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.

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).

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.

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.

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.

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).

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 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.

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.