Library / Hydraulics, open channels and fluid power / HYD-45
Hydraulics, open channels and fluid power
Blocked-port hydraulic cylinder stiffness
Stiffness of a cylinder with both ports closed, for small displacements about a pressurised point, from the effective bulk moduli B, piston areas A and trapped volumes V on the two sides.
- Ref
- HYD-45
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- kh [N/m]
With both ports closed, a small piston movement compresses the oil on one side and expands it on the other. Each side acts as a spring of stiffness B A²/V, with B the effective bulk modulus, A the piston area and V the trapped volume including the hoses. The two add: kh = B₁A₁²/V₁ + B₂A₂²/V₂. Entrained air and flexible hoses lower B a lot, so use an effective value.
Stiffness of a cylinder with both ports closed, for small displacements about a pressurised point, from the effective bulk moduli B, piston areas A and trapped volumes V on the two sides. Leakage is neglected.
Calculator
- Stiffness from side 1, B₁A₁²/V₁8,000,000 N/m
- Stiffness from side 2, B₂A₂²/V₂6,000,000 N/m
- Natural frequency with a 100 kg load59.55 Hz
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The locked cylinder acts like a spring of 1.4×107 N/m, so a 1,000 N load moves it by 0.07143 mm.
- A 100 kg load on this spring would oscillate at 59.55 Hz, which sets the speed limit of a hydraulic servo drive.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N/m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A2 Piston area, side 2 | 1e-4 to 0.5 m² | 8,007,000 | 1.667e11 | 1.667e11 | +10% | +9 % |
| A1 Piston area, side 1 | 1e-4 to 0.5 m² | 6,005,000 | 1.25e11 | 1.25e11 | +10% | +12 % |
| V1 Trapped volume, side 1 | 1e-5 to 1 m³ | 6,016,000 | 1.606e9 | 1.6e9 | +10% | −5.19 % |
| V2 Trapped volume, side 2 | 1e-5 to 1 m³ | 8,009,000 | 9.08e8 | 9e8 | +10% | −3.9 % |
| B1 Effective bulk modulus, side 1 | 1e8 to 3e9 Pa | 6,800,000 | 3e7 | 2.32e7 | +10% | +5.71 % |
| B2 Effective bulk modulus, side 2 | 1e8 to 3e9 Pa | 8,600,000 | 2.6e7 | 1.74e7 | +10% | +4.29 % |
Sweep of Trapped volume, side 1
| Trapped volume, side 1 (m³) | Hydraulic stiffness (N/m) |
|---|---|
| 1e-5 | 1.606e9 |
| 4.217e-5 | 3.854e8 |
| 1.778e-4 | 9.597e7 |
| 7.499e-4 | 2.734e7 |
| 0.003162 | 1.106e7 |
| 0.01334 | 7,200,000 |
| 0.05623 | 6,285,000 |
| 0.2371 | 6,067,000 |
| 1 | 6,016,000 |
Sweep one variable
Variables and how each one contributes
Hydraulic stiffness
What the equation gives you. Everything else feeds it.
Effective bulk modulus, side 1
The effective bulk modulus on side 1, in Pa. Raises the stiffness in direct proportion. Raising it by 10 % (to 1.1×109) moves the result by +5.71 %, from 1.4×107 to 1.48×107 N/m.
Piston area, side 1
The active piston area on side 1, in m². Raises the stiffness with the square of the area. Raising it by 10 % (to 0.0044) moves the result by +12 %, from 1.4×107 to 1.568×107 N/m.
Trapped volume, side 1
The trapped volume on side 1, in m³. More trapped volume lowers the stiffness in inverse proportion. Raising it by 10 % (to 0.0022) moves the result by −5.19 %, from 1.4×107 to 1.327×107 N/m.
Effective bulk modulus, side 2
The effective bulk modulus on side 2, in Pa. Raises the stiffness in direct proportion. Raising it by 10 % (to 1.1×109) moves the result by +4.29 %, from 1.4×107 to 1.46×107 N/m.
Piston area, side 2
The active piston area on side 2, in m². Raises the stiffness with the square of the area. Raising it by 10 % (to 0.0033) moves the result by +9 %, from 1.4×107 to 1.526×107 N/m.
Trapped volume, side 2
The trapped volume on side 2, in m³. More trapped volume lowers the stiffness in inverse proportion. Raising it by 10 % (to 0.00165) moves the result by −3.9 %, from 1.4×107 to 1.345×107 N/m.
Worked example with the values above
- Side 1: B₁A₁²/V₁ = 1×109 × 1.6×10−5 / 0.002 = 8,000,000 N/m.
- Side 2: B₂A₂²/V₂ = 1×109 × 9×10−6 / 0.0015 = 6,000,000 N/m.
- kh = 1.4×107 N/m.
Assumptions and limits
- An incompressible fluid with no leakage, except where an efficiency is given.
- Gauge pressures and steady operation, with friction and inertia not included unless stated.
Common mistakes
- Using the piston area where the annular area is needed, or the reverse.
- Mixing rev/min and rad/s.
- Forgetting back pressure on the other side of the piston.
Statistics and uncertainty
Real actuators lose force to seal friction and flow to leakage, and oil bulk modulus drops sharply with entrained air, so the stiffness and speed here are best-case figures.
Where it is used
- Sizing hydraulic cylinders, pumps and motors.
- Estimating drive speed, stiffness and natural frequency of a hydraulic actuator.
Maths you need
- Pascal’s law: pressure acts equally in all directions in a closed fluid
- Displacement: volume moved per revolution or stroke
- Bulk modulus: B, the pressure per unit volume change of the fluid