Safety couplings that operate on the ball detent principle primarily suit disengagement torque applications. But, with some modification, they can suit highly dynamic applications with resonant frequencies and torsional rigidity. Here is a brief examination of common equations used to calculate the following torques for safety coupling design in a drive system: disengagement torque, acceleration torque, acceleration and load moment, thrust force, resonant frequency, and torsional rigidity.

**Disengagement torque.** The disengagement torque must be greater than routine torque moments within a drive train. First, determine torque requirements within the drive train. In practice, a multiplication factor of 1.5 times the nominal operating torque is often adequate to accommodate acceleration moments and other influencing factors. To calculate minimum torque ratings for a drive train, use the following equation:

T_{KN} ≥ 1.5 x T_{AS}

Where:

T_{KN }= torque in the drive train (Nm)

T_{AS} = Peak torque in the drive train (Nm)

Peak torque is usually taken from the rating plate on the given drive mechanism.

You can use the number 9,550 as a constant value to convert power into Nm. Thus:

T_{KN} ≥ 9,550 x P_{AN}/*n* x 1.5

Where:

P_{AN }= Power of the driving side (kW)

*n* = speed (rpm)

**Acceleration torque**. The acceleration torque method is a more accurate technique. In addition to angular acceleration, it makes allowances for peak torque on the driving side, the mass distribution, and the moments of inertia inherent to the driving and driven ends. With the help of a correction factor (surge or load factor) established according to the machine and application, acceleration torque can be determined using this method. Normally, a distinction is made between three types of surge or load factors:

S_{A} = 1 (harmonic strain)

S_{A} = 2 (periodic strain)

S_{A} = 3-4 (non-periodic strain)

The following equation reflects these relationships:

T_{KN} ≥ α x J_{L} ≥ (J_{L}/J_{A} + J_{L}) x T_{AS} x S_{A}

α = Angular acceleration (s^{-2)}

J_{L} = Moment of inertia on the load side (kgm^{2})

J_{A} = Moment of inertia on the driving side (kgm^{2})

S_{A} = Surge or load factor

**Acceleration and load torque**. The most accurate but complex assessment of torque for the evaluation of safety couplings is the acceleration and load torque method (start-up under load). This approach simulates an application in which constant acceleration and deceleration under load conditions takes place. Load torque is used as an additive factor to acceleration torque.

The following equation, with differentiation of individual variables, describes this relationship:

T_{KN} ≥ a x J_{L }+ TAN ≥ [(J_{L}/J_{A} + J_{L}) x (T_{AS} – T_{AN}) + T_{AN}] x S_{A}

T_{AN }= Peak torque for the load side (Nm)

These three design methods are based on manufacturer data for the drive and the load components. In addition to torque moments, only moments of inertia and potentially incurred acceleration are included.

**Thrust force.** Another option for assessing application torque is the thrust force method. This method can be applied to spindle and lead screw drives as well as toothed belt drives, depending on the design of the drive system.

In addition to overall thrust force for the entire unit, thread pitch and efficiency play substantial roles in the proper design of spindle and lead screw drives. Here is the equation for the applied torque:

T_{AN }= (s × F_{v})/2000 × ∏ × η

s = thread pitch (mm)

F_{v }= thrust force (N)

η = efficiency

∏ = pi

If the drive and load are not linked by way of a spindle or lead screw, but by a toothed belt drive, use the following equation to calculate the incurred torque:

T_{AN }= (d_{0} × F_{v})/2000

d_{0 }= pinion diameter of the toothed pulley (mm)

**Resonant frequency**. Each body and component in the drive train has its own natural frequency. The resonant frequency of the coupling and the entire drive system can be approximated with the following equations. A prerequisite for the calculations is the summation of mass moments of inertia of the individual components to determine the total mass moment of inertia. The torsional rigidity of the entire drive train also has a big influence on oscillation. The equation for calculating the coupling’s resonant frequency in Hz is:

ƒ_{e} = 1/2p x √C_{T} x ((J_{A} + J_{L})/(J_{A} x J_{L}))

The equation for calculating the natural oscillation in speed is:

*n*_{e} = 30/p x √C_{T} x ((J_{A} + J_{L})/(J_{A} x J_{L}))

ƒ_{e }= resonant frequency of the system (Hz)

C_{T} = Torsional rigidity of the coupling (Nm/rad)

*n*_{e} = Natural oscillation term of the system (rpm)

**Torsional rigidity**. Whether a machine is designed to be rigid or damping depends on the respective application. The rigidity of all individual components, including the coupling, should always be taken into account. In theory, if a body twists by a defined angle if it is subjected to a certain load (torque). The degree of twist depends on the rigidity of the body (countering the torque). This relation is expressed:

φ= 180/p x T_{AS}/C_{T}

**R+W America**

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