Basic Equations for Parker Pneumatic Rotary Actuator

Mass Moments of Inertia Equations Table

Rectangular Prism

Ix = 1/12m(b2 + c2)

Iy = 1/12m(c2 + a2)

Iz = 1/12m(a2 + b2 )

 Rectangular Prism

Circular Cylinder

Ix = 1/2ma2

Iy = Iz = 1/2m(3a2 + L2)



Thin Rectangular Plate

Ix = 1/12m(b2 + c2)

Iy = 1/12mc2

Iz = 1/12mb2

 Thin Rectangular Plate

Circular Cone

Ix = 3/10ma2

Iy = Iz = 3/5m(1/4a2 + h2)




Ix = Iy = Iz = 2/5ma2


Thin Disk

Ix = 1/2mr2

Iy = Iz = 1/4mr2

Thin Circular Disk 

Parallel Axis Theorem

Ip = ⌈ + md2


Parallel Axis Theorem 

Ip = Mass moment of inertia about an axis parallel to a centroidal axis

⌈ = Mass moment of inertia about a centroidal axis

m = Mass

d = Distance between axes


t = time

θ = angular position

ωt = angular velocity at time = t

ω0 = angular velocity at time = 0

α = angular acceleration

When Acceleration Is Constant:

θ = ω0t + 1/2αt2 ;   α = 2θ/t2

θ = ω0t + 1/2ωtt ;  α = (ωt - ω0)2/2θ

ω = (ω02 + 2αθ)1/2 ; α = (ωt - ω0)/t

When Velocity Is Constant:

 θ = ωt

Basic Velocity, Acceleration, Kinetic Energy and Torque Equations

(The equations below are based on triangular velocity profile.)


Θ = Angle of rotation (degrees)

t = Time to rotate through Θ (sec)

ω = Angular velocity, radians/sec

α = Angular accelerations (radians/sec2)

WL = Weight of load (lbf)

Ta = Torque to accelerate load (lb-in)

Us = Coefficient of static friction

Jm* = Rotational mass moment of inertia (lb-in-sec2)

Tf = Torque to overcome friction (lb-in)

TL = Torque to overcome effects of gravity

* Use "I" values from the Mass Moments of Inertia table


ωmax = 0.35 x Θ/t

α = ωmax2/ (Θ/57.3)

α = ωmax/(t/2)

K.E. = 1/2 Jmω2

Ta = α x Jm

Tf = W x Us x (Distance from pivot point to center of external bearings)

TL = (Torque arm length to C.G. of load) x WL x cos (Φ)

(Where Φ = Angle between torque arm and horizontal plane)

Coefficients of Friction

Material* μs μk
Steel on steel 0.80 0.40
Steel on steel (lubricated) 0.16 0.03
Aluminum on steel 0.45 0.30
Copper on steel 0.22 0.22
Brass on steel 0.35 0.19
PTFE on steel 0.04 0.04

*Dry contact unless noted

Available Pneumatic Cylinder Styles

Tie-rod air cylinder - non-rotating Tie Rod Cylinders Guided Cylinders
Round Body Cylinders Rodless Cylinders
Compact Cylinders    


Quote by Part Number Help with Cylinder Identification

Engineering & Product Selection Information

Pneumatic Product Selection

  • Pneumatic Actuators & Air Cylinders
  • Pneumatic Cylinders
  • Automation Products: rotary actuators, grippers, slide tables, rotary tables, escapement
  • Actuator Accessories: Linear alignment couplers, flow controls, air oil tanks, rodlocks, electronic sensors, shock absorbers

Application Engineering Data


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