Drive Science

In engineering disciplines where motion is not a concern, such as building a structure to support static loads, the need to fully appreciate the relationships between mass, force and weight rarely comes into play. However, when considering the characteristics of mechanical systems and calculating the force needed to accelerate objects — the world of drive system engineers — having a clear understanding of mass, force and weight relationships is critical.

When considering them as independent parameters, most technically trained individuals appreciate the difference between force and mass. However, due to the misunderstanding of the terms used to identify the weight of a body or component, even engineers get confused as to the relationships between these parameters and the appropriate engineering units to be applied in formulas for calculating torque and power requirements.

While seemingly elementary, a failure to grasp the underlying principles of these fundamental concepts can unwittingly lead to faulty designs and project delays. To help prevent such problems, physical considerations are the starting point for building a successful coordinated drive system.

Mass is the property of a body that quantifies its linear inertia and influences the force necessary to support it in a gravitational field. The engineering units of mass are grams in the metric system and slugs in the imperial system.

Force is the influence that, if applied to a free body, results in an acceleration of the body. This is the definition of force that is of importance to drive engineers. The engineering units of force are Newtons in the metric system and pounds in the imperial system.

Clearly force and mass are two very different parameters, and interchanging their respective engineering units will result in confusion and yield unworkable results when attempting to resolve dynamic applications. Even though some conversion references imply direct relationships — and non-technical communities often refer to these comparisons — technically there are no such relationships.

The term ‘weight’, or influence, has no consistently defined engineering units, as do force and mass. This is the primary issue that contributes to the confusion when it comes to applying the proper engineering units to formulas for calculating torque and power. In the metric system, weight (or influence) of a body is typically defined as its resistance to acceleration — or its inertia. In the imperial system, weight is typically defined as the force, in pounds, to support a body in its gravitational field.

Either definition is appropriate, but it is important to understand which engineering units are implied. Are they mass units or force units? For an engineer responsible for sizing a drive to accelerate a load at a required rate, it’s important to know which units are correct.

The imperial system

To mitigate this confusion, the term ‘weight of a body’ should be interpreted as the ‘influence of a body’. The custom in the imperial system is to assign the influence (weight) of a body to be the force (in pounds) necessary to resist the acceleration-of-gravity on that particular body.

A spring-type scale is a good example of a force-measuring device, and an appropriate engineering unit for its display of applied weight (influence) would be pounds of force. However, it must be recognized that this weight characteristic, although a function of its mass, is not its mass. It is the force required to support the mass. (Note: Although somewhat customary, force-measuring spring-scales, that provide a display in kilograms, are technically incorrect.)

To establish the mass of a body, when its influence is identified as the force required to support the body, engineers need to divide by the rate of acceleration due to the gravitational field in which the body resides. On Earth, and in imperial units, the rate of the acceleration-of-gravity is 32.17 ft/sec2.

Mass (slugs) = [Weight (lb)] / 32.17 ft/sec2

The metric system

The custom in the metric system is to assign the ‘influence’ of a body to be its characteristic to resist acceleration in any direction — that is, its mass or linear inertia. Therefore, whenever the ‘weight’ of a body is quantified in units of force, it must be converted to units of mass if an analysis of dynamics is required.

A balance-beam type scale is a good example of a mass-measuring device, and an appropriate engineering unit for the indication of its ‘weight’ (influence) would be kilograms (or grams) of mass. The indication of the calibrated weights are imbedded in the standard counter-weight applied to the balance-beam. This method of measuring the ‘weight’ of a body is independent of the gravitational field and would be appropriate at any location in the universe.

Linear mechanical power

Having established the kinetic definition of force, we can now review the concept, definition and engineering units of linear mechanical power. Power mechanically is defined as the linear relationship established by the product of force times velocity. If either is zero (0), there is no power dissipated.

The engineering-unit based definitions for linear mechanical power in the metric and imperial systems are demonstrated in Figs. 1 and 2.

These engineering-unit definitions for power are two of many examples of how cumbersome the imperial-unit system is compared to the metric-unit system. For U.S.-based engineers required to function in an international community, the application of the imperial system is both awkward and often embarrassing. Eventually, the custom of using the imperial system for engineering solutions will succumb to reason.

Rotational displacement

The universal unit of angular or rotational displacement is the radian, defined as the unit angular displacement whose resultant tangential arc-distance equals the radius of the arc. By definition, the radian has no linear units of displacement.

This is important to know because the ‘radian’ is the mechanism that aids in the transition between linear and rotational physics. Translated linear distance (arc distance) is then equal to the product of the radius of the arc, times magnitude of rotation in radians.

By geometric relationship and definition, p is the numeric relationship between the diameter of a circle and its circumference:

Circumference = pi [Diameter],

and

Circumference = 2pi [Radius]

Therefore, there are 2pi radians per revolution, or about 57.3 per radian. Although these relationships may be interesting and commonly used, for the understanding of basic physics, the radian is the most important rational displacement concept, and a revolution is simply an arbitrary amount of rotational displacement.

Torque

The rotational corollary to linear force is torque. By definition, torque is a turning or twisting force that produces or tends to produce rotation or torsion. Torque consists of the product of the resultant linear tangential ‘force’ times the perpendicular distance from the line of action of the force to the axis of rotation.

If these resultant tangential linear forces are now applied to our earlier developed linear power examples, the rotational-to-linear power relationships shown in Figs. 3 and 4 can be established for both the metric and imperial systems — and the intrinsic importance of the radian as a displacement concept is apparent. The only unfortunate issue is the use of the obtuse imperial engineering units.

Radius of gyration

We have defined and used torque to determine rotational power at steady state velocity. Next we need to establish how to determine the required magnitude of torque for acceleration. For this, we first need to understand the concept of the radius of gyration.

Conceptually, the radius of gyration is the distance that, if the entire mass
of the object were all concentrated at that radius, would give the same moment of inertia as the original object. Because most drive applications involve accelerating various forms of cylinders, let’s look at the radius of gyration for a cylinder. For a cylinder, radius of gyration is represented by the formula shown in Fig. 5.

Therefore, for a very thin ring, where r1 approaches r2, the equation results in the radius of gyration, K, equaling r2 — which is intuitive. Also, for a solid cylinder, where r1 equals 0, the equation results in the radius of gyration, K, as shown in Fig. 6, or — the RMS (root-mean-square) value of the ‘circular’ characteristic of the radius of gyration — which also appears to be an intuitive solution.

Moment of inertia

Next we have to understand rotational inertia, also referred to as the moment of inertia. Moment of inertia is defined as the ratio between the torque applied to a rigid body free to rotate about a given axis — to the angular acceleration produced about that axis.

Moment of Inertia =Torque/Angular Acceleration

Moment of inertia, which depends on the shape of the object, is important when solving problems that have to do with how things rotate. From reference material we know that the moment of inertia equals the product of the body’s mass and the square of its distance from the axis of rotation.

Why do the engineering units for rotational inertia, or moment of inertia of a body, include the engineering units of distance squared, or K², while linear inertia for the same body is simply mass? Remember that the engineering units for rotational solutions must be consistent with the compatible linear solution. Here, the compatible linear solution would be for an identical magnitude of mass being accelerated tangentially at the radius of gyration.

Therefore, understanding of the function and unit-less nature of the radian, as with velocity and power, one of the Ks is to locate the mass radially and provide for the units of linear tangential displacement not included in the unit-less angular displacement of the radian.

Linear Displacement = [Radians] x [Radius]

The second K is required to factor the relationship between torque (ft-lb) and the theoretical resultant tangential linear accelerating force (lb).

Force = Torque / Radius

Because it is understood that this ‘constructed’ characteristic of rotational inertia, or moment of inertia, is used only for the analysis of accelerating rotationally free bodies around a given axis, K² — along with its engineering units — are conditionally assigned to be part of what is known as the moment of inertia or rotational inertia.

As this discussion illustrates, the ability to grasp these basic physical concepts is instrumental to making accurate calculations and solving common drive sizing equations. More specifically, the ability to effectively distinguish the relationships between force, mass and weight, particularly using the imperial system of units, is fundamental to successful drive system design.

Pete Werner is a senior principal engineer with the Global Drive Systems business of Rockwell Automation.