Deflection, rigidity and chatter

A machine can have ample power and the cutting data can be perfectly chosen, yet the part comes out tapered, oversize or covered in chatter marks. The usual cause is not the cut itself but the structure carrying it: a long end mill, a slender shaft, a boring bar reaching deep into a bore, or a part clamped where it can flex. Under cutting forces, everything bends a little. When the bending is large enough to matter, it limits accuracy and productivity before power does.

This chapter shows how to estimate deflection with the simplest structural model, why overhang and diameter dominate the result, and what chatter is and how to reduce it.

The cantilever model

Most cutting tools, and many workpieces, behave roughly like a cantilever: a beam fixed at one end with a load at the other. Standard beam theory, found in any mechanical design text, gives the deflection at the loaded end as:

δ = F × L³ ÷ (3 × E × I)

where F is the force perpendicular to the beam, L the overhang, E the elastic modulus of the material and I the second moment of area of the cross-section. For a solid round section of diameter d:

I = π × d⁴ ÷ 64

Two features of these formulas explain most deflection problems in machining:

• deflection grows with the cube of overhang, so a 33% longer overhang more than doubles it; and
• stiffness grows with the fourth power of diameter, so a modestly smaller tool is much more flexible.

Figure 1A cutting tool behaves like a cantilever

The model is approximate. Real tools are not uniform cylinders, holders and spindles flex too, and the fixed end is never perfectly rigid. But it captures the trends reliably and gives realistic orders of magnitude.

End mill deflection

A fluted end mill is less stiff than a solid bar of the same diameter. A common approximation treats it as a solid bar of about 80% of its diameter, although the true figure varies with flute count and core diameter.

This is an illustration. Harbourline’s bracket profiling job from Chapter 4 uses a Ø16 solid carbide end mill with 48 mm projecting from the holder. Take E = 600,000 N/mm² for cemented carbide, an effective diameter of 0.8 × 16 = 12.8 mm and an assumed radial cutting force of 400 N acting near the tip.

• I = π × 12.8⁴ ÷ 64 = 1,318 mm⁴
• δ = 400 × 48³ ÷ (3 × 600,000 × 1318) = 0.019 mm

Increasing the overhang by a third, from 48 to 64 mm, multiplies deflection by (64 ÷ 48)³ = 2.37. Dropping from Ø16 to Ø12 at the same overhang multiplies it by more than three. Carbide’s high stiffness, nearly three times that of steel, is a major reason solid carbide tools hold size on long reaches.

Figure 2Deflection of a Ø16 end mill under 400 N against overhang

Solid carbide, E 600 GPaSteel body, E 210 GPa

Tip deflection (mm)Overhang from holder (mm)

Deflection of a Ø16 end mill under 400 N against overhang data

Overhang from holder (mm)	Solid carbide, E 600 GPa	Steel body, E 210 GPa
32	0.0055	0.0158
40	0.0108	0.0308
48	0.0187	0.0533
56	0.0296	0.0846
64	0.0442	0.1263
72	0.0629	0.1798
80	0.0863	0.2467

A deflection of 0.019 mm may be acceptable for a bracket wall with a tolerance of ±0.05 mm. On a wall toleranced at ±0.02 mm it is not. The tool bends away from the wall most at the tip, so the wall comes out tapered, thicker at the bottom than at the top. Remedies include shorter projection, a larger cutter, lighter finishing cuts with a constant small radial engagement, or a spring pass that repeats the finish path to remove material left by deflection.

The holder and spindle bend too

The cantilever calculation treats the tool as if it were clamped in a perfectly rigid wall. In reality, the holder, the taper connection to the spindle and the spindle bearings all add their own compliance, and on a long or heavily loaded tool they can add a large share of the total deflection.

Holder choice matters. Shrink-fit and hydraulic holders grip the shank around its full circumference and usually give low runout and good stiffness. Collet chucks vary with collet quality and condition. Side-lock holders grip a flat on the shank, which can push the tool slightly off centre. Beyond the type, the holder’s own length and diameter matter in exactly the same way as the tool’s: a long, slim holder used to reach past a fixture can bend more than the cutter it carries. Keeping the total gauge length from spindle face to tool tip as short as the job allows is the single most effective stiffness measure for milling.

Runout loads the teeth unevenly

If a milling cutter runs slightly off centre, one tooth sweeps a larger circle than the others and takes a thicker chip, while the others take thinner chips. This is an illustration. With the bracket profile’s corrected feed of 0.121 mm per tooth, a radial runout of 0.02 mm at the cutting edges could make the chip on the highest tooth up to about 0.02 mm thicker than intended, roughly a sixth more, and the chip on the opposite tooth correspondingly thinner.

The tooth that cuts the thicker chip wears faster and carries more force; the one that cuts the thinner chip may rub. The cutter’s life is then set by its most heavily loaded tooth, and the surface shows a once-per-revolution pattern. Measuring runout at the cutting edges after the tool is assembled in its holder, and using holders and shanks in good condition, protects both tool life and finish. For small cutters, where the feed per tooth itself is only a few hundredths of a millimetre, a few micrometres of runout can matter as much as this example’s 0.02 mm does for the larger tool.

Workpiece deflection in turning

Slender workpieces deflect in the same way. A Ø20 steel shaft held in a chuck with 80 mm projecting, and an assumed radial cutting force of 500 N at its end, deflects by:

• I = π × 20⁴ ÷ 64 = 7,854 mm⁴
• δ = 500 × 80³ ÷ (3 × 210,000 × 7854) = 0.052 mm

The part springs away from the tool, so the diameter at the free end is left oversize by about twice that amount, around 0.1 mm, and the shaft comes out tapered. Halving the projection to 40 mm reduces the deflection to 0.0065 mm, an eighth of the original.

Supporting the free end changes the structure completely. A tailstock centre turns the cantilever into a beam supported at both ends, which is many times stiffer. A steady rest supports long parts in between. Many shops bring up the tailstock once the unsupported length exceeds roughly three to four times the diameter and use a steady rest for much longer parts, but the right limit depends on the tolerance, the cutting forces and the material. Reducing the radial force helps too: a sharp insert with a small nose radius and a 90° or greater entering angle pushes the work less.

Boring bars

Internal turning combines both problems: the boring bar must be thinner than the bore and must reach its full depth. Suppose a Ø32 bar must reach 160 mm into a bore, five times its diameter, with an assumed radial force of 300 N:

• A steel bar deflects about 0.038 mm.
• A solid carbide bar of the same size deflects about 0.013 mm.

Tool suppliers typically suggest steel boring bars for overhangs up to about four times the bar diameter, carbide bars for up to about six times, and tuned, damped bars for longer reaches, sometimes ten times or more. Those limits are about vibration as much as static deflection, because a long, flexible bar has a low natural frequency and is prone to chatter.

The Harbourline flange’s Ø40 bore is only 30 mm long. A bar of about Ø32 at little more than one diameter of overhang is very stiff, so deflection is not a concern there.

What chatter is

Chatter is a self-excited vibration. It starts when the tool or workpiece vibrates slightly and leaves a wavy surface. On the next revolution or the next tooth pass, the edge meets that wavy surface, so the chip thickness varies, so the cutting force varies, which drives the vibration further. If the system’s damping cannot absorb the energy fed in, the vibration grows until it is limited by the tool leaving the cut. The result is a loud, characteristic noise, a patterned surface, and rapid tool wear or edge chipping.

Because each pass cuts over the surface left by the previous one, this mechanism is called regenerative chatter. Whether a cut chatters depends on the depth of cut, the stiffness and damping of the tool, holder, spindle, workpiece and fixture, and the spindle speed. Research on machining dynamics, such as the work summarised by Altintas in the references, shows the stable depths of cut as a series of lobes against spindle speed: at certain speeds a much deeper cut is stable than at others.

Chatter differs from forced vibration, which comes from an outside source such as an unbalanced spindle, a damaged bearing or interrupted cuts, and repeats at a fixed frequency regardless of depth of cut.

Reducing chatter

Practical remedies, roughly in order of how often they work:

1. Reduce the depth or width of cut. Chatter is strongly linked to how much of the cutting edge is engaged.
2. Shorten and stiffen. Reduce tool overhang and workpiece projection; use the largest practical tool and holder; tighten fixtures and support thin walls.
3. Change spindle speed. Moving the speed by a modest amount, up or down, can move the cut from an unstable region into a stable lobe, particularly at higher spindle speeds. It is a trial, not a guarantee.
4. Use sharper, freer-cutting geometry. Positive rake, sharp edges and small nose radii reduce cutting and radial forces.
5. Use variable-pitch or variable-helix cutters. Uneven tooth spacing disrupts the regenerative effect.
6. Add damping. Damped boring bars and holders absorb vibration energy.

Figure 3Working through a chatter problem

1. 1Reduce engagementless depth or width of cut
2. 2Shorten and stiffentool overhang, part projection, fixture
3. 3Try another speeda modest change can reach a stable lobe
4. 4Freer-cutting toolpositive rake, sharp edge, small nose radius
5. 5Uneven pitchvariable-pitch or variable-helix cutters
6. 6Add dampingdamped bars and holders

Find stable speeds by testing

The stability lobes that research describes cannot usually be calculated in a job shop without specialised measurement, but their practical consequence can be used directly: if a cut chatters, a change of spindle speed, sometimes up rather than down, may stop it without reducing depth or removal rate.

A practical test is to take the intended cut at a range of spindle speeds, perhaps in steps of a few per cent, listening for the chatter note and looking at the surface after each. A speed band where the cut runs quietly with a clean surface can then be recorded on the cutting data sheet for that tool, holder and overhang. Specialist services and instruments can measure the system’s response directly, with an instrumented hammer tap on the tool, and predict stable speeds and depths; for high-value or high-volume work, that measurement often pays for itself.

Rigidity is a chain

Every element between the cutting edge and the machine’s base takes part in the deflection: insert seat, tool body, holder, spindle, machine structure, fixture and workpiece. Improving one link helps only if it was a weak one. A carbide boring bar does little for a part held by its tips in soft jaws; a stiffer fixture does little for a cutter projecting twice as far as it needs to.

A useful habit is to list the links for a difficult cut and estimate, even roughly, which bends most. The cantilever formula handles the tool and the workpiece; Fixtures That Repeat covers the fixture; the machine and spindle are the hardest to change and are best accommodated by choosing tools and strategies that keep forces modest. The list usually points to one or two cheap changes, such as a shorter tool, a better holder or an extra support, that do more than any change to the cutting data.

Common mistakes

• Long tools by habit. Using the tool that happens to be in the magazine, at whatever length it is set, rather than the shortest that reaches.
• Small tools where large ones fit. Stiffness falls with the fourth power of diameter.
• Fixing deflection with more finishing passes before addressing overhang or support.
• Assuming lower speed always helps chatter. Sometimes it makes it worse.
• Ignoring the fixture. A rigid tool on a flexible part or clamp still vibrates.

Questions to ask

• What is the shortest overhang that reaches the feature, and the largest tool that fits?
• How much will the tool or part deflect under the expected radial force, and how does that compare with the tolerance?
• Is the workpiece adequately supported for its length and diameter?
• Is this vibration chatter, which changes with depth of cut, or forced vibration from another source?

Exercise: estimate deflection on a real set-up

Choose a long tool or slender part in your workshop. Measure the overhang and diameter, estimate the radial force as about a third to a half of the tangential force from Chapter 6, and calculate the deflection. Then recalculate with the overhang reduced by 25%. Compare the predicted change with the size and taper of parts made each way.

Deflection and power set the limits of what a cut can be. The next chapter turns to how long the tool will survive the cut, and the relationship that F. W. Taylor discovered more than a century ago.