# DIY Planimeter and Pentograph,A Study of Basic Terms Related to Kinematics

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## Introduction: DIY Planimeter and Pentograph,A Study of Basic Terms Related to Kinematics

Introduction:Usually the way in which topics are taught in schools and colleges are not in a practical way even though use of virtual models is done in some of the schools my experience says that one needs to apply it physically to understand every topics deeply and to create questions in mind of the students which will help the students grow.This is where instructables and STEM comes into picture.With instructables we can make different wonderful things which relate to arts science or any other fields and STEM helps to teach corresponding topics related to the instructable. Hence we are going to make a mechanism which is used to calculate area of irregular shape with simple multiplication of numbers and relate it with the study of kinematics.This instructables is centered for freshers enrolled in engineering field particularly mechanical and civil engineering as this instructables involves some tools which may not be available in schools or high schools but worry not as i have also included a mini instructable which can also relate to the same topic we have chosen.

Outcomes:

• Students will be able to relate what they have studied in classrooms to the machine or mechanisms they see around them daily.
• Student will be able to work in groups understand the understand the importance of teamwork.
• Students will be able to understand topics being taught in a fun way.

### Teacher Notes

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## Step 1: SUPPLIES

For planimeter

Tools and Materials

• Ruler
• Wood sticks
• Wooden ply
• Hobby craft knife
• Pen/pencil
• Barbecue skewers/Pen refills
• Hot glue
• Super glue
• PVA glue
• Drill machine
• Hack saw
• A4 sheet
• Drill bit
• Sandpaper
• Epoxy glue
• Vernier calliper

Software

• Autocad

For mini instructable Pantograph

• Popsicle sticks(flat ones)
• Barbecue skewers
• Pencil(for pencil lead)
• Hot glue
• Super glue
• Sandpaper
• Blade
• Drill machine/Metal Poker
• Nail
• A4 sheet

## Step 2: Pantograph(mini Instructable) Part 1:Construction

What is a Pantograph ?

Pantograph is a mechanism used to produce paths on an enlarged or reduced scale as exactly as possible the path described by a given points. It is based on four bar kinematics chain.It coonsist of four links joined in such a way that it forms a parallelogram.

Construction:

Step 1: Mark center line using a pencil perpendicular to the length of the flat Popsicle stick and lines 5 mm away from the two ends of the Popsicle stick.Repeat this procedure for three other sticks.

Step 2:Now measure the length of the marked lines using a ruler and mark a point to the center of the lines.

Step 3:Drill or punch holes corresponding to the diameter of barbecue skewers on the marked points,take adults supervision while doing this step as this may lead to injuries if not done properly and without any safety precautions.

Step 4:Now mark 3 squares on Popsicle stick of 10mm*10mm(This corresponds to the breadth of the locally available Popsicle sticks,this may vary depending on he manufacturer.).Make sure the marked patterns is square.

Step 5:Draw diagonals to the marked squares and drill a hole on a point where the diagonals intersects.Cut the squares using a sharp blade.

Step 6:Cut 3 pieces of barbecue skewers of length 2mm. Stick these pieces in the hole of square pieces as shown in the picture.Use super glue to securely hold the pieces.

Step 7:Now prepare two cross linked mechanism as shown in the pictures with using the pieces made in Step 6 as hinges.

Step 8: Cut a pencil of length 50mm and split it into half using sharp blade making sure the lead does not breaks.Glue the lead to pencil hole using hot glue leaving 10 mm below the Pantograph. This will act as marker for our scaling mechanism.When gluing the lead make sure the lead is glued to the lower link of the mechanism or else both the link become fully constrained.The upper link should freely rotate along the axis of the lead.

Step 9:Cut barbecue skewer of length 50mm and glue the skewer into the guide end.This will be used to guide it on the original figure for scaling.

Step 10:Now for the fixed end use a nail and hot glue the hammering end on a flat surface and insert the Pantograph fixed end hole on the nail.

Your Pantograph is ready!

## Step 3: Pantograph(mini Instructable) Part 2:Operation and Working

Operation:

Take a blank paper sheet and place it below the pencil lead and the other sheet with pattern below the guide end making sure the papers does not slip when the guide end is moved,masking tape can be used to hold it place.Now before starting lift the guide end and place it on more than four different points to make sure the Pantograph can draw the pattern.Start tracing the pattern and observe the pattern being replicated in a scaled proportion.After every intervals check whether you have missed a spot.The pattern that is being drawn will be very faint.After the pattern is fully traced use dark pencil to darken the drawn pattern.Here i am tracing a square from my hobby mat.

Calculating the scaling factor:

Measure one of the straight side of the original pattern and the corresponding side of the scaled drawing pattern.Now divide the original measured dimension with the scaled measured dimension,reduce it to smallest factor with 1 in numerator.

For example,

Now you can skip to STEP 13 for lessons if you don't want to make Planimeter.

## Step 4: What Is Planimeter?

Planimeters are mechanical instruments which can measure the area of closed regions in the plane. Planimeters are used in medicine to measure the size of the cross-sections of tumors or organs, in biology to measure the area of leaves or wing sizes of insects, in agriculture to measure the area of forests, in engineering it is used to measure the size of profiles.
Construction and Working:
The Amsler (polar) type consists of a two-bar linkage. At the end of one link is a pointer, used to trace around the boundary of the shape to be measured. The other end of the linkage pivots freely on a weight that keeps it from moving. Near the junction of the two links is a measuring wheel of calibrated diameter, with a scale to show fine rotation, and worm gearing for an auxiliary turns counter scale. As the area outline is traced, this wheel rolls on the surface of the drawing. The operator sets the wheel, turns the counter to zero, and then traces the pointer around the perimeter of the shape. When the tracing is complete, the scales at the measuring wheel show the shape's area.

When the planimeter's measuring wheel moves perpendicular to its axis, it rolls, and this movement is recorded. When the measuring wheel moves parallel to its axis, the wheel skids without rolling, so this movement is ignored. That means the planimeter measures the distance that its measuring wheel travels, projected perpendicularly to the measuring wheel's axis of rotation. The area of the shape is proportional to the number of turns through which the measuring wheel rotates.
Now lets get started with the build process.

## Step 5: The Fixed End Shaft

The fixed end shaft has two holes on the both ends one of which is used for free rotation along the fixed end and the other for free rotation along the main body of planimeter.
I am using a rectangular shaft of 15mm*15mm and 200mm long.
On one end of the shaft drill a hole corresponding to the size of pencil and in the other end drill a hole on the same plane corresponding to the size of barbeque skewer. Make sure the hole to be drilled is perpendicular and straight.
Now if the pencil to be used for fixed end is hexagonal in shape the you will have to sand it to cylindrical shape. An easy way to shape it in a cylindrical form will be attaching the pencil in the hand power drill where the drill bit goes and holding the sand paper along the hexagonal surface. The same can be done for barbeque skewers if the hole is not wide enough.

## Step 6: The Fixed End

This end is fixed and should not move so we will have to think of a mechanism which will allow the fixed end shaft to rotate about the fixed point.
I will be using epoxy to generate a weighted block with a shaft which will aid the fixed end shaft to rotate about it.
First the center is marked on the bottom circle of the paper coffee cup and then the pencil which was sanded to the holed circle diameter of the fixed end shaft is pierced from the bottom of the cup keeping in mind that the flat end of the pencil is in the cup and the major part of the pencil is out of the cup. The pencil and the cup are glued together where the pencil has pierced so that the epoxy does not get out when it is let to cure.
Now start pouring the epoxy into the cup making sure the cup and pencil is perfectly horizontal.let the epoxy cure depending upon the curing time of the epoxy brand.
Usually a standard planimeter has pointed needle on the fixed end which is used to securely fix the fixed end but we won't be making that and try to keep it as simple as possible.

Now i understand that this instructables should be such that the materials used should be available easily and should be easy to implement,hence for alternative of epoxy you can use a big blob of clay and pierce the pencil into the blob with the blob on flat surface.Just make sure the pencil is straight and perpendicular.

## Step 7: The Rolling Element

This is the element where all of the working of planimeter is solely dependent on,hence this element should be friction less,don't know what friction is no worries we will be discussing these terms in further steps.For something to roll properly the component has to have a cylindrical shape or of spherical shape. I am using my old broken roller scale for the rolling element. You can use old marker pen or something cylindrical just make sure the diameter of the rolling element should be big enough such that we will be able to read the scale properly.Cut the cylindrical component of the length of 40mm. Now my roller scale has both ends to hold the cylindrical components properly but when you are using the board marker pen as the rolling element you will have to enclose the ends of the cylinder,use a piece of stiff cardboard and glue it on both ends and trim off the excess.

Now mark the center of the circle of the ends of cylindrical component and drill a hole corresponding to the drill bit of size equal to barbecue skewer on both ends of the cylinder.

## Step 8: Scale for Rolling Element

In this step we are going to design scale for the rolling component in Autocad and then print it.As i said earlier this instructable is centered for mechanical engineering students and hence is important for them to have at least basic understanding of Autocad. The scale can also be hand drawn but it will increase the chances of errors and will take much time as compared to Autocad.

Draw a rectangle of random size using a rectangle command or by using lines.Now measure the width of rolling element and use this dimension for the base of the rectangle,this can be done with the help of linear dimension tool in parametric tab.After selecting the tool click on two of the bottom vertices and then edit to the proper dimension.Now for the height of the rectangle we will have to measure the diameter of rolling element,Now perimeter of a circle is given by formula (π*D) where d is the diameter of roller.Repeat the steps as we did for the base of the rectangle.Provide small clearance for the scale from the left edge of the rectangle using lines tool and draw a line as shown in the images.Draw a small horizontal line on the top edge of the rectangle in my case it is 3mm.Using arraypath tool select the 3mm line as object and one of the vertical edge of rectangle as path.Now we want 100 such equally spaced lines so divide the length of the height of the rectangle by 100 and input the value in item spacing input area,input no of items as 100.Repeat the step for 5 mm vertical line with 20 equally spaced lines and for 8mm vertical line with 10 equally spaced lines.Number the 8mm line using text command as shown in images.Click on all other text which appears and delete it.Draw a reference scale of 100mm vertical and horizontal.

Now the most important task would be to convert the drawing into pdf file and printing it according to scale.Press control key and p key simultaneously on the keyboard a dialog box will appear.Follow the images properly and then select ok and save it in your folder.Open the pdf file and print with printing parameters set according to the images above.

Follow the steps as shown in the pictures for proper understanding.

If you still don't understand something you can put your queries in the comment section.

## Step 9: The Main Body

For the main body we will be using mm ply wood making sure the plywood is of the size mm*mm and perfectly rectangular stick it to the base ply of dimensions mm*mm*mm.Once the glue is properly cured use main shaft of square cross section of dimension no more than the thickness of plywood to mark on the base ply as shown in the pictures.the length of the main shaft usually doesn't matter much but the length should be at least 100mm bigger than the main body.Do not glue the shaft we are using this right now only for marking the dimensions on the base ply.After marking use another piece of plywood of same thickness and length but with a breadth of 200 mm and glue it to the base along the marked line leaving space for main shaft.Let the glue cure and slide the main shaft in the space to see if it is sliding properly.

Now according to the diameter and length of rolling component mark properly on the main body by leaving 5mm to 10mm clearance parallel to length of the main body,leave 20mm from both the sides of the main body. If you are finding it difficult to understand what is written here you can always refer to images above.Cut the mark space using hacksaw .

Now mark proper circle on the sides of the main body leaving 2mm to 3mm below the main body to roll on the floor and drill hole corresponding to hole size of barbecue skewer. Repeat this on the other side.

Now glue the upper ply with same dimensions as the lower ply and once again check whether the main shaft is sliding properly.

Now assemble the rolling component properly use the images above for proper reference.

A small square piece of popsicle stick can be used to constrain the linear motion.

## Step 10: Main Shaft

At one end of the main shaft file a groove using semicircular file as shown in the figure. Use hot glue to glue barbecue skewer into the groove. Now before glueing the skewer make sure the skewer is little below the main shaft. The depth at which the skewer should be below the main shaft can be calculated by inserting the main shaft into the main body.

## Step 11: Assembling

Before assembling we need to drill a hole on the upper face of the main body for the assembly of fixed end shaft. Mark 20mm below the main shaft space and 20mm from the right. Drill a hole corresponding to the size of barbecue skewer. Now make sure the skewer is freely rotating or else sand off the skewer as we did earlier for the pencil.A point ended barbecue skewers is glued on the main body for indication and reading of scale.
Now assemble all the parts as shown above.
My main shaft wasn't tight enough so i made a paper wedge to securely hold it in the main body.

## Step 12: Operation

After assembling place the fixed end away from the pattern such that the main shaft can freely move over the perimeter of the pattern.

Calibration or finding multiplication factor:

To find the multiplication factor of the instrument we need to have a base pattern with which we will find the factor by tracing the pattern.

Now place the main shaft's marking end at one point on the perimeter of the pattern.Make sure the the pointer on the planimeter is coinciding with initial point i.e 0 of the scale and start tracing the square pattern with 100mm*100mm dimensions in clockwise manner.After reaching the start point the scale show a reading,note down the reading.Repeat the steps and note down the reading two more times and take the average value of the readings.Now as we know the area of square is given by (side of the square *side of the square).=100*100=10000mm^2.

Therefore the formula becomes as follows

reading on scale * multiplication factor=area of square

Now we know the reading on scale and area of square,we can calculate the multiplication factor by doing simple math.

Lets take my designs example:

4.4*multiplication factor=10000

Therefore multiplication factor=2272.727

Now to calculate an irregular shapes area i traced our beloved instructables robots perimeter in the pdf given below using the same multiplication factor.It is highly possible that the are traced wont be accurate but will be near to the true value as in my case.

Note:The multiplication factor will be different for different rolling element diameter and main shaft.

This is not how a standard planimeter is operated to find area of irregular shapes,some standard planimeter is factory set and can show the area directly from the scale reading.

## Step 13: Lesson 1:Kinematics

Kinematics relates to study of relative motion between the parts of machine.

Under study of kinematics, the relative motion of various parts can be determined with respect to various positions of the other parts provided the length of various parts is known.

Kinematic link:

Each part part of a machine or instrument which has relative motion to some other part of the machine is known as kinematic link or element.

A link is not necessary a rigid body but it is resistant body. A resistant body is one which is capable of transmitting the required motion and force with negligible deformation.

## Step 14: Lesson 2:Constrained Motions

There are basically three types of constrained motions viz.

1. Completely constrained motion
2. Incompletely constrained motion
3. Successfully constrained motion

Completely constrained motion:

A completely constrained motion is one which the relative motion between the two links can be placed only in a definite direction and it can be predicted.

Example: The motion of a rectangular bar sliding in a rectangular slot as shown in the picture, in this case the motion of rectangular bar is only of sliding type in a fixed block.

Incompletely constrained motion:

When two links are connected in such a manner that their relative motion can take place in more than one direction , then the motion is said to be incompletely constrained motion. In such case the motion of the link can not be predicted.

Example:The motion of a round bar in a circular through hole of a fixed block as shown in the picture.This is a case of incompletely constrained motion since the bar can rotate or reciprocate or both motions of rotation and sliding of bar can take place simultaneously.

Successfully constrained motion:

When the relative motion between the links is not completely constrained by itself but it is made so by other means is called as successfully constrained motion.

Example:The motion of round shaft in circular blind hole as shown in the picture.The sliding motion is prevented up to a limit and only rotational motion is possible.

## Step 15: Lesson 3:kinematic Pair and Kinematic Chain:

Definition of Kinematic Pair:

When two kinematic links are connected in such a way that their motion is either completely or successfully constrained ,these two links are said to form a kinematic pair.

Definition of Kinematic Chain:

A combination of kinematic pairs in which each link forms a part of two kinematic pairs and the relative motion between the links is either completely constrained or successfully constrained is called as kinematic chain.

The kinematic pairs are classified based on the nature of relative motion between links and nature of contact between links.

Classification according to nature of relative motion between links.

1. Sliding pair or prismatic pair:

A sliding pair or prismatic pair is formed by two links in such a manner that one link is constrained to have a sliding motion relative to other link.

Example: The motion of a rectangular bar sliding in a rectangular slot as shown in the picture, in this case the motion of rectangular bar is only of sliding type in a fixed block.

2. Turning or revolute pair:

When the two links are so connected that one link is constrained to turn or revolve relative to other link they are said to form a turning pair. Sometimes they are called hinge pair.

Example:

3. Screw or helical pair:

When the nature of contact between the pair of links is such that one of the links can turn about the other link by screw threads they form a screw pair.In this case the motion of the screw will have both rotational and sliding motion.

Example:The motion of bolt in a fixed nut.it can be noted that in this case the bolt has both rotational and sliding motion but for a specific amount of rotation of bolt it results in a strictly proportionate amount of axial displacement relative to nut.

4.Rolling pair:

When two links are connected in such a way that one of the links rolls over the other link they are said to form rolling pair.

Example:In ball bearing the with its bearing forms a rolling pair.Similarly the ball and the shaft forms another rolling pair.

5.Spherical or globular pair

When one link forms a spherical shape and turns inside a fixed element,the resulting kinematic pair is called a spherical pair.

Example:A ball and socket joint found in action figure at the joints of hand and body.

Classification according to nature of contact between links.

1.Lower pair:

When the two links have surface contact or area contact while transmitting the motion, they form a lower pair. The relative motion between the links of lower pairs is purely sliding or turning type.

Example:All sliding,turning and screw pair form lower pairs.

2.Higher pairs:

When the two links are connected in such a way that they have a line or point contact while transmitting the motion these links are said to form higher pair.

Example:Motion of rolling cylinder on a flat surface.

## Step 16: Lesson 4:Joints in Kinematic Chain

Binary joint

When two links are joined at the same connection to form a kinematic chain, the joint is know as binary joint.

For example in the above image point 2 is binary joint.

Ternary joint

When three links are joined together at the same connection to form a kinematic chain, the joint is know as ternary joint.One ternary joint is equivalent to two binary joints.

For example in the above image point 5 is ternary joint.

Quaternary joint

When four links are joined together at the same connection to form a kinematic chain, the joint is known as Quaternary joint.One Quaternary joint is equivalent to three binary joints.

For example in the above image points 1,3,4 are quaternary joint.

## Step 17: Lesson 5:Mechanism

A mechanism is a kinematic chain with any one link fixed which is use to transmit the required motion.A mechanism with four links is called a simple mechanism and the mechanism with more than four links is called a compound mechanism.When a mechanism is used to transmit forces or motion to do some work,it is called as machine.

## Step 18: Lesson 6:Degree of Freedom

Degree of freedom of a point:

Degree of freedom of a point is the number of independent coordinates required to define the position and orientation of the point.Thus a point in space has only three degree of freedom or three coordinates are required to define its position.

Degree of freedom of a rigid body:

## Step 19: Lesson 7:Friction

Friction can be defined as resistance to motion of an object,body,surface,Fluid sliding against each other.

Friction is mainly classified into three types which are

Dry friction:

Dry friction is a force that opposes the relative lateral motion of two solid surfaces in contact.dry friction generally arises from the interaction of surface features, known as asperities.

Fluid friction:

Fluid friction describes the friction between layers of a viscous fluid that are moving relative to each other which in other terms is called as viscosity of the fluid.

Skin friction:

Skin friction is a component of drag, the force resisting the motion of a fluid across the surface of a body.This type of friction is usually studied in aerospace,aeronautical and hydrodynamic engineering fields.

Friction can also be classified as static and kinetic friction.Static friction acts when the body is at rest and kinetic friction when the body is moving relative to each other

We will be focusing on the first type of friction that is dry friction since majority of our instructables incorporates this type of friction.

For better understanding lets do an experiment.Take a ruler and an eraser,place the eraser on the flat side of the ruler while the ruler lays flat on a horizontal surface such as school bench.Now slowly start raising the ruler from one end while the other end is touching the base surface.You may observe that as you raise the ruler the eraser is sticking to the surface of the ruler this phenomena is due to friction but there comes a position when the eraser starts sliding.Now substitute the eraser with any other thing such as pencil sharpener the angle created by the ruler and base surface will be different for different materials.This difference of angle for eraser and sharpener corresponds to a term called coefficient of friction which is different for different material and also depends on the surface finish in contact.

Friction can constructive as well as destructive depending on the application.For example in braking system of vehicles we need material with high coefficient of friction but in application where we need smooth operation such as fidget spinner bearings we need low coefficient of friction.

Dry friction can be high for surface or area contact or kinematic link with lower pair as we have learned in earlier lessons.

Dry friction can be low for line or point contact or kinematic link with higher pair as we have learned in earlier lessons.

Several possibilities are shown in the images above.

In our instructable friction may result in errors and can deviate the readings from the true value hence we need to minimize friction as much as possible.

## Step 20: Lesson 8:Errors and Limitations

Now there are lots of possibilities that the calculated value may deviate from the original value this may be because of friction or irrelevant movement of main shaft pointer which takes the movement into consideration when showing the reading.Another type of error is parallax error this is mainly due to wrong observation such as reading the scale as 6 when the actual reading is 5 this may be possible when the viewing angle is not proper..

This diy planimeter is limited to the maximum area a rolling scale can read.In my case this can be approximated as 40000mm^2

## Step 21: STEM Analysis

STEM is a curriculum based on the idea of educating students about four basic disciplines-Science Technology Engineering Mathematics in an applied approach.STEM integrates all the subjects to teach different topics.STEM can be taught to students of different age groups.Depending on the topic and age group the level of the course can be selected.

In this instructables we have included all the topics related to STEM lets understand shortly how this instructable incorporates the subject mentioned above.

Science:This instructable teaches the students about the basics of kinematics of a machine or mechanism.

Technology:This instructables makes the student understand about the technological advancement and the technology we have far forgotten which can be used in future for same or maybe some other purpose.

Engineering:This instructables helps the student to learn the basics of computer aided drawing software and carpentry which are the base of mechanical engineering.

Mathematics:After understanding the above subjects students will be able to comprehend how some of the mathematical problems can be solved using mechanical systems.

## Step 22: Failures

This is a topic which many of the professors usually don't focus on much.It is important for students to understand that failures may come in every possible way whether you are studying working or doing anything,the important part is to embrace the mistakes and learn from it and try to rectify. Here is a quote from one of my favorite famous personality.

“When you take risks you learn that there will be times when you succeed and there will be times when you fail, and both are equally important.” —Ellen DeGeneres

Now back to the topic,i intended to use a gearing mechanism as gears are heart of mechanical engineering students.The gears would be driven from the rolling element and would have driven a circular disc which was going to act as secondary scale for larger area calculations.

I started modelling the gear pair in Autodesk Inventor with component generator tool.Sliced the 3d model in cura software and started 3d printing.I wasn't satisfied with the results as the gear turned out to be deformed.This is because the dimensions for the gear were too small to 3d print.

## Step 23: Activity for Teachers and Students

After teaching all the lessons teachers can perform group activity by providing a task of relating the terms learnt in the lessons.
For example teacher can ask a certain group to identify all the completely constrained motion

## Step 24: Conclusion

As the problem statement suggest that the instructables should be replicable in the classroom or other educational setting and teach skills related to Science, Technology, Engineering, or Math, i have tried my best to include all the topics. My mother tongue is not English hence i might have made some mistakes please help me by pointing out my mistakes. I am still learning english but struggle with punctuation and tenses if you find any errors help me out in comment section.
This instructables is easy but requires efforts and time and if done properly it will be really helpful for students to learn in a practical way. This is just a example of one topic,teachers can find other ways to interact with students to help them learn in fun way.
Finally any suggestions and tips would be really helpful.

## Step 25: Reference

A proper reference is also important for teaching students and inculcating true knowledge.There are various books and sources available to understand,teach and refer i chose this book which was locally available in book store.

Dr.F.B.Sayyad(2018).Kinematics of Machinery.Maharashtra:Tech-Max Publications

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## Discussions

Well, now I know what a planimeter is! It's a bit confusing as to how it gets the area, though by only measuring the perimeter. The pentograph is easy to understand.