<< Click to Display Table of Contents >> Navigation: General Design Information > Mechanism Synthesis > About Instantaneous Kinematics 
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 'pickup motion'.
A couplercurve 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 couplercurve that passes through the same point two times, but with a different tangent. The couplercurve has a loop, or it a figureofeight.
•StraightLines:
A couplercurve that is a straight line for a substantial section. The line is approximate or exact.
•Constant Radius:
A couplercurve that is a constant radius for a substantial section
•Symmetrical Coupler Curves:
A couplercurve 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 HernÃ¡ndez, CompMech, Department of Mechanical Engineering, UPVEHU for the permission to use the GIM® software. (www.ehu.es/compmech).)
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 InstantaneousCenter (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 outputpart. 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 InstantaneousCenters. •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. 

FixedPolode 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 FixedPolode. Each Dot on the FixedPolode is the position of the IC at a different rotation angle of the input Interesting: the FixedPolode passes through the PivotCenter of the rocker two times


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 MovingPolode. Like the FixedPolode, we cannot see the MovingPolode  we must either draw it or imagine it. The green trace is the MovingPolode. Each Dot on the MovingPolode is the position of the IC at different rotation angle of the input Interesting: the MovingPolode passes through the end of the rocker two times. 

Fixed and Moving Polode When we show the FixedPolode and MovingPolode 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 FixedPolode. At the same time, •... the same point that is the Pole is marked on the Coupler  the trace of the Pole becomes the MovingPolode


Pole Tangent and Pole Normal. The MovingPolode is always in contact with the FixedPolode 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 FixedPolode 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 FixedPolode otherwise the FixedPolode would not move from Point 1 to Point 2. The Pole also travels along the MovingPolode from Point 1 to Point 2. It also has an instantaneous velocity that is tangential to the MovingPolode 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 MovingPolode would diverge from the Pole marked on the Frame by the FixedPolode. If the velocity vector of the Pole moving along the FixedPolode is the same as the velocity vector of the Pole moving along the MovingPolode. the velocities are identical and tangential with each other. Thus, the MovingPolode rolls along on the FixedPolode without slipping. •At every moving instant, tangent to the MovingPolode and FixedPolode at the Pole is called the Pole Tangent. •The Normal (⊥) to the Pole Tangent at the Pole is called the Pole Normal. 

Coupler Curves A CouplerCurve 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 CouplerPoint is labeled as K Each point on the Coupler will trace out a different CouplerCurve.
In this image, you can see that K is a random place on the Coupler, and the CouplerCurve has no particular interest. Are there any interesting CouplerCurve?. 

Cusps When the CouplerPoint, K, on the Coupler, coincides with any point of the MovingPolode, the CouplerCurve will be at a Cusp when the CouplerPoint, K, contacts with the FixedPolode. Why is this? In this image, the 4bar has a certain posture. We have moved Coupler Point, K, to be on the MovingPolode. Remember that the MovingPolode is the trace of the Pole as marked on the Coupler. The CouplerPoint, K, is on the Coupler. Thus, the MovingPolode and the CouplerPoint move together. As the fourbar cycles, the MovingPolode and the CouplePoint, K, approach the FixedPolode. 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 PoleTangent, The Coupler Curve can only be exactly away from each other, and at the instant they separate, the movement will be normal to the PoleTangent. 



Question: Is there anything interesting about the motion of the CouplerPoint. Answer: Yes! The instantaneous motion of the CouplerPoint, K, is always perpendicular (⊥) to a ray from Pole I14 to the CouplerPoint. 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 CouplerPoint we choose on the Coupler will be rotating around the Pole As the mechanism cycles, the Pole moves. But the CouplerPoint will be moving in a direction that is perpendicular to the ray from the Pole


Osculating Circle The CouplerCurve , is a curve, and thus, by definition, all points along ANY curve has Curvature. The Curvature of the CouplerCurve is continuously changing. Thus, its RadiusofCurvature. is also continuously changing.
We can draw a circle at the Coupler Point, K. with a radius equal to its RadiusofCurvature 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. 

CouplerPoint 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 fourbar at this instant in its cycle. How do we calculate the diameter and the center of the inflection circle? 


The EulerSavary Equation Assume you have a fourbar 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 EulerSavary 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 

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 PoleTangent  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. 
