Why are spindle speeds arranged in geometric progression?

It gives the same percentage loss in cutting speed between every pair of steps

It gives equal speed differences between steps

What is the drawback of an arithmetic progression of speeds?

It gives a large loss at low speeds and wastefully close speeds at high speeds

It cannot include the top speed

It gives equal percentage losses

How does stepless regulation compare with stepped?

Stepless gives any speed in the range, stepped gives a fixed number of speeds with a positive, rigid drive

Stepless has a fixed number of speeds

Stepless is always cheaper and more rigid

Practice · 6 marks

Requirements of machine tool drives

Short answer

The drive must cover a wide speed range (Rn up to 100), maintain constant power and torque, be rigid, allow fast safe speed changes, and remain compact and stable. Stepped drives offer fixed speeds with high efficiency; stepless drives provide continuous speed but lower efficiency and higher cost. A geometric progression of speeds yields a constant percentage speed loss between steps, so the optimal speed is always close to an available one, and requires fewer gears.

Explain the requirements of a machine tool drive. Compare stepped and stepless speed regulation, and explain why spindle speeds are arranged in a geometric progression.

A machine tool drive transmits power from the motor to the spindle or slide and gives the range of speeds and feeds needed for different materials, tool materials and diameters.

Requirements

Cover the full range of cutting speeds, $v = \pi D N/1000$ , for the smallest and largest diameters. The speed ratio $R_n = N_{max}/N_{min}$ is typically 20 to 100 or more.
Enough power and torque at the spindle, constant power over most of the range.
High rigidity and low vibration, accurate running of shafts and gears, low noise.
Quick, easy change of speed, preferably during running or with levers; safe and simple to operate.
Compact, economical, easy lubrication and maintenance; minimum number of gears and shafts.
Stable speed under changing load.

Stepped vs stepless

Point	Stepped (gearbox, cone pulley)	Stepless (variable speed)
Speeds	Fixed number of speeds	Any speed in the range
Examples	Sliding gears, clutch gears, cone pulleys	Variable-frequency motor, hydraulic, friction/pulley variators
Efficiency and rigidity	High, positive drive	Lower with friction or belts
Cost and complexity	Gears needed but robust	Higher cost for control
Use	Lathes, milling machines, drills	CNC machines, grinding, special drives

Why geometric progression

If speeds are chosen as $N_1,\ N_1\varphi,\ N_1\varphi^2, \dots$ the percentage loss in cutting speed between steps, $(N_{j+1}-N_j)/N_{j+1} = (\varphi-1)/\varphi$ , is the same at every step. The ideal speed for a job is therefore never more than a constant fraction away from an available speed. An arithmetic progression gives a large loss at low speeds and a wasteful closeness at high speeds. The GP also needs fewer gears and fits the standard preferred-number (Renard) series R5, R10, R20, R40 with $\varphi = 1.58, 1.26, 1.12, 1.06$ .

✓ Answer checked by Claude Opus 5.5 (high effort)

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Requirements of machine tool drives

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