Cam with oscillating roller follower - position and displacement equations

 Cam with oscillating roller follower - position and displacement equations

Cam with oscillating roller follower - Position and displacement equations derived using analytical method.

In the below picture shown is a cam with oscillating roller follower. Various dimensions are as shown. 

The above cam follower is converted to an equivalent mechanism, in which crank length is L2, angle between crank and horizontal axis is theta 2. Coupler link length is L3,  and the angle is theta 3, fixed link length is L1. angle between follower link and horizontal is theta 4. Angle between follower link and coupler is gamma. From the below picture it is evident that the equivalent mechanism is a four bar mechanism, fixed link is inclined at an angle theta with the horizontal.

In the below pictures described the procedure to calculate various angles.

In the below picture described the procedure to calculate position of the point C on the mechanism at any given crank angle theta 2.

To find the velocity and acceleration, treat the mechanism same as a four bar mechanism and follow the procedure described in the below links.

Four bar linkage / Mechanism velocity equations

https://www.kinematics-mechanisms.com/2022/05/four-bar-linkage-four-bar-mechanism_27.html

Four bar linkage / mechanism Acceleration equations

https://www.kinematics-mechanisms.com/2022/05/four-bar-linkage-four-bar-mechanism_65.html

Four bar linkage / mechanism problem - toggle postures

 A crank-rocker four-bar linkage is shown in one of its two toggle postures. Find θ2 and θ4 corresponding to each toggle posture. What is the total rocking angle of link 4? What are the transmission angles at the extremes?

In the below picture shown is a crank rocker four bar mechanism, r1 is the fixed link length, r2 crank length, r3 is coupler length and r4 is follower (rocker) link length.

Toggle position occurs when crank and couple links align, i.e. the angle between  crank is either 180 degrees or zero degrees.

First toggle position is shown in the below picture O2A'B'O4. In this position angle between crank and coupler links is 180 degrees.

From the below picture, it is evident that angles theta 2 and theta 3 are equal.

From triangle O2B'O4, Theta2 is calculates as described below and from triangles O2B'O4 and O4B'C',  rocker angle theta 4 is calculated.

In the below picture shown the second toggle position of the mechanism, O2A"B"O4, in this position angle between crank and coupler is zero degrees.

From the above picture, it is evident that, Crank angle theta2 is 180 degrees plus theta3. From triangle O2B"O4, theta 3 is calculated as described below. and from triangles O2B"O4 and O2B"C", rocker angle theta 4 is calculated.

After having derived all the necessary equations, now let us solve a problem using Microsoft excel. In this problem, link lengths r1 is 16 in, r2 is 8 in, r3 is 20 in and r4 is 16 in taken as inputs. Theta 3, theta 2, theta 4 and gamma are calculated for two toggle positions. Values are listed in the below picture. Total rocking angle is 135.59 - 57.91 = 77.68 degrees.

Differences between Lower pair and Higher pair

 Differences between Lower pair and Higher pair

1. Lower pair: Kinematic pair in which there is a surface/area contact between the contacting elements is called a lower pair.

Examples of lower pair: All revolute pairs, sliding pairs, screw pairs, globular pairs, cylindrical pairs and flat pairs

Lower pairs can carry heavy loads since they have surface contact.

Friction is high in lower pairs.

Wear is low in lower pairs, because load is distributed over a large contact area.

Contact stresses are low in lower pairs, because load is distributed over a large contact area.

2. Higher pair: Kinematic pair in which there is a point or line contact between the contacting elements is called a higher pair.

Examples of higher pair: Meshing gear teeth, cam and follower pair, wheel rolling on a surface, ball and roller bearings, pawl and ratchet 

Higher pairs cannot carry heavy loads since they have point or line contact.

Friction is low in higher pairs.

Wear is high in higher pairs, because contact area is very less (point or line contact)

Contact stresses are high in higher pairs, because contact area is very less (point or line contact)

What is a kinematic element?

  What is a kinematic element?

The kind of relative motion between links of a mechanism is controlled by the form of the contacting surfaces of the adjacent links. These contacting surfaces may be thought of as working surfaces of the connection between adjacent links. 

For instance the working surfaces of an internal combustion engine piston and connecting rod at piston pin are so shaped that relative motion of rotation alone is possible. Each of these working surface is called as a kinematic element. 

What is a kinematic link?

What is a kinematic link?

A kinematic link is a member or a combination of members of a mechanism, connecting other members and having motion relative to them. Thus, a link may consists of one or more resistant bodies.

For example, a slider crank mechanism consists of four links, frame and guides, crank, connecting rod and slider. However, the frame may consists of bearings for the crank shaft. The crank link may have a crank shaft and flywheel also, forming one link having no relative motion of these.

Links can be classified into Binary, Ternary and Quaternary depending upon their ends on which revolute or turning pairs can be placed.

 

What is a toggle/limit position of a four bar linkage?

 What is a toggle/limit position of a four bar linkage?

In a four bar mechanism toggle position occurs when the crank and coupler lie along a straight line i.e. the angle between the crank and coupler links is either 0 degrees or 180 degrees. 

From the above mechanical advantage equation, at toggle position, Mechanical advantage will become infinity, that means a slight force is sufficient to release the mechanism from toggle position.

  

What is a dead center posture/position of a four bar linkage?

 What is a dead center posture/position of a four bar linkage?

A double rocker four bar linkage has a “Dead-center posture, when coupler and follower links lie along a straight line, i.e. the angle between coupler link and follower link is either 0 degrees or 180 degrees.

In the dead center posture, the transmission angle is either 0 degrees or 180 degrees (transmission angle is the angle between coupler and follower links).

From the above mechanical advantage equation, at dead center posture, Mechanical advantage is zero. That means, the mechanism is in the locked position. An external force such as spring must be used to unlock the linkage from this locked posture.

What are Class I, Class II and Class III kinematic chains?

 What are Class I, Class II and Class III kinematic chains?

Class I kinematic chain: A four bar linkage which satisfies the condition S + L < P + Q is called as a Class I kinematic chain or Grashof’s linkage.

Where S is the shortest link, L is the longest link, P and Q are the lengths of other two links.

If the shortest link is fixed, the linkage becomes a double crank, if the link adjacent to shortest link is fixed, the linkage becomes a crank rocker and if link opposite to shortest link is fixed, the linkage becomes a double rocker.

Class II kinematic chain: A four bar linkage, which satisfies the condition S + L > P + Q is called as a Class II kinematic chain or non-Grashof  linkage.

None of the links of a non-Grashof linkage will be capable of completing a full revolution, which causes all inversions to be triple rocker mechanisms.

Class III kinematic chain: A four bar linkage, which satisfies the condition S + L = P + Q is called as a Class III kinematic chain or a special case of Grashof linkage.

In this chain also four inversions possible similar to Class I kinematic chain however, in this Class III kinematic chain collinear arrangements, called  change points will occur at two locations during one complete revolution of the crank.

A typical example is shown in the below picture, where S = P and L = Q, so S + L = P + Q. in this linkage, two change points occur at crank angles 0 and 180 degrees respectively. 

One of the change points is explained in the below pictures (crank angle is 0 degrees), Motion of the linkages after change points will have two possible outcomes, as shown in the below pictures. 

First possible position

Second possible position

Because of the uncertainty, change points should be avoided or controlled.

Four bar mechanism - problem 1

 Four bar mechanism - problem 1

Q) In the below figure shown three four bar mechanisms, in which the figures indicate the dimensions in standard units of length. Indicate the type of each mechanism, whether it is a crank rocker, double crank or double rocker.

Let us discuss each mechanism.

a) S = 3, L = 7, P = 6 and Q = 5.      

        Grashof’s criteria,

                    S + L <= P + Q

                     3 + 7 < 6 + 5

                         10 < 11

It is a class I kinematic chain. Link adjacent to shortest link is fixed, So it is a crank rocker mechanism. 

b) S = 4, L = 8, P = 6 and Q = 7. 

        Grashof’s criteria,

                  S + L <= P + Q

                   4 + 8 < 6 + 7

                       12 < 13

It is a class I kinematic chain. Shortest link is fixed, So it is a double crank mechanism. 

c) S = 4, L = 8, P = 6 and Q = 5. 

        Grashof’s criteria,

                S + L <= P + Q

                  4 + 8 > 6 + 5

                       12 > 11

It is a class III kinematic chain. Longest link is fixed, It is a double rocker mechanism. 

Offset Slider crank mechanism - Problem 1

 Offset Slider crank mechanism - Problem 1

Q) Determine the advance-to-return ratio for the slider-crank linkage with the offset e. Also, determine in which direction the crank should rotate to provide quick return.

Ans.) In the below picture shown is an offset slider crank mechanism. Crank length is r2, coupler length is r3 and slider position is above the crank axis by a distance e. 

First extreme position of the slider will occur when crank and coupler fall in line like shown in below figure as O2AB and second extreme position will occur when crank and coupler assume the position O2A'B' as shown. When crank rotates clockwise from O2A' position, slider moves towards right, let us call it forward stroke. Angle subtended by crank during this forward stroke is theta. Similarly when crank rotates from O2A position clockwise, slider moves from right most position to its left most position, let us call it return stroke. The angle subtended by crank during return stroke is beta.

Ratio of forward stroke angle to return stroke angle is called "time ratio" or "quick return ratio". In the below pictures described the procedure to calculate time ratio. Crank should rotate in clockwise direction.

Crossed Four bar mechanism Mechanical Advantage Calculation

 Crossed Four bar mechanism Mechanical Advantage Calculation

In the below picture is shown a typical crossed four bar mechanism (Crank rocker), crank length is L2, coupler link length is L3, Follower link length is L4 and fixed link length is L1. A torque of T2 is applied on the crank and resting torque on follower link is L4. Mechanical advantage is T4/T2.

Before calculating mechanical advantage, we need to first calculate angles Gamma, Phy and Beta. In the below pictures shown the procedure to calculate the angles Gamma, Phy and Beta.

In the below pictures shown the procedure to find mechanical advantage.

After having derived all the necessary equations, now let solve the problem, in which crank length is 100mm, coupler length is 200 mm, follower length is 150 mm and fixed link length is 220 mm. Torque on the crank is 100 Nm clockwise. Here clockwise direction is considered negative. Using Microsoft excel the problem is solved. Mechanical advantage calculated for crank angles 0 to 360 degrees at an interval of 15 degrees.

In the below graph plotted crank angle versus mechanical advantage. crank angle along horizontal and mechanical advantage along vertical axis. It can be observed from the below graph that at about 30 degrees crank angle mechanical advantage is maximum and at about 210 degrees mechanical advantage changes from a certain maximum value to minimum value. These two angles represent toggle positions of the mechanism i.e. crank and coupler fall inline (angle between crank and coupler will be either 0 degrees or 180 degrees).

Four bar mechanism Mechanical Advantage Calculation

 Four bar mechanism Mechanical Advantage Calculation

In the below picture is shown a typical four bar mechanism (Crank rocker), crank length is L2, coupler link length is L3, Follower link length is L4 and fixed link length is L1. A torque of T2 is applied on the crank and resting torque on follower link is L4. Mechanical advantage is T4/T2.

Before calculating mechanical advantage, we need to first calculate angles Gamma, Phy and Beta. In the below pictures shown the procedure to calculate the angles Gamma, Phy and Beta.

In the below pictures shown the procedure to find mechanical advantage.

After having derived all the necessary equations, now let solve the problem, in which crank length is 100mm, coupler length is 200 mm, follower length is 150 mm and fixed link length is 220 mm. Torque on the crank is 100 Nm clockwise. Here clockwise direction is considered negative. Using Microsoft excel the problem is solved. Mechanical advantage calculated for crank angles 0 to 360 degrees at an interval of 15 degrees.

In the below graph plotted crank angle versus mechanical advantage. crank angle along horizontal and mechanical advantage along vertical axis. It can be observed from the below graph that at about 30 degrees crank angle mechanical advantage is maximum and at about 210 degrees mechanical advantage changes from a certain maximum value to minimum value. These two angles represent toggle positions of the mechanism i.e. crank and coupler fall inline (angle between crank and coupler will be either 0 degrees or 180 degrees).

What is a kinematic chain?

 What is a kinematic chain?

A Kinematic chain is an assembly of links in which the relative motions of the links is possible and the motion of each relative to the others is definite.

In the below picture shown a four bar linkage, in which motion a link results in a definite motion of the other links.

In case the motion of a link results in indefinite motions of other links, it is a non-kinematic chain.

In the below picture shown is a six bar linkage in which motion of a link results in an indefinite motions of the other links.

A redundant chain does not allow any motion of a link relative to the other. In the below picture shown is a three bar linkage, in which motion between links is not possible.