Basic Equations for Parker Pneumatic Rotary Actuator
Mass Moments of Inertia Equations Table
Rectangular Prism I_{x }= 1/12m(b^{2 }+ c^{2}) I_{y} = 1/12m(c^{2} + a^{2}) I_{z }= 1/12m(a^{2} + b^{2 }) 
Circular Cylinder I_{x }= 1/2ma^{2} I_{y} = I_{z }= 1/2m(3a^{2} + L^{2})


Thin Rectangular Plate I_{x }= 1/12m(b^{2 }+ c^{2}) I_{y} = 1/12mc^{2} I_{z }= 1/12mb^{2} 
Circular Cone I_{x }= 3/10ma^{2} I_{y} = I_{z }= 3/5m(1/4a^{2} + h^{2})


Sphere I_{x }= I_{y} = I_{z }= 2/5ma^{2} 
Thin Disk I_{x }= 1/2mr^{2} I_{y} = I_{z }= 1/4mr^{2} 

Parallel Axis Theorem Ip = ⌈ + md^{2}

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 
Where:
t = time
θ = angular position
ω_{t} = angular velocity at time = t
ω_{0} = angular velocity at time = 0
α = angular acceleration
When Acceleration Is Constant:
θ = ω_{0}t + 1/2αt^{2 };^{ }α = 2θ/t^{2}
θ = ω_{0}t + 1/2ω_{t}t ; α = (ω_{t}  ω_{0})^{2}/2θ
ω = (ω_{0}^{2} + 2αθ)^{1/2} ; α = (ω_{t}  ω_{0})/t
When Velocity Is Constant:
θ = ωt
Basic Velocity, Acceleration, Kinetic Energy and Torque Equations
(The equations below are based on triangular velocity profile.)
Where:
Θ = Angle of rotation (degrees)
t = Time to rotate through Θ (sec)
ω = Angular velocity, radians/sec
α = Angular accelerations (radians/sec^{2})
WL = Weight of load (lbf)
T_{a }= Torque to accelerate load (lbin)
U_{s} = Coefficient of static friction
J_{m}* = Rotational mass moment of inertia (lbinsec^{2})
T_{f} = Torque to overcome friction (lbin)
TL = Torque to overcome effects of gravity
* Use "I" values from the Mass Moments of Inertia table
Equations:
ωmax = 0.35 x Θ/t
α = ωmax^{2}/ (Θ/57.3)
α = ωmax/(t/2)
K.E. = 1/2 J_{m}ω^{2}
T_{a} = α x J_{m}
T_{f} = W x U_{s} 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*  μ_{s}  μ_{k} 
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 Cylinders  Guided Cylinders  
Round Body Cylinders  Rodless Cylinders  
Compact Cylinders 
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
 Operating Principles and Construction
 Fluids and Temperature
 Push and Pull Forces
 Mounting Information
 Ports
 Tie Rod Supports, Stroke Data & Stroke Adjusters
 Mounting Classes
 Stop Tubing
 Stroke Selection Chart
 Deceleration Force and Air Requirements
 Cushion Ratings and Air Requirements
 NFPA Rod End Data and Piston Rods
 Modifications, Special Assemblies, Tandem Cylinders, Duplex Cylinders
 Rotary Actuator Torque Requirements
 Rotary Actuator Basic Equations
 Conversion Factors
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