Dynamic Mechanisms

Kinematics, the Unified Plotting Method, and Applications

A locus is the path traced by a point moving under defined geometric or mechanical constraints. While curves such as cycloids, involutes, and spirals look completely different, you draw them using the same step-by-step approach on your drawing sheet:

1. Establish the datum geometry: Draw the fixed pivots, base circles, guiding tracks, or camshaft centres accurately using light construction lines.
2. Subdivide the driving movement: Divide the driving rotation or slide into equal steps. For a full revolution, 12 equal intervals of 30° are standard. For a partial swing, use at least 5 to 6 equal steps.
3. Coordinate dependent positions: For each step, locate the moving parts using compass arcs of fixed link lengths, unwound arc lengths, or advancing wheel centres.
4. Trace the curve: Mark each position of the tracing point ($P_0, P_1, P_2 \dots$) and join them with one smooth, firm curve.

The syllabus expects you to know real-world engineering applications for every curve you construct:

Follower Displacement Diagrams: UV, SHM, and UAR

A plate cam converts continuous rotary motion into precise reciprocating linear movement. Before drawing the cam, you must construct a displacement diagram, which plots camshaft rotation ($0^\circ$ to $360^\circ$) along the horizontal axis against linear follower lift (stroke $S$) along the vertical axis.

Types of Motion

• Uniform Velocity (UV): The follower moves at a constant speed, shown as a straight sloping line. Sudden shifts from UV to dwell cause sharp velocity steps and theoretical shock loading.
• Simple Harmonic Motion (SHM): Delivers smooth acceleration and deceleration from rest. On the vertical axis for that motion interval, draw a semicircle whose diameter equals the lift stroke. Divide this semicircle into the exact same number of sectors as the horizontal angular interval (for example, 6 divisions of $30^\circ$ for a $180^\circ$ span, or 4 divisions of $45^\circ$ for a $120^\circ$ span). Project horizontal lines across from each point on the semicircle to meet the matching angle ordinates.
• Uniform Acceleration and Retardation (UAR): The follower accelerates uniformly for the first half of the stroke and retards uniformly for the second half, forming two parabolas that meet at mid-stroke, where the follower is moving fastest and the two curves share the same slope. With 6 equal intervals, the distances moved in successive intervals are in the ratio $1 : 3 : 5 : 5 : 3 : 1$ — small steps at the start while the follower is speeding up, large steps at mid-stroke, small steps again as it slows. The sum of these 6 increments is 18 units:

The cumulative heights plotted at successive stations are $1, 4, 9, 14, 17,$ and $18$ units. If 8 divisions are required, the increments are $1:3:5:7:7:5:3:1$ (totaling 32 units), giving cumulative heights of $1, 4, 9, 16, 23, 28, 31,$ and $32$ units.

• Dwell: The follower remains stationary at a constant height, represented by a horizontal line.

Working Backwards: Displacement Diagram from a Given Cam Profile

When an exam question supplies a finished cam profile and asks for the displacement diagram:

1. Find the cam centre and draw radial lines at regular 30° intervals, numbered in the direction of follower rotation relative to the cam (opposite to cam rotation).
2. Draw the base circle (the smallest circle concentric with the shaft that touches the profile).
3. For a knife-edge follower, measure the distance from the cam centre to the profile along each radial line. The lift is that measured distance minus the base circle radius.
4. For a roller follower, draw the roller circle touching the profile at each radial station, mark its centre, and measure from the cam centre to that centre. The lift is the measured distance minus the prime circle radius. For a flat follower, draw the flat face tangent to the profile perpendicular to each radial line and measure to the contact line.
5. Set out the horizontal axis from $0^\circ$ to $360^\circ$, plot the measured lifts as vertical ordinates, join them smoothly, and label the motion types.

Radial Plate Cam Profile Construction

Cam profile layout relies on kinematic inversion: imagine the cam held stationary while the follower rotates around it in reverse. If the camshaft turns clockwise, number your radial divisions in an anticlockwise sequence around the cam centre.

Follower Types and Reference Geometry

• Base Circle: The smallest circle concentric with the camshaft that touches the working cam profile.
• Prime Circle: The reference circle passing through the centre of a roller follower at its nearest approach, concentric with the camshaft:

• In-Line Knife-Edge Follower: Divide the base circle into radial lines spaced at the motion increments, numbered opposite to rotation. Transfer the vertical lift from the displacement diagram directly outward along each radial line, starting from the base circle. Draw a smooth curve through these points.
• In-Line Roller Follower (HL): Divide the prime circle into radial lines numbered opposite to rotation. Step each displacement outward from the prime circle along its radial line to locate the roller centres (this path is the pitch curve). At each centre, draw a light circle with the roller radius. Finish the cam profile as one smooth, firm curve touching the inside of every roller circle.
• In-Line Flat Follower (HL): Measure displacements outward from the base circle along each radial line. Through each plotted point, draw a line perpendicular to that radial line, representing the flat follower face. The cam profile is the smooth continuous envelope curve drawn tangent to the inside of these flat face lines.

Rolling Curves: Cycloids, Epicycloids, Hypocycloids and Trochoids

Rolling curves are traced by points on a generating circle that rolls without slipping along a datum line or arc.

• Cycloid: Traced by a point on the rim of a circle of radius $r$ rolling along a straight line. For one full turn, the circle advances a baseline distance equal to its circumference, $2\pi r$. The centre moves along a line parallel to the base at height $r$. Divide both the circumference and baseline into 12 equal parts. From each advancing centre $C_n$, swing an arc of radius $r$ to cut the horizontal guide line projected from division $n$ on the circle.
• Epicycloid and Hypocycloid: An epicycloid forms when a circle rolls on the outside of a circular arc of radius $R$; a hypocycloid forms when it rolls on the inside. To find the angle subtended on the directing arc for one complete revolution, equate arc lengths:

The path of centres is an arc concentric with the directing arc, drawn at radius $R + r$ for an epicycloid, or $R - r$ for a hypocycloid. Divide $\theta$ into 12 equal sectors to locate centres $C_0$ to $C_{12}$. Draw concentric guide arcs from the main centre through each division of the rolling circle. From each centre $C_n$, strike an arc of radius $r$ to cut guide arc $n$, taking the intersection on the trailing side of $C_n$ (the side the wheel has already rolled over).

• Trochoids (Superior and Inferior): Traced by a point rigidly fixed to the wheel at a distance $r_p$ from the centre. An inferior trochoid ($r_p < r$, such as a spoke reflector) produces a wave-like curve. A superior trochoid ($r_p > r$) forms loops extending beyond the track. When rolling on an arc, use the same subtended angle $\theta = 360^\circ \times (r / R)$, but strike arcs of radius $r_p$ from each centre $C_n$ onto concentric guide arcs spaced at radius $r_p$.
• Tangents and Normals from the Instantaneous Centre: At any instant, the contact point $N$ between the rolling wheel and the directing path is the instantaneous centre of zero velocity. The line joining point $P$ on the curve to $N$ is the normal. The tangent is perpendicular to $PN$ through $P$.

Finding the Contact Point at Any Given Point P

If point $P$ is marked on a finished curve and you do not know the wheel centre:

1. On a cycloid, swing an arc of radius $r$ from $P$ to cut the line of centres at $C$. Drop a perpendicular from $C$ to the directing line to find contact point $N$. Line $PN$ is the normal; the tangent is perpendicular to $PN$ at $P$.
2. On an epicycloid or hypocycloid, swing radius $r$ from $P$ to cut the arc of centres (radius $R + r$ or $R - r$) at $C$. Join the directing arc centre $O$ through $C$; where $OC$ crosses the directing arc is $N$. Line $PN$ is the normal.
3. On a trochoid, use radius $r_p$ from $P$ to find centre $C$, then locate $N$ as above.

Spirals, Helices, and Space Curves

Archimedean Spiral

A point moves outward at uniform speed along a radius vector while that vector rotates at constant angular speed ($r = a + b\theta$). The distance between successive convolutions along any ray is constant. Divide the rotation and the radial gain into the same number of equal parts (typically 12) and connect the corresponding intersections.

Tangent to an Archimedean Spiral (HL)

The spiral constant $c$ is the radial gain per radian:

To construct the normal and tangent at point $P$:

1. Join pole $O$ to $P$.
2. At $O$, draw a line perpendicular to $OP$ and step off $OQ = c$, turning from $OP$ through 90° in the same rotational sense in which the spiral is generated (the direction the radius sweeps). If the spiral uncoils anticlockwise and $P$ is at 150°, $Q$ lies on the 240° ray.
3. Join $Q$ to $P$: line $QP$ is the normal. The tangent is perpendicular to $QP$ through $P$.

Logarithmic Spiral (HL)

The radius grows in geometric progression for equal angular steps ($r = a e^{b\theta}$). The curve cuts all radius vectors at a constant angle. If two consecutive radii $r_0$ and $r_1$ are known, the next radius is $r_2 = r_1 \times (r_1 / r_0)$.

Constructing a Cylindrical Helix

A helix winds around a cylinder at uniform angular and axial speed:

1. Draw the plan (a circle of diameter $D$) and elevation (a rectangle of height equal to the pitch) of the cylinder.
2. Divide the plan circle into 12 equal parts and number them in the direction of rotation.
3. On the elevation, divide the pitch height into the same 12 equal parts and draw horizontal lines.
4. Project each numbered plan point up to its matching horizontal height line.
5. Join the points with a smooth curve. Draw the front half firm and the back half dashed as a hidden line.
6. A single helix advances by its pitch in one revolution. On a multi-start thread the axial advance in one turn is called the lead, and it equals the number of starts multiplied by the distance between adjacent thread crests (so a double-start thread advances twice that crest spacing per turn).

Constructing a Conical Spiral

A conical spiral winds around a cone from base to apex:

1. Draw the plan circle and the elevation triangle of the cone.
2. Divide the plan base circle into 12 equal parts and draw 12 radial lines.
3. Divide the cone height in elevation into 12 equal parts and draw horizontal level lines.
4. Where each horizontal line cuts the cone outline in elevation, read off the radius at that height and swing that radius onto the matching radial line in plan.
5. Connect the plan points to produce an Archimedean spiral.
6. Project each plan point vertically up to its height line in elevation and join them smoothly.

Involutes, Linkages, Trammels, and Moving Circles

Involute of a Circle and Regular Polygon

An involute is traced by the end of a taut string unwinding from a shape.

• Circle: Divide the circumference into 12 equal parts and draw tangents at each station pointing in the unwinding direction. Along tangent $n$, step off length $(n / 12) \times \pi D$. Join points smoothly. The normal at any point $P$ is the tangent drawn from $P$ to the base circle touching at $T$; the tangent to the involute is perpendicular to $PT$ at $P$.
• Regular Polygon: The string unwinds from sharp corners, so the curve is a chain of circular arcs. Extend each side of the polygon in the unwinding direction. From vertex $A$, swing an arc from the free end with radius equal to the unwound side length to hit the extension of side $B$. With centre at $B$ and radius equal to two side lengths, swing to the next extension. Continue round, increasing the radius by one side length at each vertex.

Loci from Link Mechanisms

A link mechanism is an assembly of rigid bars pinned together. Since links cannot stretch, every pin position can be plotted with a compass:

1. Draw the mechanism in its starting position and identify fixed pivots, sliders, and guided pins.
2. Divide the movement of the driving crank into equal steps (12 steps of 30° for a full turn, or 5 to 6 steps for a partial swing).
3. For each step, locate the driven pin by swinging compass arcs equal to true link lengths from known pin positions onto guide tracks.
4. If a link slides through a swivel, draw a line from the pivot through the swivel centre and step the link length along that line.
5. Mark every position of the tracing point ($P_0, P_1, P_2 \dots$) and join them with a smooth curve.

Using a Trammel

A trammel is a straight strip of paper or card on which true pin distances are marked. Mark tracing point $P$ and the constrained pins on the strip edge. Lay the strip on the drawing so the pin marks lie on their respective guide tracks, and prick point $P$. Repeat for 12 positions and join the points. For an ellipse, mark semi-minor distance from $P$ to $M$, and semi-major distance from $P$ to $N$; keeping $M$ on the major axis and $N$ on the minor axis guides $P$ along the ellipse.

Loci of Moving Circles

When tracking a moving circle, focus on its centre:

• Rolling on a line: centre moves parallel to the line at distance $r$.
• Rolling on a circle of radius $R$: centre moves on a circle of radius $R + r$ (exterior) or $R - r$ (interior).
• Circle touching a line and a fixed circle (centre $F$, radius $R$): its centre is equidistant from focus $F$ and a line parallel to the datum at distance $R$, forming a parabola.
• Circle touching two fixed circles externally: difference of distances to centres is constant, forming a hyperbola. If the moving circle lies inside one fixed circle, touching it internally, while touching the other externally, the sum of the distances to the two centres is constant and equal to R₁ + R₂, so the locus is an ellipse with those centres as foci.

Gear Profiles: Involute and Epicycloidal Teeth (Higher Level)

Gears transmit positive drive and constant velocity ratios between rotating shafts. Higher Level candidates must know both involute and epicycloidal tooth constructions.

Standard Involute Proportions

Tooth sizes are defined by the metric module ($m$):

• Pitch Circle Diameter (PCD): Theoretical rolling diameter where mating gears touch: $\text{PCD} = m \times T$, where $T$ is tooth count.
• Circular Pitch ($p$): Distance between corresponding points on adjacent teeth: $p = \pi m$.
• Addendum: Radial height above the pitch circle: $\text{Addendum} = m$.
• Dedendum: Radial depth below the pitch circle: $\text{Dedendum} = 1.25m$.
• Pressure Angle ($\phi$): Angle between the tooth normal and pitch tangent, standardised at $20^\circ$.
• Base Circle Diameter: The circle from which the involute unwinds:

• Tooth Thickness: Measured along the pitch circle: $t = p / 2 = 0.5\pi m$.

Always check question data: if an exam specifies different proportions (such as a $14.5^\circ$ pressure angle or dedendum of $1.157m$), use the stated values.

Constructing the Involute Tooth

1. Draw concentric addendum, pitch, base, and dedendum circles.
2. Unwind an involute from the base circle outward to the addendum circle.
3. Step off tooth thickness $t$ along the pitch circle.
4. Mirror the involute across the tooth's radial centreline.
5. Below the base circle, draw straight radial lines down to the root circle, blending with fillets.

Epicycloidal (Cycloidal) Gear Teeth

Common in horology (clocks and watches), cycloidal teeth use a separate generating circle:

1. Draw root, pitch, and addendum circles.
2. Roll the generating circle along the outside of the pitch circle from the pitch point to create an epicycloid up to the addendum circle. This curve forms the tooth face.
3. Roll the same generating circle along the inside of the pitch circle to create a hypocycloid down to the root circle. This forms the tooth flank.
4. Mirror both curves across the tooth radial centreline. If the generating circle diameter equals the pitch circle radius, the hypocycloid becomes a straight radial line.

Key terms

Locus

The path traced out by a geometric point moving according to fixed mathematical or mechanical constraints.

Displacement Diagram

A linear graph plotting camshaft rotation angle (0° to 360°) on the horizontal axis against follower linear displacement on the vertical axis.

Base Circle

The smallest circle concentric with the camshaft that touches the working profile of a plate cam.

Prime Circle

The reference circle passing through the centre of a roller follower at its nearest approach, concentric with the camshaft.

Pitch Curve

The locus traced by the centre of a roller follower relative to a stationary cam.

Kinematic Inversion

The drawing technique of holding the cam stationary and rotating the follower in reverse around it to establish the required profile.

Instantaneous Centre

The stationary pivot point about which a moving body rotates at a specific instant; for a rolling wheel, it is the contact point on the track.

Involute

The planar curve traced by the free end of a taut string as it unwinds from the perimeter of a base circle or polygon.

Pitch of a Helix

The axial distance advanced by a helix during one complete revolution of 360°.

Archimedean Spiral

A planar curve traced by a point moving outward along a radius vector at constant speed while the vector rotates at constant angular speed.

Logarithmic Spiral

An equiangular spiral whose radius grows in geometric progression for equal angular steps, cutting all radius vectors at a constant angle.

Cycloid

The locus traced by a point on the circumference of a circle rolling without slipping along a straight line.

Epicycloid

The locus traced by a point on the circumference of a circle rolling without slipping along the outside of a circular directing arc.

Hypocycloid

The locus traced by a point on the circumference of a circle rolling without slipping along the inside of a circular directing arc.

Module

The standard metric unit of gear tooth size, calculated as the pitch circle diameter divided by the number of teeth.

Pressure Angle

The angle between the common normal to two meshing gear teeth and the common tangent to their pitch circles, standardised at 20°.