When sizing a motion system with a ball or lead screw drive, the first step is to determine the screw diameter and lead that can meet the application requirements for thrust force, speed, and compressive (buckling) loads. Once the screw diameter and lead are determined, the next step is to choose a motor that can deliver the required torque and speed, while also providing sufficient control of the moved load. One of the key factors when sizing and selecting the motor is the inertia of the load.
Newton’s First Law — sometimes referred to as the law of inertia — explains that an object will resist a change in its state of motion unless acted upon by an external force. For a linear motion system, this means that in order to accelerate or decelerate a load, the motor must overcome the inertia, or resistance to change in motion, of the object that it’s trying to move – in this case, the external load plus the screw itself.
The ratio of the load inertia (external load + screw) to the motor inertia also plays a significant role in determining how well the motor can control the load during acceleration and deceleration and, in turn, how precisely the motor can position the load without overshoot (or undershoot) or excessive settling times.
Note that in this context, we’re referring to mass moment of inertia, as opposed to planar moment of inertia. Mass moment of inertia describes an object’s ability to resist angular acceleration, while planar moment of inertia describes an object’s resistance to bending. Planar moment of inertia is commonly used in linear motion design for determining the deflection or bending of a cantilevered axis.
Inertia is relatively simple to calculate. For a point mass, it’s simply the mass of the object multiplied by the square of its distance from the axis of rotation:
I = mass moment of inertia (kgm2)
m = mass (kg)
r = radius, or distance from axis of rotation (m)
The inertia of a ball or lead screw can be sufficiently approximated by using the formula for inertia of a solid cylinder:
The mass of the screw depends on its volume (radius and length) and material density:
V = volume (m3)
ρ = material density (7810 kg/m3 for bearing steel)
And volume is determined by the radius and length of the screw:
L = length (m)
In many sizing guides, the equation for ball screw inertia includes the variables of length, material density, and radius since these will vary depending on the application:
Note that the letter “I” is typically used as the symbol for inertia, although some manufacturers and reference texts — especially those for motion applications — use “J” for mass moment of inertia and “I” for planar moment of inertia.
Applications that require long stroke lengths and/or large screw diameters can result in very high screw inertia, which may require a larger motor and make it more difficult to precisely control the load. One advantage of a rotating nut ball screw assembly — also referred to as a driven nut — is that it has relatively low inertia, since the screw shaft remains stationary and only the nut rotates.