Exercises: Gears, belts and chains

Explain what the transmission ratio $i$ is in a system of two gears and write its formula $i = \frac{Z_1}{Z_2}$, indicating what $Z_1$ and $Z_2$ represent.

On a bicycle, the chainrings are attached to the pedals and the sprockets to the rear wheel, connected by the chain. Explain which combination you choose (large or small chainring, large or small sprocket) if you want to travel a long distance with each pedal revolution, and which combination you choose to climb a steep hill, justifying your answers.

The figure shows two meshed gears: the input gear has $Z_1 = 12$ teeth and the output gear has $Z_2 = 36$ teeth, and the input gear rotates in the direction of the arrow, as shown in the figure.

Without doing calculations, reason: (a) does the output gear rotate faster or slower than the input gear? (b) In the same direction or in the opposite direction? (c) How many revolutions does the output gear make for each complete revolution of the input gear?

Explain what a pinion is and how it differs from the gear wheel with which it meshes (size, number of teeth and role in the transmission).

Two meshed gears rotate in opposite directions. Explain why this happens and what you could add to the system if you need the output to rotate in the same direction as the input (an idler gear).

Name the parts that make up a bicycle's chain drive system (chainring, chain, sprockets, derailleurs) and explain the function of each in a single sentence.

Explain a practical difference between belt drives and chain drives (slippage, noise, maintenance) and reason why a car engine's timing system uses a chain or a toothed belt, never a smooth belt.

A drill has a gear reducer between the motor and the chuck. Explain what this reducer does to the speed and torque, and why the drill's motor is not connected directly to the drill bit.

A motor has a pinion with $Z_1 = 20$ teeth that meshes with a wheel with $Z_2 = 60$ teeth. (a) Calculate the transmission ratio $i = \frac{Z_1}{Z_2}$. (b) If the motor rotates at $120\ \text{rpm}$, calculate the rotational speed of the output wheel, given that $n_2 = n_1 \cdot i$. (c) State whether this is a speed-reducing or speed-increasing system and justify your answer.

A pulley with a diameter of $D_1 = 10\ \text{cm}$, connected to a motor rotating at $900\ \text{rpm}$, drives a pulley of $D_2 = 30\ \text{cm}$ via a belt. Given that for pulleys $i = \frac{D_1}{D_2}$ and $n_2 = n_1 \cdot i$, calculate the transmission ratio and the speed of the driven pulley, including their units.

Compare gears, belt and pulleys, and chain with chainring and sprocket: state one advantage and one disadvantage of each system (precision, slippage, maintenance, noise, centre distance...), and give a real-world application where each one is particularly suitable.

In a reduction drive system, what is lost in speed is gained in torque (ideally, the output torque is the input torque divided by the transmission ratio). A motor provides a torque of $2\ \text{N} \cdot \text{m}$ at $900\ \text{rpm}$ and drives a reduction gear of $i = \frac{1}{3}$. (a) Calculate the output speed. (b) Calculate the available output torque. (c) Using this result, explain why a tightly fastened screw is loosened more easily using a drill's low-speed setting than its high-speed setting.

A shaft needs to rotate at $200\ \text{rpm}$ using a motor running at $1200\ \text{rpm}$. Calculate the required transmission ratio and propose a specific pair of gears (with integer numbers of teeth) that achieves it, verifying your proposal numerically.

A pulley of $D_1 = 8\ \text{cm}$ drives another pulley of $D_2 = 24\ \text{cm}$ via a belt, with the motor rotating at $600\ \text{rpm}$. Calculate the transmission ratio and the speed of the driven pulley, and reason what would happen to that speed if the belt slipped slightly (difference compared to a gear).

An industrial fan needs to rotate faster than its motor, so a step-up (speed multiplier) gear system is fitted: the input has $Z_1 = 60$ teeth and the output has $Z_2 = 20$ teeth. Calculate the transmission ratio and the output speed if the motor rotates at $50\ \text{rpm}$, and name another appliance that uses a speed multiplier.

A bicycle has a 50-tooth chainring and a 25-tooth sprocket. (a) Calculate how many revolutions the rear wheel makes for each full revolution of the pedals. (b) If the wheel has a circumference of $2\ \text{m}$, calculate how many metres the bicycle travels per pedal revolution. (c) Reason what changes when switching to a 20-tooth sprocket while keeping the same chainring.

The reduction mechanism shown in the figure is part of a hoisting winch driven by a motor that rotates at $2000\ \text{rpm}$ and delivers a useful torque of $1,8\ \text{Nm}$.

(a) Deduce the total transmission ratio $i_\text{total}$ and calculate with justification the rotational speed (in $\text{rpm}$) and the torque available at the output shaft ($Z_4$), considering a mechanical efficiency of the reducer $\eta = 85\,\%$.

(b) A technician argues that it would be cheaper and simpler to replace this compound train with a simple single-stage reducer (one pinion and one gear) while keeping the driving pinion $Z_1 = 10$. If all gears use a standardized module $m = 2,5\ \text{mm}$, calculate the pitch diameter that this single driven gear should have and technically refute the technician's proposal by providing at least two design or kinematic drawbacks compared to the current assembly.

(c) Determine with reasoning the direction of rotation of the output gear relative to the driving pinion. What minimal kinematic modification (without altering the speeds or shifting the input and output shafts) would reverse the output's direction of rotation?

You need to design a two-stage spur gear reduction unit to drive the drum of a concrete mixer at $50\ \text{rpm}$ from a motor rotating at $1500\ \text{rpm}$.

Due to chassis design and manufacturing constraints, the following restrictions are imposed:

• No driving pinion can have fewer than $14$ teeth (to prevent interference during cutting) or more than $22$ teeth.

• No driven gear can have more than $90$ teeth due to housing volume restrictions.

• The reduction ratio per stage ($i_k = Z_\text{driven} / Z_\text{driving}$) must not exceed $6:1$ to balance dimensions and stresses between stages.

(a) Determine the integer numbers of teeth ($Z_1, Z_2, Z_3, Z_4$) that strictly meet all the above constraints and achieve the required drum speed (allow a maximum error of $\pm 2\,\%$, justifying it with the calculation of the actual speed obtained).

(b) If the motor supplies a power of $1.5\ \text{kW}$ and each gear stage has a mechanical efficiency $\eta_1 = 0.94$ and $\eta_2 = 0.96$, calculate the effective torque that the reduction unit delivers to the concrete mixer drum.

(c) Technically justify what would happen to the mechanical stresses (torque and wear) if an operator decided to reverse the order of the stages by placing the one with the larger reduction at the beginning instead of at the end.

A vehicle with a mass of $1200\ \text{kg}$ is fitted with wheels of effective radius $R = 0.30\ \text{m}$. Its engine delivers a constant maximum useful power in the range of $3000$ to $4500\ \text{rpm}$, but its torque drops significantly at low revs.

(a) To start on a steep incline, it is calculated that the driving wheels require a tangential traction force of $3600\ \text{N}$. Determine the required torque at the wheel axle and deduce the minimum total transmission ratio required in 1st gear ($i_1 = \omega_\text{wheel} / \omega_\text{engine}$) if the engine starts off at $2000\ \text{rpm}$ delivering an engine torque of $180\ \text{Nm}$, considering an overall transmission efficiency $\eta = 88\,\%$.

(b) If that same transmission ratio calculated in (a) were kept fixed while driving on a flat motorway at $120\ \text{km/h}$, at how many $\text{rpm}$ would the engine be forced to rotate? Interpret the result and justify technically and thermodynamically why a multi-ratio gearbox (including tall gears such as 5th) is essential instead of a fixed-ratio transmission.

A compound gear train consists of $Z_1 = 12$ at the input, which meshes with $Z_2 = 48$; rigidly mounted on the intermediate shaft rotates gear $Z_3 = 12$, which meshes with the final gear $Z_4 = 36$. All gears have a module $m = 2\ \text{mm}$ and the motor rotates at $1800\ \text{rpm}$.

(a) Following a breakdown on the intermediate shaft, a technician suggests swapping the positions of gears $Z_2$ and $Z_3$ on that shaft (so that $Z_1$ meshes with $Z_3$ and $Z_2$ meshes with $Z_4$) while keeping the shaft positions unchanged. Quantitatively demonstrate, by calculating the theoretical center distances of both stages, whether the technician's proposal is geometrically viable or if it will cause backlash/mechanical jamming.

(b) If instead of swapping them it is decided to completely decouple the intermediate shaft and directly connect the drive pinion $Z_1$ to the output gear $Z_4$ by adjusting the center distance, calculate the new total transmission ratio and explain how the useful output torque and direction of rotation would be affected compared to the original assembly.

The shaft of a $6000\ \text{rpm}$ motor with a power of $450\ \text{W}$ must drive the blades of an industrial food processor at $600\ \text{rpm}$.

(a) Design a two-stage reduction train with cylindrical spur gears of module $m = 1.5\ \text{mm}$ that exactly satisfies this transmission ratio under the following construction specifications:

• No driving pinion ($Z_1, Z_3$) can have fewer than $15$ teeth (to avoid interference).

• The pitch diameter of neither driven gear ($Z_2, Z_4$) can exceed $80\ \text{mm}$.

• Numerically verify your combination of teeth ($Z_1, Z_2, Z_3, Z_4$) and the resulting speed.

(b) Calculate the theoretical torque available at the blades (assuming negligible losses) and numerically justify why a single-stage design with the same input pinion $Z_1$ would be inadmissible for the cylindrical housing of the appliance.

A bicycle has two chainrings ($P_1 = 50\ \text{teeth}$ and $P_2 = 34\ \text{teeth}$) and a rear sprocket cassette with $Z = [11, 12, 14, 16, 18, 21, 24, 28, 32]$ teeth. The outer diameter of the wheel with the tire is $D = 680\ \text{mm}$.

(a) Calculate the gear development (meters traveled per complete pedal stroke) for the two combinations with maximum "cross-chaining": large chainring with large sprocket ($50 \times 32$) and small chainring with small sprocket ($34 \times 11$).

(b) Find an alternative "non-cross-chained" combination in the transmission for each of the two previous cases that provides an equivalent development with a discrepancy of less than $5\,\%$. Justify your answer with comparative calculations.

(c) Explain, using the analysis of forces and contact between link and tooth, why the severe angular misalignment typical of cross-chaining causes a drop in the cyclist's energy efficiency and a drastic increase in wear on the tooth flanks and chain pins.

To drive the drum of a workshop hoist at $29\ \text{rpm}$ from an electric motor at $1450\ \text{rpm}$ with a useful torque of $20\ \text{Nm}$, the technical team evaluates two reduction alternatives:

• Option A: A two-stage helical gear reducer (total efficiency $\eta_A = 0.94$).

• Option B: A worm gear reducer with a single-start worm ($z_1 = 1$) and a worm wheel of $z_2 = 50$ teeth (mechanical efficiency $\eta_B = 0.45$).

(a) Critically analyse what will happen in each case if, while the load is suspended at mid-height, an unexpected power outage occurs at the motor without a secondary brake acting. Justify the geometric or tribological condition on which the behaviour of the self-locking option is based.

(b) Calculate the torque available at the drum shaft in both options and explain the trade-off between operational safety and energy efficiency that the designer must weigh when choosing between A and B.

(c) Regarding Option A, justify why helical gears are quieter than spur gears for this application and explain what parasitic force on the shaft bearings they introduce and how it is usually solved without losing their acoustic advantages.

A simple gear train mounted on a fixed frame consists of three gears meshed in a straight line: driving pinion $Z_1 = 14$, idle intermediate gear $Z_2 = 42$, and driven output gear $Z_3 = 28$, all with module $m = 2\text{ mm}$.

(a) An operator claims that if the intermediate gear is replaced with another one of $Z'_2 = 21$ teeth, the output gear will rotate at twice the speed and deliver half the torque. Rigorously refute this claim by deriving the algebraic expression for the total transmission ratio and justifying the role of the intermediate gear in the kinematics and torque of the system.

(b) If the shafts of the driver ($Z_1$) and the output ($Z_3$) are rigidly fixed to the frame at a centre distance of $126\text{ mm}$, demonstrate quantitatively whether it is physically possible to replace $Z_2$ with $Z'_2$ without relocating the shafts, calculating the centre distances that the new gear would require.

(c) Determine the relationship between the direction of rotation of the input and the output in this train. Generalize by deriving what mathematical condition the total number of gears $N$ in a simple train must satisfy for the output shaft to rotate in the opposite direction to the driving shaft.

Unit challenge. How would you optimize a bicycle's mechanical transmission to improve your effort during daily commutes? In this challenge, you will analyze and improve the kinematic system of a domestic or city rental bicycle.

1. Real data collection: Count the teeth of the largest and smallest chainrings ($Z_1$), as well as the largest and smallest sprockets ($Z_2$). Measure the length of the crank (lever) and the radius of the rear wheel ($R$, in metres) using a ruler or tape measure.

2. Kinematic analysis: In LibreOffice Calc, tabulate and calculate the gear ratios ($i = Z_1 / Z_2$), the mechanical advantage of the crank-pedal lever, and the linear development per pedal stroke ($d = 2\pi R \cdot i$, using $\pi \approx 3,14$). Graph the development according to the gear combination.

3. Reasoned technical proposal: Imagine you must tackle a $7,5\,\%$ urban gradient. Propose a system redesign (varying the number of teeth on chainrings/sprockets or adding a tensioner/reduction pulley) that increases mechanical advantage by $25\,\%$ while maintaining a comfortable cadence.

Assessment criteria:

• Rigour in obtaining empirical data and calculating parameters ($i$, moments, and developments).

• Appropriate use of LibreOffice Calc to model and graph the mechanism.

• Technical coherence and kinematic justification in the redesign proposal.