Cutting Forces in Milling: The Formula for Successful Workholding

One of the basic rules of engineering design is that it’s “done with numbers”, rather than with vague assertions about the properties of the system being designed. There isn’t much use in saying that we want our workholding systems to be “really stiff and strong” without specifying what that means.

It's not very helpful if a customer comes into your shop asking for quotes and says that they wanted the part dimensions to be “very precise” and the surface finish to be “super smooth." Machinists need a set of accurate numbers to guide the process and ensure what the customer wants is what is delivered. The same is true when you shop for workholding systems. Quantifying the level of performance you want will help you select the system that will make your vision of the part become a reality. Yes, there's some math involved, but luckily, math is fun. Let’s go through it.

Workholding 101

The first step in becoming an informed workholding buyer is to learn how to estimate the magnitude of cutting forces during a machining operation. Using that information, we can specify what performance we require from the components selected for our workholding system. Let's start with a basic question: What is it we want a workholding system to do?

Two of the most important requirements of workholding systems are:

1. Hold the workpiece against the cutting forces
2. Minimize deflections due to those forces  

The first requirement simply means that however the workpiece is held in the fixture, it will not loosen or be pulled out of the fixture by the cutting forces. The second requirement recognizes that to achieve the proper dimensions and surface finish, tolerances must be held without variation. This means the workholding system cannot deflect or vibrate excessively when exposed to the forces generated by the cutting process.

Of course, if we are trying to design or select a workholding system intelligently, we will need to know just how large those cutting forces are so we can evaluate how the workholding system responds. 

Cutting forces in milling

In milling, the edge of a rotating tool is forced through a stationary workpiece to form a chip. Most milling cutters have multiple cutting edges, and as the tool rotates each edge engages the workpiece, removes a chip, and exits the workpiece. 

The figure below shows a top view of a slotting operation using a 2-fluted cutter. The cutter is rotating clockwise and is being fed into the workpiece towards the right. The crescent-shaped area bounded by the dashed line is the chip that will be removed by this edge. The next edge will remove a similarly shaped region. The thickness of the chip as it is removed is zero when the edge first engages the work, grows to its maximum value after 90 degrees of cutter rotation, and then drops back to zero when it exits after 180 degrees of rotation.

At any instant during the rotation, there is a cutting force component known as the tangential cutting force, F_t, acting on the edge of the cutter that is tangent to the cutter body. The magnitude of the tangential force is proportional to the axial depth of cut, b, the instantaneous chip thickness, t, and is dependent on the type of material being cut.

The maximum value of the chip thickness, f, is the feed-per-tooth, or chip load, selected by the programmer. Therefore, the cutting force will reach its maximum value when the edge is at 90 degrees, and the chip thickness is maximum.

There are also components of the cutting force directed from the edge towards the tool’s center, and along its axis. However, in practice these radial and axial cutting force components are normally a small fraction of the tangential force; so we will concentrate only on the tangential cutting force, calling it the cutting force.  

As this figure shows, the cutting force is not constant in either magnitude or direction. This is a characteristic of almost all milling operations and is different from lathe operations where the single cutting edge is constantly engaged with the workpiece as it rotates, and the resulting cutting force is nominally constant.

Another thing to consider is if the tool has more than two cutting edges, each contributes its own cutting force, and the workholding system must be designed to counteract their combined total. If the cut uses a partial radial engagement (less than a full slot), the situation changes again with regard to when edges are in or out of the workpiece. If the radial engagement is less than half the cutter diameter, the maximum chip thickness never reaches the “feed-per-tooth” value, f.

Estimating the maximum cutting force

It might seem hopelessly complex to calculate cutting forces. Since they vary continuously during a cut, the workpieces held may vary substantially in size, shape, and material, and each different part being machined may use many different cutters. Fortunately, in engineering design, we don’t generally try to analyze every single loading situation a structure might encounter in use. Instead, we want to create our design so that it is capable of withstanding the highest loads expected.  

For example, a bridge designer doesn’t bother to calculate the stresses of every weld in the bridge for the case of a small child on a bicycle crossing alone. Instead they base their analysis on their estimate of the largest loads that might ever be encountered, like a closely spaced military convoy of tanks crossing the bridge. Let’s do the same for workholding systems.

Machining is a subtractive process in which every workpiece starts out with an initial volume, and ends with a smaller, final volume. Generally speaking, the larger the volume of material to be removed, the longer it takes to machine it away. One key to minimizing cycle time is to maximize the material removal rate, or MRR; and high MRRs generally mean higher cutting forces and require higher spindle powers.  

Most cuts do not consume the full spindle power, although it is good practice to maximize MRR as much as possible, considering constraints on tool breakage and wear, and surface finish. Nonetheless, it makes sense to see how high the cutting forces are when the spindle is operating at its full capacity and MRR is maximized and use those values when we analyze the workholding system components.

For most machining operations it is easy to calculate the maximum value the total cutting force can reach using the rated power of the spindle motor. The simple formula below uses only three inputs, the spindle power, P, in horsepower, the spindle speed, n, in rpm, and the tool radius, rtool, in inches; and gives the cutting force, F_t, in pounds. This relationship holds despite the material being cut, the type of cutter used, the tool path, or the selected feeds and speeds.

For instance, suppose your milling center has a spindle that is rated at 30 HP and you are cutting with a 1-inch diameter tool at a spindle speed of 2500 rpm. The tangential cutting force will be F_t,max=1512 lbs when the spindle operates at max power, and somewhat less than that for any operation that does not consume the full power of the spindle. The workholding system must be sturdy enough so that forces of this magnitude don’t loosen the part, and any resulting deflections or vibrations don’t cause problems with tolerances or surface finish.  

The formula above shows us that for a given spindle power and speed, as the tool diameter increases, the maximum cutting force the spindle can generate decreases, so a 2-inch diameter cutter can only create cutting forces that are half those for a 1-inch cutter, or 506 lbs. Increasing the spindle speed can also reduce the maximum cutting force, especially for smaller diameter tools. However, at some point, as we try to maximize MRR for smaller diameter tools, the cutting forces can become so large that the tool either breaks or wears out prematurely.  

In this instance, the limit on MRR is not caused by insufficient spindle power; and the cutting force will generally be lower than for the max power case. Nevertheless, we should design our workholding system to be capable of handling the cutting forces caused by the max power case.  

Spindle power vs. spindle torque

Variable speed electric motors, like machine tool spindles, aren’t usually capable of delivering their full-rated power at all speeds. Similarly, they usually cannot deliver their full-rated torque at all speeds. 

Not all machine tool OEMs provide information about spindle torque to their customers, and in that case, we will need to rely on the max power to estimate the maximum cutting force that might be exerted on the part, and thus on the workholding system.  

If you know the maximum torque rating of your spindle motor, T_max, it is easy to calculate the max cutting force using the formula below, where T_max is in inch-lbs, and r_tool is in inches.

If the maximum cutting force calculated from the spindle torque rating is higher than from the maximum power rating, then we will use that value. In practice, the two estimates are usually similar, and since we will be using generous safety factors either is suitable for our purposes.

What about vibrations?

From this diagram, we also see that a single cutting edge cannot engage the workpiece for more than 180 degrees of tool rotation in a slotting cut. Most milling operations use less than a slotting cut, so the tooth’s angle of engagement in the work is even less.  

If the radial depth-of-cut is low, the tooth may only engage for a small part of a tool revolution. For example, consider a finishing operation using a one-inch diameter end mill with four teeth, as shown in the figure below. If the finishing pass removes 0.010” of material, the cutting edge enters the workpiece at Position A and leaves it at Position B. A little trigonometry shows us that the tooth is only engaged with the workpiece for 11.5 degrees of rotation. The next tooth doesn’t engage the workpiece for another 78.5 degrees of rotation, meaning that there is no contact with the workpiece for approximately 88% of the time. During that time the cutting force is zero.

The cutting force here is actually a series of short pulses that occur at the tooth passing frequency, and anyone who has operated a milling machine will now understand that the process’ distinct sound is actually caused by the varying force as each tooth engages and leaves the workpiece. This is part of all milling operations and can lead to excessive vibrations if the workholding system is not properly designed. 

In extreme cases, if the workholding system is too flexible the vibrations can turn into full-blown chatter, resulting in poor surface finish and even damage. The dynamic vibrational response of the workholding system to the periodic variations in the cutting force should ideally never be the limiting factor in how aggressive the machining operation can be in pursuit of maximizing MRR to lower cycle time.  

Summary

In this post, we learned how to use the maximum rated spindle power of a machine to estimate the maximum cutting force that the machine tool can ever generate and therefore exert on the workpiece and its workholding system. We also saw that the cutting force is not constant in milling operations, but varies at the tooth passing frequency, potentially causing vibrations that may lead to poor surface finish.

Download the pdf here.

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