1-2). The calculation results are shown in Table 1-2.
Table 1-2 Theoretical output force of double-acting single-piston-rod hydraulic cylinder
For the main parameter of hydraulic presses (rated force) expressed in tons, in engineering, it can be converted using 1 N = 10⁻⁴ t.
(2)Example of cylinder barrel tube bore diameter calculation
1.If parameters such as the nominal (or rated) pressure of the hydraulic system and the main parameter (nominal force) of the hydraulic press are given, the inner diameter of the hydraulic cylinder barrel tube can be designed and calculated.
For example, given the nominal pressure of the hydraulic system as pn (p) = 16 MPa, and the nominal force of the powder product hydraulic press as F = 400 kN, use the formula derived from Eq. (1-1):
Where:
- D is the cylinder barrel tube inner diameter, mm;
- F is the theoretical output push force of the hydraulic cylinder, N;
- p is the nominal pressure or rated pressure, MPa.
Calculate the cylinder barrel tube inner diameter:
According to the specification “Hydraulic and pneumatic systems and components Cylinder bores and piston rod diameters”, round the calculated value. Thus, the cylinder inner diameter for this hydraulic press can be selected as D = 180 mm.
If the selected rated pressure is lower than the nominal pressure (for example, if the selected rated pressure is 14 MPa) or if a certain margin (degree) of cylinder output force is required, one could further consider selecting a cylinder barrel tube inner diameter of D = 200 mm. However, the pros and cons must be carefully weighed. Nevertheless, a hydraulic cylinder barrel tube with an inner diameter of D = 160 mm cannot be selected for this hydraulic press. This is because, under the nominal pressure of the hydraulic system, its cylinder output push force would only be:
which cannot meet the required nominal force of F = 400 kN for this powder product hydraulic press.
The nominal force of the aforementioned powder product hydraulic press is provided by the output push force of one double-acting single-piston-rod hydraulic cylinder. However, the nominal force of most hydraulic presses is provided collectively by the output forces of two or more hydraulic cylinders (in special hydraulic press structures, the nominal force can also be provided by the output pull of hydraulic cylinders). Therefore, when calculating the cylinder barrel tube inner diameter based on the nominal force of the hydraulic press, attention must be paid to converting the nominal force appropriately.
Furthermore, here disagrees with the statements and practices found in some references that suggest increasing the hydraulic cylinder output force by raising the hydraulic system working pressure (i.e., exceeding the nominal or rated pressure). This approach poses risks to the hydraulic cylinder, the hydraulic system, and even the hydraulic press itself. Moreover, such practice is prohibited by the safety technical requirements for hydraulic presses.
2.If parameters such as the input flow rate from the hydraulic system to the hydraulic cylinder, the cylinder extending speed, or the cylinder extending time are given, the inner diameter of the hydraulic cylinder barrel tube can be calculated.
For example, given the cylinder extending speed (working stroke) as v ≤ 8 mm/s, and the input flow rate from the hydraulic system of a 100T hydraulic press brake to a single hydraulic cylinder as Q = 12.125 L/min, use the formula derived from Eq. (1-5):
Where:
- D is the cylinder barrel tube inner diameter, mm;
- Q is the hydraulic system input flow rate to the rodless chamber of the cylinder, L/min;
- v is the cylinder extending speed, mm/s.
Calculate the cylinder barrel tube inner diameter:
According to the specification “Hydraulic and pneumatic systems and components Cylinder barrel tube bores and piston rod diameters”, round the calculated value. Thus, the cylinder inner diameter for this hydraulic press can be selected as D 180 mm.
(3) Example of piston rod diameter calculation
1.For a double-acting single-piston-rod hydraulic cylinder, if parameters such as the nominal (or rated) pressure of the hydraulic system and the output pull force of the hydraulic cylinder are given, the piston rod diameter can be designed and calculated after selecting the cylinder barrel tube bore diameter.
The formula for calculating the piston rod diameter can be derived from Eq. (1-2), as follows:
Where:
- d is the piston rod outer diameter, mm;
- D is the cylinder barrel tube bore (inner) diameter, mm;
- F is the theoretical output pull force of the hydraulic cylinder, N;
- p is the nominal pressure or rated pressure, MPa.
If the nominal pressure of the hydraulic system is given as p = 16 MPa, the theoretical output pull force of the hydraulic cylinder is F = 200 kN, and the cylinder barrel tube bore diameter is selected as D = 160 mm, then:
According to “Hydraulic and pneumatic systems and components – Cylinder barrel tube bores and piston rod diameters” and by rounding the calculated value, the piston rod diameter for this hydraulic cylinder can be selected as d = 90 mm.
Generally, a piston rod diameter of d=100 mm cannot be selected. This is because, under the nominal pressure of the hydraulic system, the cylinder output pull force would only be:
This cannot meet the design requirement of a theoretical output pull force of F = 200 kN for the hydraulic cylinder.
2.If the reciprocating speed ratio or the area ratio of the two chambers of the hydraulic cylinder is given, the piston rod diameter can be designed and calculated after selecting the cylinder barrel tube bore diameter.
The calculation formula for the piston rod diameter can be derived from Eq. (1-8), as follows:
Or
Where:
In the formula:
- d ——Piston rod outer diameter, mm;
- D ——Cylinder barrel tube bore (inner) diameter, mm;
- φ——Ratio of the extending speed to the retracting speed for a double-acting single-piston-rod hydraulic cylinder;
- ϕ Area ratio of the two chambers.
Since ϕ= 1/φ, when the speed ratio φ is known, the area ratio ϕ can be calculated. The calculated value can then be adjusted according to the specifications of “Area ratios of two chambers for single-rod hydraulic cylinders”, and the piston rod diameter can be selected according to “Hydraulic and pneumatic systems and components – Cylinder barrel tube bores and piston rod diameters”. Alternatively, the piston rod diameter can be selected directly based on the area ratio.
For example, given the area ratio of the two chambers as ϕ = 1.46, and having selected the cylinder barrel tube bore diameter as D = 160 mm, then according to Eq. (1-23):
Adjusting the calculated value according to “Hydraulic and pneumatic systems and components – Cylinder barrel tube bore and piston rod diameter”, the piston rod diameter for this hydraulic cylinder can be selected as d = 90 mm.
3.If parameters such as the input flow rate from the hydraulic system to the hydraulic cylinder, the cylinder retracting speed, or the cylinder retracting time are given, the piston rod diameter can be calculated after selecting the cylinder barrel tube bore diameter.
For example, given the cylinder retracting speed as v ≥ 70 mm/s, and again using the 100T hydraulic press brake hydraulic system as an example, where the input flow rate from the hydraulic system to a single hydraulic cylinder is Q = 12.125 L/min, use the formula derived from Eq. (1-6):
Where:
- d is the piston rod diameter, mm;
- D is the cylinder barrel tube bore diameter, mm;
- Q is the input flow rate from the hydraulic system to the cylinder’s rod side chamber, L/min;
- v is the cylinder retracting speed, mm/s.
Calculate the piston rod diameter:
After rounding the calculated value, the piston rod diameter for this hydraulic cylinder can be selected as d = 170 mm (non-standard).
The following main issues should be noted when selecting the piston rod diameter:
a. The area ratios of the two chambers (1.06, 1.12, 1.25, 1.4, 1.6, 2, 2.5, 5) given in standard are preferred numbers, not actual values.
b. When the area ratio of the two chambers is large (e.g., ϕ ≥ 5), care must be taken to prevent pressure intensification from exceeding the rated pressure or nominal pressure limit. This can occur due to the large difference between the effective area of the piston side chamber and the effective area of the rod side chamber, i.e., the small difference between the cylinder barrel tube bore diameter and the piston rod diameter.
c. When the area ratio of the two chambers is large (e.g., ϕ≥ 5), avoid using contact between the piston and other cylinder components as the stroke limit for the extending motion.
d. When the area ratio of the two chambers is small (e.g., ϕ≤ 1.06), care must be taken to avoid bending or buckling of the piston rod. This can result from the small difference between the effective area of the piston side chamber and the effective area of the rod side chamber, i.e., the large difference between the cylinder bore diameter and the piston rod diameter. If necessary, strength, stiffness, and buckling stability calculations should be performed for the piston rod.
For the selection of non-standard piston rod diameters, consideration should be given to the selection of the piston rod sealing device or the various seals within the system. A safety valve should be installed in the rod side chamber of hydraulic cylinders with a large area ratio between the two chambers.
4.Piston rod strength calculation (check).
a. When the piston rod is in a stable state and subjected only to axial loads (ignoring its own gravity), the simple tensile (compressive) strength condition is applied:
Taking solid piston rods made of normalized #20 steel, normalized 45 steel, and quenched and tempered 45 steel as examples, the piston rod outer diameters that meet the strength conditions under nominal pressures of 16 MPa and 25 MPa, respectively, are shown in Table 1-3.
Table 1-3 Piston rod outer diameters that meet strength requirements mm
Note: Because the piston rod outer diameter selected for a product is typically at least one size larger within the piston rod outer diameter series specified in standard and listed in Table 1-3, medium-carbon steel piston rods subjected only to unidirectional loads (e.g., push force output) do not require quenching and tempering. Under certain conditions, normalizing or normalizing + tempering may be used in place of quenching and tempering. Table 1-3 serves not only as the basis for selecting piston rod outer diameters, but also as the foundation for these piston rod technical requirements.
b. When the piston rod is in a stable state and subjected to eccentric loads (ignoring its own gravity), meaning under the action of a bending moment, the eccentric tension (compression) strength condition is applied:
Where:
- σ: Maximum eccentric tensile (compressive) stress on the critical cross-section under normal temperature and static load, MPa;
- F: External load parallel to but not coincident with the piston rod axis, equal to the theoretical output thrust or pull of the cylinder, N;
- A: Cross-sectional area of the critical section, mm²;
- M: Bending moment generated by the eccentric load on the piston rod, N-m;
- W: Section modulus of the piston rod, m³.
The others are the same as above.
(4) Calculation of cylinder stroke limit
Under certain conditions, cylinder stroke has a limit. When the cylinder stroke exceeds this limit, the cylinder and piston rod may experience strength, stiffness, and rod stability issues.
① Double-acting, single-piston-rod hydraulic cylinder stroke limits. The stroke limits for double-acting, single-piston-rod hydraulic cylinders specified in various hydraulic cylinder product standards are shown in Tables 1-4 and 1-5.
Table 1-4 Double-acting single-piston-rod hydraulic cylinder stroke limits (PN ≤ 16 MPa)
Note: 1. Excerpted from Table 1 of “Hydraulic Cylinders for Metallurgical Equipment (PN ≤ 16 MPa).” The corresponding mounting types for each mounting type code and Table 1-5, “Stroke Limits for Double-Acting, Single-Piston-Rod Hydraulic Cylinders (PN ≤ 25 MPa),” are provided in that standard.
2. “Certain conditions” should also include at least piston rod material, structural type, heat treatment, and surface treatment.
Table 1-5 Double-acting single-piston-rod hydraulic cylinder stroke limits (PN ≤ 25 MPa)
Note: 1. Excerpted from Table 1 of “Hydraulic Cylinders for Metallurgical Equipment (PN ≤ 25 MPa).” The corresponding mounting types and dimensions for each mounting type code are listed in that standard.
2. “Certain conditions” should also include at least piston rod material, structural type, heat treatment, and surface treatment.
3. Except for hydraulic cylinders specified in “Hydraulic Cylinders for Metallurgical Equipment (PN ≤ 25 MPa).”, all other conditions are for reference only.
② Stroke Limit Determined by the Critical Load of a Strut.
Based on the strut stability condition:
Where:
Where:
For example, if selecting a hydraulic cylinder with spherical hinge constraints at both ends (using spherical bearings at both ends, such as type d specified in CB/T 3812), where n = 1, then:
Table 1-6 Sum of installation distance and travel limit under several selected safety factors
Since l = L + Smax (where the installation distance is L, and the stroke limit is Smax), once the installation distance is given a value, the stroke limit can be determined according to Table 1-6.
For example, the installation distance (minimum) for the marine hatch cover hydraulic cylinder 220/125x specified in standard is 680 mm. According to Table 1-6, when nk = 5, the l ≤2.228 m, thus Smax ≤ 2228 – 680=1528mm specified in standard. This is basically consistent with the 1600 mm stroke
A few points are explained as follows:
a. Strictly speaking, there are two different types of stability calculations: stability check calculations and stability design calculations. The calculation according to Eq. (1-31) belongs to the stability design calculation.
b. Equations (1-30) and (1-31) are applicable to the calculation of long, slender struts with a uniform cross-section, where the hydraulic cylinder and piston rod are subjected to axial (non-eccentric) loads and the stress does not exceed the proportional limit.
c. The reference values in Table 1-6 are sourced from Table 9 of standard “Marine Hatch Cover Hydraulic Cylinders”.
d. The selectable ranges for the strut stability safety factor in various references may differ, such as nk = 2~4 or nk = 3.5 ~ 6, etc. The article recommends selecting it as nk = 4 ~ 6.
