TORQUE is a force that produces rotation of a shaft. It is measured by the product of the force (F) and the perpendicular distance from the line of action of the force to the centerline of rotation (r).
T = F r
ROTATION results when an unbalanced torque acts on a body producing an angular acceleration. The torque to accelerate is the product of the body's moment of inertia about its axis of rotation (J) and the angular acceleration (a).
T = J a
MOMENT OF INERTIA of a body is determined by the distribution of its mass about the axis of rotation. It tends to resist any change in angular velocity.
J = m r2
ANGULAR ACCELERATION is the rate of change of angular velocity and is expressed in radians per second per second. If angular velocity changes from wo at time o to wt at time t in time (t), the average angular acceler- ation is:
a = (wt - wo)/t
ANGULAR VELOCITY is the rate of angular rotation about an axis and is expressed in radians per second. If a body moves through a rotation of q radians in a time of t seconds, the average angular velocity is:
a = q/t
ANGULAR ROTATION is the arc traveled in rotary motion and can be expressed in degrees, revolutions or radians. One radian is the angle defined from the center of a circle by an arc that is equal in length to the radius.
KINETIC ENERGY is the energy of a mass in motion. It is a function of the moment of inertia (J) and the square of the angular velocity (w) expressed as:
Ek = 1/2 J w2
EQUATIONS FOR ANGULAR MOTION are analogous to those for linear motion:
| v = a t | s = 1/2 a t2 | v2 = 2 a s |
| w = a t | f = 1/2 a t2 | w2 = 2 w f |
If vo and wo denote the initial linear and angular velocity, then
| v = vo + a t | s = 1/2 j w2 | v2 = vo2 + 2 a s |
| w = wo + a t | f = wo + 1/2 a t2| w2 = wo2 + 2 a f |
|
| Linear Displacement | s | s = q r | Angular Displacement | q | |
| Linear Velocity | v | v = w r | Angular Velocity | w | |
| Linear Acceleration | a | a = a r | Angular Acceleration | a | |
| Mass (Inertia) | m | J = m r2 | Moment of Inertia | J | |
| Force | F | T = F r | Torque | T | |
| Linear: | F = m a | Ek = 1/2 m v2 | Work = F s | Power = F v | |
| Angular: | T = J a | Ek = 1/2/ J 22 | Work = T q | Power = T w |
| Symbol | Quantity | Units of Measure |
| a | Linear Acceleration | in/sec2 |
| Cf | Friction Coefficient | (dimensionless) |
| CT | Torque Coefficient | lb-in/psi |
| Ec | Cushion Energy | lb-in |
| Ef | Friction Energy | lb-in |
| Eg | Gravitational Energy | lb-in |
| Ek | Kinetic Energy | lb-in |
| Ep | Propelling energy | lb-in |
| ET | Total Energy | lb-in |
| F | Force | lb |
| g | Acceleration due to Gravity | 386 in/sec2 |
| J | Moment of Inertia | lb-in-sec2 |
| m | Mass (W/g) | lb-sec2/in |
| P | Pressure | psi |
| r | Radius | in |
| ra | Radius Arm Length | in |
| rb | Radius of Bearing | in |
| s | Linear Displacement | in |
| T | Torque | lb-in |
| Ta | Torque of Acceleration | lb-in |
| Td | Torque of Deceleration | lb-in |
| Tf | Torque of Friction | lb-in |
| Tp | Torque of Propulsion | lb-in |
| t | Time | sec |
| v | Linear Velocity | in/sec |
| W | Weight | lb |
| a | Angular Acceleration | rad/sec2 |
| q | Angular Displacement | rad |
| qa | Angle of Acceleration | rad |
| qd | Angle of Deceleration | rad |
| f | Angle of Arm to Vertical | deg |
| Average Angle from Vertical | deg | |
| wa | Angular Velocity | rad/sec |
Caution: Formulas given on this page, Rotary Motion, and Moment of Inertia and Cushion Data must be applied to all applications to assure proper selection of the actuator and system accessories.