Exercises: Cams and ratchets

Explain what a cam is and what motion transformation it performs together with its follower. Give an example of an everyday machine that uses cams and state what it achieves by using them.

Explain what a ratchet is and how it manages to permit rotation in one direction while blocking it in the other. Give two examples of objects that use it (tools, toys, lifting systems...).

The figure shows an eccentric cam rotating about its shaft and pushing a vertical follower resting on it, as shown in the figure.

Observing the cam profile and the rotation arrow, explain: (a) what type of motion the follower performs while the cam rotates at constant speed; (b) at which position on the profile (the point furthest from or closest to the axis) the follower reaches its highest point; (c) in which direction (clockwise or anticlockwise) the cam is rotating at the moment depicted, which is what the question mark asks.

Explain what a cam follower is and what three contact tip shapes it can have (knife-edge, roller, and flat-face), stating one advantage of a roller follower.

In an internal combustion engine, what is the camshaft and what components does it actuate? Explain what would happen to the engine valves if the camshaft stopped rotating.

Explain what a ratchet is and why a car's handbrake lever makes a "click, click" sound when pulled up and stays locked when released.

Explain what an eccentric is (a disc mounted off its centre of rotation) and what motion it produces in a component resting on it as it rotates. Give an application (vibratory sander, construction plate compactor).

Name three everyday objects that use a ratchet (socket wrench, lashing strap, zipline, winch) and explain, choosing one of them, what would happen if its ratchet failed.

The profile of a cam determines the law of motion of the follower. Explain the relationship between the distance from the profile to the axis of rotation at each point and the position of the follower, and reason what shape the profile should have so that the follower rises slowly during half a revolution and drops suddenly.

In an internal combustion engine, cams open and close the intake and exhaust valves at precisely the right moment. Explain why the profile of these cams is critical to the operation of the engine and what would happen if a cam were worn down or displaced from its position.

A hoist for lifting loads and a storage unit with an up-and-over door require that the load or the door does not fall back when the force is released. Analyse how a ratchet with its pawl solves this problem and compare it with using a simple brake: advantages and disadvantages of each solution.

Compare a roller follower with a knife-edge follower in a cam that rotates for many hours a day: wear, friction, movement precision, and cost. Which would you choose for an industrial machine and why?

The camshaft must rotate in synchronisation with the crankshaft, which is why they are connected by a timing belt or timing chain. Explain what misalignment causes the belt to jump a tooth (the valves open out of time) and what consequences this can have in an engine where valves and pistons share space (interference engines).

In a winch with a ratchet, the load does not fall even if you let go of the crank handle. Explain the mechanism that prevents this (toothed wheel and pawl with its spring), why the pawl must always be under tension, and what you would inspect on this assembly before using it with a load.

A mechanical music box has a drum with pins that lift tuned metal strips. Explain how this drum resembles a set of cams, how the position of the pins "programmes" the melody, and what advantage being able to swap the drum offers over a fixed drum.

The displacement diagram of a follower represents its height as a function of the angle rotated by the cam. A cam has a profile such that, during the first half-revolution, the distance from the profile to the axis increases continuously, and during the second half-revolution it decreases symmetrically until it returns to the initial height, as shown in the figure.

(a) Describe what the second half of the curve, marked with a question mark, should look like, and draw it in your notebook. (b) Explain the relationship between the shape of the cam profile (its distance to the axis at each angle) and the shape of this curve. (c) Reason what would change in the diagram if the cam rotated at twice the speed.

For an industrial punching machine, a snail cam rotating at $120\ \text{rpm}$ is proposed to lift a punch follower of mass $m = 1,5\ \text{kg}$ to a height of $30\ \text{mm}$ during $270^\circ$ of rotation at constant velocity, descending over the remaining $90^\circ$.

1. a) Qualitatively draw the displacement diagram $s(\theta)$ and the velocity diagram $v(\theta)$ of the follower, justifying the shape of each section.
2. b) A technician claims that «the sharper the drop (smaller return angle), the greater the impact force of the punch against the workpiece, so it is advisable for the return to take place in $0^\circ$ (vertical radial step)». Technically refute this claim by analyzing what would happen to the follower (bounce, loss of contact, and destructive stresses at the edge of the cam).

A manual winch lifts a load of $150\ \text{kg}$ using a drum of $20\ \text{cm}$ in diameter driven by a crank of $40\ \text{cm}$ in radius. To cut costs, it is installed without an anti-reversal ratchet, relying solely on the operator's strength ($g = 9,8\ \text{m/s}^2$).

1. a) Calculate the torque exerted by the load on the drum and the tangential force that the operator must exert at the end of the crank to keep it in equilibrium. If the operator releases the crank unexpectedly, predict kinematically and dynamically what will happen to the crank and what immediate mechanical hazards are triggered.
2. b) Design the integration of a ratchet mechanism for this winch: determine the orientation of the teeth relative to the lifting direction, technically justify the choice between a torsion spring or gravity to keep the pawl in contact with the wheel in a dirty construction environment, and explain what additional mechanical element would be required to allow the controlled lowering of the load without releasing the ratchet by hand.

In a high-rate industrial packaging line ($600\ \text{cycles/min}$), there is an ongoing debate between redesigning the pushers using conjugate mechanical cams or using «electronic cams» (independent programmable linear servomotors synchronized via a control network).

1. a) Analyze the behavior of both systems in the event of a sudden electrical power cut at maximum speed: which one guarantees kinematic synchronization and prevents physical collision between pushers? Justify your answer based on the operating principle of each alternative.
2. b) If the factory must change package format (different stroke and velocity profile) twice a week, evaluate the impact of each system on line downtime and operating costs.
3. c) Provide a reasoned decision on which technology you would recommend for a line dedicated to a single product for years at an extreme rate, and which one in a flexible manufacturing plant with small batch sizes.

A flat disc cam with a roller follower must be designed to actuate the pusher of an assembly cell. The cycle for each turn of the cam ($360^\circ$) requires:

• From $0^\circ$ to $120^\circ$: rise of $h = 20\ \text{mm}$ at uniform velocity.

• From $120^\circ$ to $240^\circ$: top dwell (rest at the maximum elevation level).

• From $240^\circ$ to $360^\circ$: return with constant acceleration and deceleration (parabolic motion: acceleration during the first $60^\circ$ and deceleration during the last $60^\circ$).

The radius of the cam's base circle is $R_b = 40\ \text{mm}$.

1. a) Analytically determine the radial distance of the cam profile relative to its rotation axis for $\theta = 60^\circ$, $\theta = 180^\circ$, and $\theta = 300^\circ$.
2. b) Technically justify why a constant acceleration profile is used for the return instead of an abrupt linear drop, relating it to kinematic shock ("jerk" or infinite jerk) and surface fatigue wear.
3. c) If the radius of the follower roller is $15\ \text{mm}$, predict what geometric problem (undercutting) would occur if we reduced the base radius $R_b$ to only $10\ \text{mm}$.

In the locking ratchet mechanism of a mechanical rack jack, the active flank of the tooth forms an angle $\alpha$ with respect to the line perpendicular to the pawl's line of action.

1. a) Draw the free-body diagram of the contact point between the pawl and the tooth under load, representing the normal force $N$ and the friction force $F_r = \mu N$. Derive the algebraic condition that angle $\alpha$ and the coefficient of friction $\mu$ must satisfy for the ratchet to be self-locking (that is, so that the load tends to draw the pawl into the tooth root rather than pushing it outward).
2. b) The pawl and the tooth crest have become severely rounded due to abrasive wear. Predict the catastrophic failure mechanism that will occur upon applying a load of $15\ \text{kN}$ if the effective contact angle exceeds the static friction angle.
3. c) Justify from mechanism theory why a rigid mechanical jack stand with a solid shear cross-pin provides intrinsic safety ("fail-safe"), whereas a hydraulic system or a worn ratchet must not be considered passive retention devices for human safety.

An internal combustion engine with a fixed camshaft presents a thermal and fluid-mechanical compromise: if the cam is ground for wide valve overlap (a large angular overlap in which intake and exhaust are open at the same time), the engine performs optimally at $6000\ \text{rpm}$, but at idle ($800\ \text{rpm}$) it is unstable, loses volumetric efficiency, and pollutes excessively.

1. a) Explain, by referring to the inertia of the gas column in the manifolds, why this wide overlap is beneficial at high revolutions but causes backflow of burned gases into the intake manifold at low revolutions.
2. b) Variable valve timing (VVT) systems allow continuous modification of the camshaft's angular phasing relative to the crankshaft. With justification, predict whether the intake camshaft actuator should advance or retard the valve opening timing during: (1) sharp acceleration at low engine speeds ($1500\ \text{rpm}$, maximum engine torque), and (2) steady high-speed highway cruising with part-throttle (seeking to reduce pumping work).

While performing timing system maintenance on an inline 4-cylinder four-stroke engine, a mechanic mistakenly installs a $38$-tooth pulley on the intake camshaft instead of the standard $40$-tooth pulley, keeping the original $20$-tooth crankshaft sprocket.

1. a) Calculate the original theoretical transmission ratio and the new anomalous transmission ratio caused by the mistake.
2. b) Mathematically demonstrate what will happen to the synchronization between the piston and the intake valves as the crankshaft completes successive rotations. Will the valves open at the same angle in the cycle or will there be continuous phase drift? Calculate how many degrees of phase shift the camshaft will accumulate after just 10 revolutions of the crankshaft.
3. c) Deduce the mechanical effect this anomaly will have on the valves and piston crowns within the first few seconds of operation.

A ratchet anti-reverse mechanism is designed for a retractable terrace clothesline that lifts a total mass of $50\ \text{kg}$ via a drum of radius $r = 40\ \text{mm}$. The ratchet wheel is rigidly attached to the drum shaft and has a tooth root radius $R_t = 60\ \text{mm}$ ($g = 9,8\ \text{m/s}^2$).

1. a) Calculate the torque generated by the load on the shaft and the tangential compressive force that the locking pawl must withstand when the mechanism engages.
2. b) The mechanism will be exposed to the elements (marine humidity, acid rain, and continuous solar radiation). Make a reasoned selection of the material for: (1) the wheel-pawl assembly, and (2) the spring that biases the pawl. Rule out uncoated carbon steel and standard polymers such as polyethylene or PVC with technical arguments.
3. c) It is proposed to eliminate the spring and rely solely on the pawl's own weight (gravity engagement). Critically evaluate under what mounting conditions and environmental situations (gusty wind, dust deposits, or incipient corrosion at the pivot) this gravity-based design will cause catastrophic failure of the system.

Unit challenge. Design the mechanical system of a wave energy buoy to harness wave power along the Spanish coast.

1. Real data collection: Go to the Puertos del Estado or AEMET website and retrieve the significant wave height ($H$) and mean wave period ($T$) from an actual buoy (for example, $H = 1,2\text{ m}$ and $T = 5,5\text{ s}$).

2. Kinematic design: Layout the transformation train in Tinkercad, FreeCAD, or LibreOffice Draw:

3. A float coupled to a slider-crank or crankshaft mechanism that transforms the vertical back-and-forth motion into rotation.

4. A ratchet that prevents the drive shaft from reversing, ensuring continuous unidirectional rotation toward the generator.

5. A cam attached to the shaft that periodically operates a lubricant dosing plunger once per cycle.

6. Technical proposal: Prepare a brief report calculating the system stroke ($c = 2 \cdot r$) suitable for your value of $H$, estimating the mean angular velocity ($\omega = 2\pi / T\text{ rad/s}$), and justifying the cam profile and the ratchet orientation.

Assessment criteria:

• Functional and coherent integration of the slider-crank/crankshaft, cam, and ratchet.

• Rigour in kinematic calculations using the obtained marine data.

• Graphical quality of the design or schematic and clarity of the explanatory report.