About Instantaneous Kinematics

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About Instantaneous Kinematics

Instantaneous Kinematics (IK)

Instantaneous Kinematics, IK, is helpful when you need a mechanism for a particular design application.

For example, you can use IK to help you search for:

Cusps: sometimes called a 'pick-up motion'.

A coupler-curve that comes to a Point. The coupler point is on the moving polode, such that it approaches the fixed polode and 'kisses' it (osculates), and then reverses away from it. It is generated when the coupler point coincides with the instantaneous center of rotation of the coupler.

Crunodes: Double Points

A coupler-curve that passes through the same point two times, but with a different tangent. The coupler-curve has a loop, or it a figure-of-eight.

Straight-Lines:

A coupler-curve that is a straight line for a substantial section. The line is approximate or exact.

Constant Radius:

A coupler-curve that is a constant radius for a substantial section

Symmetrical Coupler Curves:

A coupler-curve that is symmetric about an axis line. These can be used for indexing mechanisms and, by adding two more links, to give a mechanism with two dwells.


From this point onwards, I am using the Mechanism Synthesis software, from University of Basque Country, Spain. I am using their software to show the 'classic' instantaneous curves that are often very useful for mechanism synthesis.

I must acknowledge Alfonso Hernández, CompMech, Department of Mechanical Engineering, UPVEHU for the permission to use the GIM® software. (www.ehu.es/compmech).)


KinSyn02

Terminology - from the Beginning


Note - although not necessary, we will assume that the input to the mechanism can rotate fully, and we call it the crank. It is the left of the two grounded parts in the image. The other link that is connected to ground we will call the rocker. The line between the two fixed joints will be called the fixed link. A posture of the mechanism is its position at a particular crank angle.


The Instantaneous centers (IC) - Pole

The orange dot , I14, is the Instantaneous-Center (IC) of rotation of the coupler relative to the fixed frame.

We also call this IC the Pole, sometimes called the the Velocity Pole

It is at the intersection of two rays.

One ray is the along the axis of the crank, or input link,, and the other ray is along the axis of the rocker, or output-part. Clearly, the Pole, I14, moves as the mechanism cycles.

The orange dot , I14, is the point around which the coupler is rotating at the particular posture of the mechanism. If we stood on the Coupler at the Pole, we would only be rotating relative to the frame.


Interesting: There are five other Instantaneous-Centers.

Four ICs at each of the four joints.

One IC is at the intersection of the ray along the axis of the coupler with the ray along the axis of the frame.


KinSyn05

Fixed-Polode

As the mechanism cycles, the Pole ,, also moves. We can plot the position of the Pole on to the Frame, at each step of the input rotation.

The trace of the Pole on the frame is the Fixed Polode.

We cannot see the Fixed Polode - we must either draw it, or imagine it.

The purple trace is the Fixed-Polode.

Each Dot on the Fixed-Polode is the position of the IC at a different rotation angle of the input


Interesting: the Fixed-Polode passes through the Pivot-Center of the rocker two times


 

KinSyn03

Moving Polode

We can also plot the position of the Pole on the Coupler at each step of the input rotation.

The trace of the Pole on the Coupler is the Moving-Polode.

Like the Fixed-Polode, we cannot see the Moving-Polode - we must either draw it or imagine it.

The green trace is the Moving-Polode.

Each Dot on the Moving-Polode is the position of the IC at different rotation angle of the input


Interesting: the Moving-Polode passes through the end of the rocker two times.


KinSyn04

Fixed and Moving Polode

When we show the Fixed-Polode and Moving-Polode you can see that they are in contact at the Pole as the mechanism makes a cycle.

At each angle of the input crank the ...

the point that is the Pole is marked on the Frame - the trace of the Pole becomes the Fixed-Polode.

At the same time,

... the same point that is the Pole is marked on the Coupler - the trace of the Pole becomes the Moving-Polode

 

KinSyn06

Pole Tangent and Pole Normal.

The Moving-Polode is always in contact with the Fixed-Polode at the position of the Pole, Thus, the Polodes must have a common tangent line at the Pole.

Why a Tangent?

Logically, imagine two instants, close together. The Pole travels along the Fixed-Polode from Point 1 to Point 2 - its velocity must be in the direction from Point 1 to Point 2. tn the limit, the velocity is tangential to the Fixed-Polode otherwise the Fixed-Polode would not move from Point 1 to Point 2.

The Pole also travels along the Moving-Polode from Point 1 to Point 2. It also has an instantaneous velocity that is tangential to the Moving-Polode but in the Coupler.

The absolute velocity vector of the Pole must be the same (relative to each other) otherwise the Pole marked on the Coupler by the Moving-Polode would diverge from the Pole marked on the Frame by the Fixed-Polode.

If the velocity vector of the Pole moving along the Fixed-Polode is the same as the velocity vector of the Pole moving along the Moving-Polode. the velocities are identical and tangential with each other.

Thus, the Moving-Polode rolls along on the Fixed-Polode without slipping.

At every moving instant, tangent to the Moving-Polode and Fixed-Polode at the Pole is called the Pole Tangent.

The Normal () to the Pole Tangent at the Pole is called the Pole Normal.

KinSyn08

Coupler Curves

A Coupler-Curve is the trace that a particular point on the Coupler, called the Coupler Point, makes on the fixed frame by a mechanism with defined link lengths.

The Coupler-Point is labeled as K

Each point on the Coupler will trace out a different Coupler-Curve.

 

In this image, you can see that K is a random place on the Coupler, and the Coupler-Curve has no particular interest.

Are there any interesting Coupler-Curve?.

KinSyn09

Cusps

When the Coupler-Point, K, on the Coupler, coincides with any point of the Moving-Polode, the Coupler-Curve will be at a Cusp when the Coupler-Point, K, contacts with the Fixed-Polode.

Why is this?

In this image, the 4-bar has a certain posture. We have moved Coupler Point, K, to be on the Moving-Polode.

Remember that the Moving-Polode is the trace of the Pole as marked on the Coupler.

The Coupler-Point, K, is on the Coupler. Thus, the Moving-Polode and the Coupler-Point move together.

As the four-bar cycles, the Moving-Polode and the Couple-Point, K, approach the Fixed-Polode. to roll over each other. They do not slide.

The Coupler Curve can only be exactly towards each other, and at the instant they touch the movement will be normal to the Pole-Tangent,  

The Coupler Curve can only be exactly away from each other, and at the instant they separate, the movement will be normal to the Pole-Tangent.

 

'Expand' then 'Play'

Moving a Coupler Point along the Moving Polode to show that the Coupler Point moves towards the Fixed Polode with a Cusp

 

You can see in the Video as I move the Position of the Coupler Point along the Moving Polode, then Coupler Point will move along the Fixed Plane until it touches the Fixed Polode.

KinSyn10

Question: Is there anything interesting about the motion of the Coupler-Point.

Answer: Yes!

The instantaneous motion of the Coupler-Point, K, is always perpendicular () to a ray from Pole I14 to the Coupler-Point.

See the Image. The Blue line is the direction of the velocity vector of K, a tangent to the coupler curve, and it is perpendicular to the ray from I14  

Why is this the case?

Because the Pole I14 is the center of rotation of the Coupler Plane. Thus, ANY Coupler-Point we choose on the Coupler will be rotating around the Pole

As the mechanism cycles, the Pole moves. But the Coupler-Point will be moving in a direction that is perpendicular to the ray from the Pole


 

 

KinSyn11

Osculating Circle

The Coupler-Curve , is a curve, and thus, by definition, all points along ANY curve has Curvature. The Curvature of the Coupler-Curve is continuously changing.

Thus, its Radius-of-Curvature. is also continuously changing.

 

We can draw a circle at the Coupler Point, K. with a radius equal to its Radius-of-Curvature

We call the Circle the Osculating Circle

It is of great interest to note that the:

the Coupler Point, K

the center of the Osculating Circle, R

AND the Pole

...are always along a single ray.

One ray joins the Pole, the center of the Osculating Circle, R,  and the Coupler Point, K.

More Detailed Definition of Osculating Curve

There is intimate contact between Osculating Circle and the Coupler Curve with three infinitesimally separated points!. Imagine a point to each sides of K, called -K, and +K, on the Coupler Curve. Draw Perpendicular Bisectors between the secants -K,K and K,+K. These will intersect at the center of the Osculating Circle in the limit as -K, K and +K become close together. Point -K, K and +K are the three infinitesimally separated Points.

KinSyn11a

Coupler-Point at which the Radius of Curvature is Infinite.

Question: Why might this be interesting?

Answer: Because a Coupler Curve that has an infinite radius is in fact a straight line.

There is, in fact, a circle of points that move in a straight line for each position of the Mechanism.

The circle is called the Inflection Circle.

All of the Points on the Cyan circle move along a straight line.

Remember, this is the Inflection Circle for this four-bar at this instant in its cycle.

How do we calculate the diameter and the center of the inflection circle?

 

KinSyn12

The Euler-Savary Equation

Assume you have a four-bar mechanism, and you want to find its inflection circle at a certain posture.

We can find the Pole. The Pole lies on the Inflection Circle.

The Euler-Savary Equation...

1/P.A - 1/P.A0 = 1/P.AJ

We know distance P.A - it is the distance from the Pole to the End of the Crank, A

We know distance PA0 - it is the distance from the Pole to the center of the Crank

We can calculate the distance P.AJ - the distance from the Pole to a Point on the inflection circle.

All three distances measured from P and a Positive dimension  in the same direction as P to A.

We can do the same for the ray from the Pole through the Rocker - or P.B and P.Bo

KinSyn07

There is a different way to find the Pole Tangent

Draw a Line from the Instant center I14 (which is the Pole in the image above, to confuse!) to I13

This is called the Collineation Axis

Angle β is from the Collineation Axis to the center of rotation of the Crank

Angle β is then used to add the Pole-Tangent - it is the same angle measured from the ray that is I14 to center of the Rocker

You can then draw in the Pole Normal - note by chance it is near to the Collineation Axis.

 

MORE INFORMATION TO FOLLOW WITH TIME.