Learning Outcomes - Students should be able to: An ...dcgnotes.weebly.com/uploads/2/4/0/6/24061727/... · Plot the involute of point P as it is unwound from the regular hexagon in

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________

Learning Outcomes - Students should be able to:

1

The human inner ear, the cochlea, is displayed in figure 3. The shape of the

cochlea is based on an Archimedean spiral of multiple convolutions.

Draw one and a half convolutions of the spiral given the smallest radius vector

|OA| is 15mm and the largest |OB| is 75mm.

An image of the spiral ramp from the Vatican museum is displayed in figure 1.

This magnificent spiral ramp, designed by Giuseppe Momo in 1932, transports

visitors from street level up to the floors of the museum. The plan view of the

ramp is in the shape of an Archimedean spiral, figure 2.

(a) Construct one convolution of an Archimedean spiral representing the

balustrade, given that the shortest radius vector is 0mm and the longest

radius vector |OA| is 110mm.

(a) Draw a tangent to the spiral at a point 95mm from the pole

An Archimedean spiral is characterised by ________

movement towards or away from the pole during

___________angular movement about the pole. Thus it

is a spiral of _______________ progression

Figure 1 - Spiral Ramp, Vatican Museum Figure 2 - Plan view of ramp Archimedean spirals in nature

Figure 3 - Cochlea

ARCHIMEDEAN SPIRAL

• Construct an Archimedean Spiral from given data

• Construct a tangent at a point on an Archimedean Spiral

ARCHIMEDEAN SPIRAL

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


2

The lengths of consecutive radii of the Logarithmic spiral

increase by a constant _____________ thus forming a spi-

ral of ______________

Figure 2 shows a satellite picture of hurricane Ike. The hurricane hit the coast of the United States

in September 2008. Storms like this hurricane appear as a logarithmic spiral when viewed from

above.

Draw one convolution of the logarithmic spiral given that the longest radius vector OA is 100mm

and the lengths of the preceding radii at 30° intervals decrease in the ratio of 7:8.

Triangle ABC is shown in figure 1. x and y divide the line AB in the ratio of 3:2:1.

Two lines are constructed parallel to BC through x and y to intersect AC at u and v.

Using the table below record |Au|, |uv| and |vC|.

What do you notice about the ratio of the divisions.?

Figure 2 - Satellite picture of Hurricane Ike.

Figure 1 - Triangle ABC

Spirals are naturally occurring in nature. The image above, figure 3, is a common

sea creature known as a Nautilus. The shape of the shell is an approximate loga-

rithmic spiral.

(a) In the space provided construct a logarithmic spiral of one convolution from

a shortest radius vector, OA, of 20 mm and successive radii increase in the

ratio of 5:4 at 50° intervals

(b) Draw a tangent to the spiral at a point P, 45 mm from the pole.

Figure 3 - Nautilus sea creature

• Construct a Logarithmic Spiral from given data

• Construct a tangent at a point on a Logarithmic Spiral

LOGARITHMIC SPIRAL

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


3

The Pitch is the distance travelled in one ______________ measured parallel to the ___________. The development

of a helix is a ___________________________. Other examples of everyday uses of helices are;

1. __________________, 2.____________________

A spiral staircase of one complete revolution is shown in the photograph.

The handrail of the stairs is in the shape of a helix of one complete revolution

The plan and elevation of the cylinder which forms the basis for the handrail as well

as the relative positions of the ends of the handrail, a and b, are shown below.

(a) Draw the elevation and plan of the handrail.

(The thickness of the handrail may be ignored)

(b) Determine the length of material required to make the handrail

A water tower is shown in Figure 2. The cylindrical portion is

divided into levels for the safe storage of the water .

The plan and elevation of the cylindrical portion of the tower is

shown above. Points a and b are two access points at different levels

Draw the projections of the shortest pipe that may be used to join

points a and b.

Human DNA, Fig. 3, is double stranded and takes the form of a double helix.

The partial plan and elevation of a cylinder are shown across. The cylinder

forms the basis of a left hand helix of one and a half revolutions.

Draw the projections of the helix starting at pt. P at the base.

Figure 2.—Water Tower

• Construct the helix from given data

HELICAL SPIRAL

Figure 3 Human DNA

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


4

Vor Frelsers Kirke spire, Copenhagen is displayed in figure 1. The

conical spiral contains an external staircase that may be used to climb

to the top.

The elevation and plan of a cone, which represents a portion of the

spire, are shown below.

Draw the projections of a conical spiral of one revolution from point

0 on the base to the apex of the cone.

A conical spiral is the locus of a _________ located on the

surface of a __________, as it moves, both around and in the

direction of the axis at a _______________ rate.

The plan of a conical spiral is an ______________ spiral.

Figure 1. - Vor Frelsers Kirke Spire

• Construct a Conical Spiral from given data

CONICAL SPIRAL

Figure 2 - Divining pendu-

Divining is a method employed to locate ground water and many other

objects and materials without the use of scientific apparatus.

A pendulum suspended from a chain is commonly used. Figure 2.

shows one such pendulum called a spiral.

The plan and elevation of an inverted conical outline of the pendulum is

shown. The projections of point P are also included.

Draw the projections of a conical spiral of one revolution from the base

to the apex of the cone passing through point P.

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


5

The word involute means to unfold

An involute is the _________ traced by

the end of a taut piece of string when it

is ____________from a basic shape

Figure 2 displays an image of a wind breaker that is used to shelter people

from wind when at the beach. The plan of the windbreaker, in a given

position., is represented by a square of 25mm side.

(a) Plot the locus of the point P as the windbreaker is ‘unwound’ in an

anticlockwise direction, in a taut fashion, to the open position displayed

in figure 3.

(b) Construct a tangent to the curve at a point of your choice.

Figure 2 - Wind Breaker Figure 3 - Wind breaker in open position

A Toblerone bar is displayed in figure 1. The packaging has a

triangular cross section. A perforated strip must be torn away as

illustrated, to remove the bar from its packaging . The path of the

endpoint of the strip forms an involute.

Using the triangle below, plot the locus of the endpoint P of the

strip as it is torn away to reveal the chocolate.

Figure 1 - Toblerone Bar

• Construct the involute of a circle and regular polygons

• Construct a tangent at a point on an involute

INVOLUTES 1

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


6

An image of a yo-yo is displayed in figure 2. First made popular in the 1920s, yo-yoing is enjoyed by children and

adults. The string is initially wound, in a taut manner, around the spool by hand; the yo-yo is then thrown down-

wards so that it first descends unwinding the string, then (by inertia) climbs back winding it up; and finally the yo-

yo is grabbed, ready to be thrown again.

(a) The spool and string of a yo-yo are represented above. Plot the locus of point P as the string is wound

around the spool prior to playing. (Note: The length of string will require more than one revolution)

(b) Draw a tangent to the involute at a point 93mm from the centre of the circle.

Figure 1 shows an interlocking barrier system which may be used in event man-

agement or pedestrian control usage. The barriers link together through pivot

points which allow them to rotate about each other. The plan of the layout of six

barriers, in a regular hexagonal shape, is displayed below. The barriers are to be

repositioned in a straight line from P without dismantling.

Plot the involute of point P as it is unwound from the regular hexagon in a clock-

wise direction.

Figure 2 - Yo-yo

Figure 1 - Interlocking barrier system.

• Construct the involute of a circle and regular polygons

• Construct a tangent at a point on an involute

INVOLUTES 2

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


7

A cycloid is the path traced out by a ________________ on the circumference of a circle as it

rolls along a fixed ____________ ______________ without slipping.

Figure 1 shows an image of a racing bicycle. The tyres are inflated

through a valve located on the rim of the wheel. A circle and a line

representing the wheel and the ground are displayed in Figure 2. Point

‘P’ represents the valve on the wheel.

Plot the locus of point P as the circle rolls along the line for one com-

plete revolution.

The Roman architect Vitruvius developed machines for

measuring distance, one such machine was an Odometer.

The diameter of a chariot wheel was 4 feet. If a mark is

placed on a wheel where it touches the ground, and the

chariot then moves until the mark comes to the ground

again, the chariot will have moved 12.6 feet. A device was

attached to the axle of the chariot such that when a wheel

made one revolution, a drum with 400 teeth on it moved

round one notch. A complete rotation of the drum was

equivalent to a motion of 400(12.6)= 5,040 feet by the

chariot. At this point the device dropped a stone into a

bronze bowl, 5040 feet was one Roman mile.

Figure 3 represents the wheel of a Roman chariot. The lines AB and BC

represent the ground on which the chariot is travelling.

Plot the locus of point P as the circle rolls along AB and BC, without

slipping, until point P comes in contact with the line BC.

Figure 3

• Construct Standard Cycloids from given data

• Construct a tangent at a point on a cycloid.

CYCLOIDS

Figure 1 - Racing Bicycle

Figure 2

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


8

An image of a snooker table is displayed in figure 4 along with an enlarged image of the pocket

mechanism for storing the balls. Figure 5 represents one of the guide bars used to support the balls,

as well as a ball (A) which is resting in the pocket. Another ball (B) along with a point P on its sur-

face is also indicated.

(a) Plot the locus of point P on ball (B) as it rolls along the guide bar until it comes to rest against

the other ball (A).

(b) Construct a tangent to the locus from a point 70mm from P.

Crazy golf is a pastime which may be enjoyed by all of the

family. A garden crazy golf set is displayed in figure 1. One

particular piece, shown in figure 2, is an arch based on an arc

of a circle. The goal is to roll the ball over the arch.

A sectional elevation of a portion of the arch and ball are

shown below. Point P is located on the circumference of the

circle.

(a) Draw the locus of point P, for one complete revolution

of the golf ball as it rolls along the arch.

(b) Draw a tangent to a point on the curve located 120mm

from P.

An Epicycloid is a plane curve, produced by tracing the path of a chosen ________ on a generating circle, as the circle

rolls without slipping around the __________ of the circumference of a ___________.

Figure 1

Figure 2

Figure 3 - Cross section through the arch and the golf ball

Figure 4 - Snooker table with enlarged view of the pocket

• Construct Epicycloids and Hypocycloids from given data

• Construct a tangent at a point on an Epicycloid and Hypocycloid

HYPOCYCLOID & EPICYCLOID

Figure 5

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


9

The curves obtained in the previous worksheet involving wheels are called cycloids. If we plot the path traced out by a reflector on the spoke of

a bicycle wheel, we obtain a different curve, called a trochoid.

A trochoid is a curve traced by a point on the radial spoke of a circle as the circle is rolled along a line. Figure 1 shows an image of a mountain

bike with a reflector mounted on the spokes of the wheel. Figure 2 shows an enlarged detail of a reflector on a bicycle wheel.

A circle and a line representing the wheel and the ground, respectively are shown in Figure 3. Point ‘P’ represents the reflector on the wheel.

Plot the locus of point P as the circle rolls along the line for one full revolution.

The path traced out by a point that lies on the outside of a circle which is rolled along a straight line is called a superior trochoid.

A superior trochoid is traced by a pedal when a bicycle is pedalled along a straight line. Figure 4 shows an image of a bicycle pedal and the front

chain wheel. Figure 5 shows a circle and a line representing the chain wheel and the direction of travel of the bicycle, respectively. Point ‘Q’

represents one of the pedals. Plot the locus of point Q as the circle rolls along the line for one full revolution.

Figure 2 - Reflector

Figure 1 - Mountain bike

The path traced out by a point that lies on the inside of a circle which is rolled along a straight line is called an inferior trochoid. Figure 3

Figure 4 - Bicycle pedal

Figure 5

• Construct inferior and superior trochoids from given data.

TROCHOID 1

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


10

• Construct an inferior epitrochoid and an Archimedian spiral from given data

TROCHOID 2

Spirograph is a geometric drawing toy that uses gears and wheels of different sizes to produce mathematical curves. To use it, a smaller gear is

placed inside or outside a larger one and lined up with interlocking teeth. There are holes in the gears to place the point of a pen. Different

designs can be created by rolling the gears alongside each other using different shapes or sizes of gears.

The thunbnail across shows a spirograph design obtained by tracing the path of a point P attached to a circle rolling around the outside of a fixed

semicircle for one complete revolution. The point P is positioned at the midpoint of the radius.

(a) The drawing above shows the the semicircle. Determine the radius of the rolling circle and complete one half of the design showing all

construction lines. The resulting curve is called an epitrochoid.

(b) Redraw the circle after it has rotated through one complete revolution. Draw one half of one convolution of an Archimedian spiral that is

unwound about C from P to A in an anti-clockwise direction.

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


11

LOCI 1

A ladder is shown leaning against a brick wall. An eyelet for attaching a safety rope is

labelled as point P on the ladder. If the ladder were to slide down the wall and come

to a rest on the ground as indicated, plot the locus of the point P for this movement.

P

A gardener wishes to mark out an elliptical shaped flowerbed similar

to the one in the photograph across. The gardener will use a trammel

to mark out the curve.

The major and minor axes are 1.5m and 1.2m respectively. To a scale

of 1:10 mark out the ellipse using a trammel on the centrelines drawn

below.

P

PA 45 PB 75

P1A1 P1B1

P2A2 P2B2

Shown below is an ellipse with major and minor axes of 150mm and 90mm, respectively. The line

PB is equal in length to half the major axis and is positioned with P on the curve and B on the

minor axis. A is the point where the line intersects the major axis.

The measurements PA and PB are recorded in the table across. Fill in the remaining measurements

in the table. Do you notice any pattern emerging? What can we conclude from this information?

• Construct the locus of a point defined by the movement of a line relative

to another line

• Construct an ellipse using a trammel

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


12

LOCI 2

• Construct the locus of a point in a link mechanism

An interactive toy for a child is pictured in Figure 1. The toy incorporates a

mechanism which causes the front loader to raise and lower when the back

wheels rotate. Figure 2 shows the front loader in its lowest position. The line

diagram for the mechanism is shown below.

(a) Draw the front loader when it is in a fully extended position.

(b) Draw the locus of point P for this movement Figure 2 - Loader in lowest position

Figure 1 - Interactive Toy

Figure 3 - Elliptical

Exercise Machine

Daily exercise is an essential component of a healthy lifestyle.

Figure 3 displays an image of a ski exercise machine commonly

found in the gym. A line diagram representing the movement of

the ski machine is captured in figure 4. Point P represents the

front of the foot platform and is fixed on BC. Points A and O are

fixed, B is a pin joint and C is constrained to travel about O

Plot the locus of point P for the movement of the mechanism.

Figure 4

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


13

LOCI 3

Construct the locus of a point in a link mechanism

Fig. 1

A

P

B

C

slider

fixed rail

Shown in the photograph across is a PVC window. Fig. 1 shows a plan drawing of the mechanism used to

open this window. The main dimensions are listed in the table in Fig. 2. Point A is fixed. As the window

opens, point C is constrained to travel along the fixed rail (slider).

Construct a line diagram of the window mechanism and plot the locus of point P as the window moves

from a closed position to an open one until the point C is 135mm from A.

Note: Point A has been setup on the baseline shown below.

A

Fig. 2 Window mechanism in closed position

B C

AB 115 PB 100 BC 70

The mechanism for an up-and-over garage door is shown in the 3D graphic below.

The point A travels in a horizontal direction and the point B is constrained by an

arm to rotate about the point C which is fixed.

Plot the locus of the bottom of the door (point P) from the time it is closed in a

vertical position until it is open and in a horizontal position.

Fig.3

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


14

CAMS 1

• Construct cam profiles and displacement diagrams

• Construct radial plate cams of given uniform velocity, simple harmonic

motion, uniform acceleration and retardation to in-line knife edge followers

A toy train is shown in Figure 1.The toy train has a cam attached to the front axle which

causes the smokestack to rise and fall as the train moves. Figure 2 shows a sectional

view with the cam and in line knife edge follower in mutual contact.

(a) Using the cam profile below plot the displacement diagram for one complete

clockwise revolution of the cam.

(b) Indicate the maximum displacement of the follower on the displacement diagram.

Figure 1 - Toy Train

Figure 2 - Cam & Follower

Figure 3. - Bingo Machine

A Bingo Machine is shown in Figure 3. The ball is presented to the bingo caller by

a controlled movement using a cam and follower mechanism. The cam rotates in

an anticlockwise direction.

Plot the displacement diagram and determine the cam profile to impart the follow-

ing motion to an in-line, knife-edge follower:

0° to 180° simple harmonic motion rise of 30mm

180° to 210° dwell

210° to 360° uniform velocity fall of 30mm

The nearest approach of the follower to the cam centre is 16mm

A cam is a shaped component generally used to change rotary movement into

____________ movement.

A _________ presses against the curved surface of the cam.

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


15

CAMS 2

Figure 1, across, shows a pinball machine commonly found in amusement arcades. A cam and

follower mechanism is used to release the ball to the player.

When a coin is inserted a cam rotates anticlockwise and the follower mechanism raises the ball

to the launch pad in front of the plunger. The ball is held momentarily for the plunger to make

contact with it; then the mechanism drops back down to receive the next ball.

(a) Plot the displacement diagram for the cam given that it imparts the following motion on

the follower;

0° to 180° rise of 30mm with uniform acceleration and retardation

180° to 270° dwell

270° to 360° fall of 30mm with uniform velocity

The nearest approach of the follower to the cam centre is 18mm.

(b) Construct the corresponding cam profile to impart this motion

Shown in figure 2 is a model of a cam operated sledge hammer

based on an idea by Leonardo Da Vinci. The cam, based on an

Archimedean spiral, is designed to raise the hammer slowly and

then drop it from a height.

Construct the cam profile given a minimum radius vector of 12mm

and a maximum radius vector of 60mm.

Figure 2 - Cam operated Sledge Hammer

The original Da Vinci sketch of the cam

operated hammer. It is believed that he

developed the idea of an automatic

hammer to help blacksmiths, who used

repeated blows of a hammer in the

process of forging steel.



motion, uniform acceleration and retardation to in-line knife edge followers.

Figure 1 - Pinball Machine

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


16

CAMS 3

Figure 2 - Cam shafts



motion, uniform acceleration and retardation to in-line roller and flat followers

CAMS 3

Figure 1 - ‘Twin Cam’ Corolla



motion, uniform acceleration and retardation to in-line roller and flat followers

A model of a fairground carousel is shown in figure 3. As the carousel turns

the animation of the horses is achieved by means of a cam and roller

follower mechanism. Construct the displacement diagram and cam profile

using the following displacement data:

0° to 120° uniform acceleration and retardation rise of 20mm

120° to 180° uniform velocity rise 10mm

180° to 360° simple harmonic motion fall of 30mm

Cam rotation is clockwise. The roller follower diameter is 12mm. Mini-

mum distance between the cam centre and roller centre is 26mm.

Figure 3 - Fairground Carousel

Figure 1 shows an image of a ‘Twin Cam’ Corolla performing in the Rally of the Lakes, Killarney. ’Twin Cam’ refers to the camshaft (figure 2),

which opens and closes the valves allowing fuel and air to enter and exhaust fumes to exit the engine. Most modern engines use overhead cams;

the cam shaft is located directly above the valves. Many high performance engines have four valves per cylinder thus requiring two camshafts,

hence the term “twin cam”.

Construct a cam profile to impart the following motion on an inline flat follower.

0° to 150° Simple Harmonic Motion rise of 28mm

150° to 210° Uniform Velocity rise of 12mm

210° to 360° Uniform Acceleration and Retardation fall of 40mm

Cam rotation is clockwise. The follower extends 7mm either side of the centreline.

The nearest approach of the follower to the cam centre is 15mm

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


17

GEARS 1

• Construct involute gear profiles

Given the following data:

• Pitch Circle diameter = 200mm

• Module = 10

• Pressure angle = 20°

(a) Complete the caluclations for the gear data in table 2.

(b) Construct the spur gear, using centre O below, showing at

least four teeth.

A

B

C D

E

G

F

H

I

Figure 1

The involute gear profile is the most commonly used gear profile. In the involute gear the tooth shape is generated by

the involute of a circle. This profile gives very smooth movement between the two meshing gears.

Plot the involute of the semicircle C as P is unwound in an anticlockwise direction.

Table 1

Table 2

Gears are widely used in the world around us today. They are found in many everyday objects such as clocks,

watches, DVD players & toys. Two meshing spur gears are displayed in figure 1. Spur gears are used to transmit

motion between two parallel shafts.

A portion of a spur gear is displayed in figure 2. Using table 1, identify the parts of the gear labelled A to I.

DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


18

GEARS 2

CD’s do not operate efficiently when scratched or dirty. A device for cleaning CDs

is displayed in figure 1. A cleaning solution is added to the surface of the disc and it

is loaded into the disk cleaner. Meshing spur gears (figure 2) are employed to rotate

the CD against a soft pad, thereby removing light scratches and cleaning the surface

of the disk.

Given the following data:

• Gear Ratio = 4:3

• Driver Gear Details:

• Module = 8

• Niumber of Teeth = 24

• Pressure Angle = 20º

(a) Complete the calculations for the gear data in table 1.

(b) Using centre O, draw the two involute gears in mesh showing at least four

teeth on each gear.

Figure 1—Disk Cleaner Figure 2 - Gear Mechanism

GEAR RATIO

If a driver gear with 10 teeth meshes with a driven gear with 20 teeth the gear ratio is

_____ : _____

The driven gear rotates at ___________ the speed of the driver gear.

Driven Driver


DYNAMIC MECHANISMS

NAME: ______________________________ DATE: _____________


19

GEARS 3

For decades cameras have been used to capture images and record

moments and memories. Many cameras can be mounted on tripods

similar to the one in figure 1. Camera height may be adjusted by

winding a handle which causing the centre portion of the stand to

move upwards. (figure 2) This mechanism is based on a rack and

pinion gear system.

Given the following gear data for the pinion:

• Number of teeth = 20

• Pressure Angle = 20º

• Module = 10

(a) Complete the calculations for the gear data in table 1.

(b) Using centre O, draw the gear and rack in mesh, showing four

teeth on the gear and an equivalent on the rack

Figure 1

Figure 2

Other objects that employ a rack and pinion mechanism

A rack and pinion mechanism converts _________________ motion into ______________

motion or vica-versa.

The rack may be considered a spur gear with a diameter equal to ________________ .


Documents

Learning Outcomes - Students should be able to: An ...dcgnotes.weebly.com/uploads/2/4/0/6/24061727/... · Plot the involute of point P as it is unwound from the regular hexagon in