Hydraulic and Pneumatic Cylinders

Hydraulic and Pneumatic Cylinders

Hydraulic and Pneumatic Cylinders: Method and Sources

A cylinder turns fluid pressure into a straight-line force. Everything on this page follows from four relations: force is pressure times area, speed is flow over area, power is pressure times flow, and a confined gas obeys P Vⁿ = constant. The tool applies them to five tasks under one select. The relations and their worked examples are those of two public U.S. training manuals, Army FM 5-499 Hydraulics (1997) and Navy NAVEDTRA 12964 Fluid Power (1990, later reissued as NAVEDTRA 14105), with fluid properties from the Army handbook AMCP 706-123 Hydraulic Fluids (1971). Makers' catalogues and the NFPA and ISO cylinder standards are cited only: their ratings, stroke charts and dimension tables are not reproduced, and where they would supply a number (a load ratio, a design factor, a cushion capacity) the tool takes an entered value with an illustrative default.

Force, speed and power

Areas and forces. The cap end of the piston presents the full bore area A_c = πD²/4 and the rod end the annulus A_a = π(D² − d²)/4; A_r = πd²/4 is the rod and φ = A_c/A_a the area ratio. With supply pressure p, back pressure p_b on the exhausting side and a force efficiency η_f for seal and bearing friction, the extend force is η_f (p A_c − p_b A_a) and the retract force η_f (p A_a − p_b A_c). FM 5-499 states F = PA (par. 1-1b) and works 50 lb/in² on 100 in² = 5,000 lb (p. 1-4) and Pascal's two-piston apparatus (p. 1-5); NAVEDTRA 12964 works the same law on pp. 2-6 to 2-8 and p. 2-15, including the differential-area piston of its Figure 2-12 (6 in² and 4 in² faces at 20 lb/in²: 120 lb against 80 lb, a net 40 lb). That net force on the rod area is the regenerative circuit: both ends see the supply, the rod-end oil joins the pump flow, and the cylinder extends with the force η_f p A_r at the speed Q/A_r. The test reproduces every one of these printed figures.

Sizing. For a load W and a load factor LF (the share of the available force the load may take, the rest being the reserve for acceleration and line losses) the area required is W/(LF η_f p) and the bore √(4A/π); the pressure an entered bore needs is W/(η_f A_c) + p_b A_a/A_c.

Speed and flow. By continuity v = Q/A: Q/A_c extending and Q/A_a retracting, so a single-rod cylinder retracts φ times faster than it extends. The flow leaving the other end is Q/φ while extending and Q φ while retracting, more than the pump delivers, and the return line, filter and valve must pass it. In US units v [in/s] = 231 Q [gal/min] / (60 A [in²]). The cycle time is the two stroke times plus an entered dwell.

Power. Hydraulic power is p Q; FM 5-499 (par. 6-4c, p. 6-13) writes it HP = f p / 1,714 with the flow in gal/min and the pressure in lb/in². The exact constant is 550 × 12 × 60 / 231 = 1,714.29. The pump shaft takes p Q/η_p and the drive p Q/(η_p η_m); the rod returns F v.

Meter-out intensification. With a flow control throttling the oil that leaves the rod end, the piston is in balance when p_a A_a = p A_c − F_L, so the rod-end pressure is (p A_c − F_L)/A_a: φ p with no load and more with an overrunning load. A 2:1 cylinder at 3,000 lb/in² puts 6,000 lb/in² on its rod-end seals this way.

Pneumatic cylinders

Thrust. The theoretical thrust is the gauge pressure times the piston area. A moving air cylinder delivers less, because air is compressible and the piston moves only while there is a pressure difference to drive the flow: the tool multiplies by an entered load ratio. Catalogues recommend loading a moving cylinder to roughly half to seven tenths of its theoretical thrust and a clamping cylinder higher; that guidance is cited, not sourced to a public document, and the default 0.7 is illustrative.

Free-air consumption. Each stroke fills the swept volume A S to the absolute supply pressure. By Boyle's law (NAVEDTRA 12964 eq. 11-6; its example of 4 ft³ at 100 lb/in² gauge expanding to 6 ft³ gives 76.47 lb/in² absolute, 61.77 gauge, reproduced by the test) that air occupied A S (p + p_atm)/p_atm as free air. A double-acting cylinder uses A_c S on the extend stroke and A_a S on the return; the average flow is the volume per cycle times the cycles per minute. The volume of the lines between valve and cylinder is filled each stroke as well and is not included.

Valve flow coefficient. C_v is defined with water: Q [gal/min] = C_v √(Δp [lb/in²]/G). Written in consistent units for any fluid of density ρ the same definition is a mass flow W = C_v k_q √(Δp ρ), with k_q = (1 gal/min) √(ρ_water/(1 lb/in²)). For a gas the tool takes ρ at the outlet pressure, ρ₂ = p₂ M/(R T). In customary units this is Q [standard ft³/min] = K C_v √(Δp p₂/(T G)), Δp in lb/in², p₂ in lb/in² absolute, T in °R, G the gas gravity; K is 22.67 for a standard state of 60 °F and 14.73 lb/in² (one sixtieth of the 1,360 of the ISA gas equation, computed here from the definition) and 23.07 for 68 °F and 14.696 lb/in². Fluid-power sizing guides print the same form with a constant of about 22.5; that figure is recalled, not verified against the NFPA document. The form is the low-pressure-drop limit of the ISA equation used by the Control Valve Sizing tool: with x = Δp/p₁, the outlet-density form carries the factor √(1 − x) where the ISA form carries the expansion factor Y = 1 − x/(3 F_k x_T), and the two agree to first order in x when x_T = 2/3 (within 1.2 % at x = 0.2 for x_T = 0.7). Past the critical pressure ratio (2/(k+1))^k/(k−1) = 0.528 the flow is sonic and the outlet pressure is held at that ratio. The required coefficient is the mass of one stroke over the stroke time divided by the mass flow of a valve of unit coefficient at the entered valve drop; a second figure fills the chamber only to the pressure downstream of the valve. The result is a first estimate: it treats the stroke as steady flow at a constant drop and leaves out the exhaust side, which needs a similar coefficient.

Cushion energy. The cushion absorbs E = ½ m v² + p A_c s_c + m g s_c sin θ: the kinetic energy at entry, the work the supply pressure keeps doing over the cushion length s_c, and the potential energy on an incline. The mean retarding force is E/s_c and the mean cushion pressure that force over the annulus. The capacity is a catalogue figure for the bore and is entered; the independent check integrates the equation of motion and finds the piston stopped in exactly s_c under the mean force.

Structural checks

Rod buckling. The extended rod is treated as a column of length L (between the pivots, or from the mounting face to the rod-end attachment) with the classical end conditions of the mounting: pivoted both ends K = 1, rigid mount with a pivoted guided rod end K = 0.699 (the root of tan x = x), rigid mount with a rigidly guided rod end K = 0.5, rigid mount with a free rod end K = 2. For a solid round rod I = πd⁴/64 and r = d/4. Above C_c = √(2π²E/S_y) the critical load is Euler's π²EI/(KL)²; below it the Johnson parabola A S_y [1 − S_y (KL/r)²/(4π²E)]. The allowable push is the critical load over the entered factor, and the tool also solves the longest column and the smallest rod for the push. The K values are checked against the characteristic equations and a finite-difference eigenvalue solution, the loads against the library of the Column Buckling Tables tool. What the check leaves out matters for long cylinders: the tube and its bearing clearances make the real column more flexible than the rod alone between rigid ends, the weight of a horizontal cylinder and any misalignment add bending, and makers fit a stop tube to long strokes to spread the bearing loads; their stroke charts include these effects and their own factor and take precedence for a catalogue cylinder.

Rod in tension. Retracting at pressure pulls the rod with p A_a across the rod-end thread, whose tensile-stress area is 0.7854 (d − 0.9743 P)² for a unified thread (NBS Handbook H28) or 0.7854 (d − 0.9382 P)² for an ISO metric thread; the stress is compared with S_y/n.

Tube. The thin-wall hoop stress is p D/(2t). Lamé's thick-cylinder solution gives at the bore σ_θ = p (D_o² + D²)/(D_o² − D²) and σ_r = −p, with σ_z = p D²/(D_o² − D²) when the tube carries the end load; the thin-wall figure is low by about t/D. The burst pressure is taken where the Lamé bore hoop stress reaches the ultimate strength, S_u (D_o² − D²)/(D_o² + D²), and the allowable working pressure is that over the entered burst factor. Four to one on burst is a common basis of industrial ratings; it is cited as a convention and the factor is an input. A ductile tube bursts somewhat above this elastic estimate, and ports, welds, threads and grooves burst below it, which is why ratings rest on tests. The growth of the bore, (D/E)(σ_θ − ν(σ_r + σ_z)), is printed for the seal clearance. The Lamé stresses are checked against a numerical solution of the thick-cylinder boundary-value problem.

Tie rods. The pressure load on the cap, p A_c, is shared by the tie rods; each is preloaded to an entered multiple of its share so that the heads stay seated, and the stretch F L/(A E) is printed for tightening by elongation. The joint diagram, the torque and thread stripping belong to the Bolt Preload and Torque and Thread Engagement and Strength tools.

Dynamics

Acceleration. F = m a + m g sin θ + F_f with a = v/t; the pressure is F/A_c.

Oil-column natural frequency. Oil of bulk modulus β trapped in a volume V behind a piston of area A is a spring of stiffness β A²/V: moving the piston by x changes the volume by A x, the pressure by β A x/V and the force by β A² x/V. With both ends closed, k = β (A_c²/V_c + A_a²/V_a), where V_c = A_c x + V_dc and V_a = A_a (S − x) + V_da include the dead volumes of ports and lines up to the valve, and the load of mass m oscillates at f = (1/2π) √(k/m). Setting dk/dx = 0 gives the softest position, V_c/V_a = (A_c/A_a)^3/2, which for a double-rod cylinder with equal dead volumes is mid-stroke. The closed form is checked against a numerical minimisation. This is the standard hydraulic-spring model of the servo literature (Merritt, cited). The default bulk modulus, 212,000 lb/in², is the isothermal secant value given in the worked example of AMCP 706-123 par. 3-2.11.4.3 (p. 3-50; 1,000 lb/in² gauge, 100 °F); entrained air, hoses and temperature reduce the effective value, often severely.

Locked oil. Oil locked between closed valves expands with temperature more than its steel cavity. Equating the thermal volume change (α_v − 3α_s) ΔT to the compression of the oil Δp/β plus the stretch of a thin tube Δp D/(E t) gives Δp = (α_v − 3α_s) ΔT / (1/β + D/(E t)); a rigid cavity gives β (α_v − 3α_s) ΔT. The default oil coefficient, 0.00045 per °F, lies in the 0.00042 to 0.00058 range AMCP 706-123 Fig. 3-38 (p. 3-40) plots for hydraulic fluids. With the default cylinder the rise is 83 lb/in² per °F with the tube elasticity and 91 rigid, which is why locked circuits carry thermal relief valves.

Gas-charged accumulators

On absolute pressures the gas follows P Vⁿ = constant. With the pre-charge P₀ filling the size V₀, the gas volume at the maximum pressure is V₂ = V₀ (P₀/P₂)^1/n_c and at the minimum pressure V₁ = V₂ (P₂/P₁)^1/n; the oil delivered is V₁ − V₂ and the size for a demand ΔV is ΔV divided by the delivered fraction (P₀/P₂)^1/n_c [(P₂/P₁)^1/n − 1]. n = 1 is Boyle's law; n = 1.4 is the adiabatic exponent of nitrogen or air at moderate pressure. AMCP 706-123 par. 2-7.2 (p. 2-23) says the expansion can be taken as adiabatic when the discharge is rapid (less than about a minute) and the compression as isothermal when it is slow, which is the default (isothermal charge n_c = 1, entered n on the discharge); the table prints all three combinations. A pre-charge set at one temperature is corrected to the operating temperature at constant volume (P ∝ T; the NAVEDTRA example of 1,800 lb/in² gauge at 70 °F becoming 2,020 lb/in² absolute at 130 °F is reproduced). The pre-charge must not exceed the minimum pressure; makers recommend somewhat less (about nine tenths for bladder types, a convention cited from memory). The ideal gas is assumed; at high pressure nitrogen departs from it (AMCP 706-123 eq. 2-1).

What is not covered

Telescopic and tandem cylinders, the bending of the tube and rod as a stepped column, seal friction models, cushion pressure histories, the charging transient of an air cylinder and the exhaust-side restriction, valve dynamics, and the fatigue of threads and welds. Related tools: Compressed Air System Sizing (receiver and compressor), Control Valve Sizing (the full ISA equations), Column Buckling Tables, Bolt Preload and Torque, Thread Engagement and Strength, Pressure Vessel.