Sizing Sheets

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Frequently used

V_{T} = [\frac{Δ V}{{(p_{0})}^{\frac{1}{γ}}}] \cdot [\frac{1}{{(\frac{1}{p_{M i n}})}^{\frac{1}{γ}} - {(\frac{1}{p_{M a x}})}^{\frac{1}{γ}}}]

Spring Stiffness

k = \frac{P_{m a x} \cdot (\frac{π}{4}) \cdot {d_{p}}^{2}}{L_{s} + L}

M = \frac{P \cdot (\frac{π}{4}) \cdot {d_{p}}^{2}}{g}

Maximum Stored Energy

E_{a c c; M a x} = \frac{1}{2} \cdot k \cdot (2 \cdot L_{s} \cdot L + {(L)}^{2})

Maximum Stored Energy

E_{a c c; M a x} = M \cdot g \cdot L

Precharge (Absolute) at the minimum working temperature

p_{0; T_{M i n}} = p_{0; T_{M a x}} \cdot \frac{T_{M i n}}{T_{M a x}}

Component Sizing

Piston Side Area

A_{p} = π \cdot {(\frac{d_{p}}{2})}^{2}

A_{r} = π \cdot [{(\frac{d_{p}}{2})}^{2} - {(\frac{d_{r}}{2})}^{2}]

R = \frac{{d_{p}}^{2}}{{d_{p}}^{2} - {d_{r}}^{2}}

Piston side pressure - Resistive force

P_{e} = \frac{F_{e}}{A_{p}}

Piston side pressure - Driving mass

P_{p} = \frac{M \cdot g \cdot \sin (α)}{A_{p}}

Piston side pressure - Spring return

p_{p} = \frac{K_{s} \cdot (L + L_{k o})}{A_{p}}

Stem pressure - Resistive force

P_{r} = \frac{F_{r}}{A_{r}}

Stem pressure - Driving mass

P_{r} = \frac{- M \cdot g \cdot \sin (α)}{A_{r}}

Stem pressure - Output spring

P_{r} = \frac{K_{s} \cdot (L + L_{k o})}{A_{r}}

Extend Pressure

P_{e} = \frac{4 \cdot (f_{e} + M \cdot g \cdot \sin (α) + k_{s} \cdot (L + L_{k o}))}{π \cdot ({d_{p})}^{2}}

Total Extend Load

F_{e} = P_{e} \cdot \frac{π \cdot {d_{p}}^{2}}{4}

Total Retract Load

F_{r} = P_{r} \cdot \frac{π \cdot ({d_{p}}^{2} - {d_{r}}^{2})}{4}

Rod Critical Force

F_{c} = \frac{π^{3} \cdot E}{4 \cdot λ^{2} \cdot {L_{e}}^{2} \cdot S} \cdot \frac{{d_{r}}^{4} - {d_{i}}^{4}}{16}

Average extend speed (with Piston Side Area)

v_{e} = \frac{Q_{e; p}}{A_{p}}

Average extend speed (with Stroke)

v_{e} = \frac{L}{t_{e}}

Average retract speed (with Rod Side Area)

v_{r} = \frac{Q_{r; r}}{A_{r}}

Average retract speed (with Stroke)

v_{r} = \frac{L}{t_{r}}

t_{e} = \frac{π \cdot {d_{p}}^{2} \cdot L}{4 \cdot Q_{e; p}}

t_{r} = \frac{π \cdot (d_{p}^{2} - d_{r}^{2}) \cdot L}{4 \cdot Q_{r; r}}

Spring Stiffness

K_{S} = \frac{P_{e} \cdot π \cdot ({d_{p})}^{2} - 4 \cdot f_{e} - 4 \cdot M \cdot g \cdot \sin (α)}{4 \cdot (L + L_{k o})}

Spring Stiffness

K_{s} = \frac{P_{r} \cdot π \cdot ({d_{p}}^{2} - {d_{r}}^{2}) - 4 \cdot f_{r} + 4 \cdot M \cdot g \cdot \sin (α))}{4 \cdot (L + L_{k o})}

Mechanical Output Power (Extension)

P_{e; o} = (f_{e} + M \cdot g \cdot \sin (α)) \cdot v_{e}

Mechanical Output Power (Retraction)

P_{r; o} = (f_{r} - M \cdot g \cdot \sin (α)) \cdot v_{r}

Component Sizing

Fluid Line Velocity

v_{f} = \frac{4 \cdot Q}{π \cdot D^{2}}

Laminar Flow Pressure Loss

Δ p_{l} = \frac{128 \cdot L \cdot ν \cdot ρ \cdot Q}{π \cdot D^{4}}

Turbulent Flow Pressure Loss

Δ p_{t} = \frac{8 \cdot L \cdot ρ \cdot Q^{2}}{π^{2} \cdot D^{5} \cdot {[- 1.8 \cdot \log_{10} (\frac{1.725 \cdot π \cdot D \cdot ν}{Q} + {(\frac{ϵ}{3.7})}^{1.11})]}^{2}}

Component Sizing

Loss Coefficient

K_{L} = \frac{2 \cdot Δ p}{ρ \cdot {(\frac{4 \cdot Q}{π \cdot D_{i}^{2}})}^{2}}

Component Sizing

C = \frac{Q}{ω \cdot η_{v}}

Overall Efficiency (with Mechanical Efficiency)

η_{t} = η_{v} \cdot η_{m}

Overall Efficiency (with Shaft Input Power)

η_{t} = \frac{P_{h}}{P_{i}}

T = \frac{C \cdot (p_{o} - p_{i})}{η_{m}}

Effective Hydraulic Power

P_{h} = Q \cdot Δ p

Shaft Input Power

P_{i} = T \cdot ω

Hydraulic Output Power

P_{o} = Q \cdot p_{o}

Component Sizing

C = \frac{T \cdot η_{v}}{Δ p \cdot η_{t}}

Q = \frac{C \cdot ω}{η_{v}}

ω = (\frac{Q}{C}) \cdot η_{v}

Shaft Output Power

P_{o} = Q \cdot η_{v} \cdot Δ p \cdot η_{m}

Effective Hydraulic Power

P_{h} = \frac{C \cdot ω \cdot (p_{i} - p_{o})}{η_{v}}

Component Sizing

Discharge Coefficient

C_{d} = (\frac{Q}{(\frac{π}{4}) \cdot {d_{o}}^{2}}) \cdot \sqrt{\frac{ρ \cdot (1 - β^{4})}{2 \cdot Δ p}}

Component Sizing

q = ϵ \cdot \frac{U \cdot A}{N T U} \cdot (T_{h, i} - T_{c, i})

Component Sizing