Crank and slotted lever quick return mechanism - Displacement, Velocity and acceleration analysis

 Crank and slotted lever quick return mechanism - Displacement, Velocity and acceleration analysis

In the below picture shown is a crank and slotted lever quick return mechanism. Where crank length is L1, fixed link length is L2, oscillating link length is L4, coupler link length is L5 and offset of the slider is L6.

In the below picture shown the procedure to find angles theta 2, theta 3 and theta 5 is described. Clock wise angles considered negative and counter clock wise angles considered positive. Angular velocity of the crank omega 1 taken counter clockwise positive and angular acceleration of the crank alpha 1 counter clockwise taken as positive.

In the below pictures shown procedure to find displacement of the slider, Velocity of the slider and acceleration of the slider described. To find velocity of the slider V, differentiate displacement equation S once with respect to time and to find acceleration of the slider A, differentiate displacement equation S twice with respect to time.

In the below pictures described the procedure to find angular velocity of the coupler link Omega 5 and angular acceleration of the coupler link Alpha 5

In the below pictures described the procedure to find angular velocity of oscillating link Omega 3, angular acceleration of the oscillating link Alpha 3, Velocity of the sliding block along the oscillating link L3 dot., and acceleration of the sliding block along the oscillating link L3 double dot..

What is the best working range of compression spring?

 What is the best working range of compression spring?

Why we should use compression springs only between 30% to 70 % of its deflection range?

Lf is the free height of the spring and Ls is the solid height. L1 is the first working height/ assembled height and L2 is second working height.

 % Deflection X Spring stiffness

In the above table shown the actual test data of a compression spring.

Percentage deflection versus spring stiffness is drawn from the above test data. From this graph it is evident that stiffness of the spring is fairly linear between 20% and 80% of the spring deflection.

Spring stiffness is not linear below 20% of deflection, because all the coils of the spring are not deflected in this range (only top and bottom coils deflect).

Spring stiffness is not either linear beyond 80% of the deflection, this is because, some coils start touching each other in this region. That means spring getting compressed near to its solid height.

Near the end of the deflection, that is closer to its solid height, spring stiffness increases very rapidly and will be equal to spring material stiffness at spring solid height.

If spring linearity is not the criteria, then the compression spring can be used between 5% to 95% of deflection.

If spring linearity is more important, then it is advised to use the spring between 20% and 80% of deflection. 

For more precise operation and long life of the spring, it is advised to use between 30% and 70% of the deflection.

Offset Slider crank mechanism 2 - Torque Vs Force Calculator

 Offset Slider crank mechanism 2 - Torque Vs Force Calculator

In the below picture shown is an Offset slider crank mechanism with length of the crank L2, length of the coupler link is L3, offset 'e' and angle between crank and horizontal axis is theta 2. The equation given below shows the relationship between Applied load P and torque T on the crank in terms of crank length L2, coupler length L3, offset 'e' and crank angle theta 2.

To find torque on the crank for a given crank angle theta 2 and applied load P, enter crank length L2, coupler length l3, crank angle theta 2, offset 'e' and applied load p in the below calculator spread sheet.

Offset Slider crank mechanism 1 - Torque Vs Force Calculator

 Offset Slider crank mechanism - Torque Vs Force Calculator

In the below picture shown is an Offset slider crank mechanism with length of the crank L2, length of the coupler link is L3, offset 'e' and angle between crank and horizontal axis is theta 2. The equation given below shows the relationship between Applied load P and torque T on the crank in terms of crank length L2, coupler length L3, offset 'e' and crank angle theta 2.

To find torque on the crank for a given crank angle theta 2 and applied load P, enter crank length L2, coupler length l3, crank angle theta 2, offset 'e' and applied load p in the below calculator spread sheet.

Inline Slider crank mechanism - Torque Vs Force Calculator

 Inline Slider crank mechanism - Torque Vs Force Calculator

In the below picture shown is an Inline slider crank mechanism with length of the crank L2, length of the coupler link is L3 and angle between crank and horizontal axis is theta 2. The equation given below shows the relationship between Applied load P and torque T on the crank in terms of crank length L2, coupler length L3 and crank angle theta 2.

To find torque on the crank for a given crank angle theta 2 and applied load P, enter crank length L2, coupler length l3, crank angle theta 2 and applied load p in the below calculator spread sheet.

Offset Slider crank as Quick return mechanism - Calculator 2

 Offset Slider crank as Quick return mechanism - Calculator 

In the below picture shown is an offset slider crank mechanism. Equations to find forward stroke angle theta, reverse stroke angle beta, theta 1, theta 2 and ratio of forward stroke angle to reverse stroke angle 'Q' are shown. 

In the below calculator, enter link lengths L2, L3 and offset 'e' to find theta 1, theta 2 and time ratio 'Q'.

Offset Slider crank as Quick return mechanism - Calculator 1

 Offset Slider crank as Quick return mechanism - Calculator

In the below picture shown is an offset slider crank mechanism. Equations to find forward stroke angle theta, reverse stroke angle beta, theta 1, theta 2 and ratio of forward stroke angle to reverse stroke angle 'Q' are shown. 

In the below calculator, enter link lengths L2, L3 and offset 'e' to find theta 1, theta 2 and time ratio 'Q'.

Offset Slider Crank mechanism | Transmission angle calculator 2

 Offset Slider Crank  mechanism | Transmission angle calculator 2

In the below picture shown are equations for an offset slider crank mechanism (Slider is below crank axis) transmission angle, minimum transmission angle and maximum transmission angle. To calculate minimum and maximum transmission angles, just enter crank length L2, coupler length L3 and offset e in the below calculator.

Difference between Lower pair and higher pair

 Difference between Lower pair and higher pair

Lower pair: 

• Kinematic pair in which there is an area/surface contact between the contacting elements is called Lower pair.
• All Sliding pairs, revolute pairs, screw pairs, cylindrical pairs, globular pairs and flat pairs are lower pairs.
• A lower pair can be inverted.

Higher pair: 

• Kinematic pair in which there is a point or line contact between the contacting elements is called higher pair.
• All Meshing of gears, cam and follower, ball and roller bearings, wheel rolling on a surface and pawl and ratchet are higher pairs. 
• A higher pair can not be inverted.

Offset Slider Crank mechanism | Transmission angle calculator 1

 Offset Slider Crank  mechanism | Transmission angle calculator 1

In the below picture shown are equations for an offset slider crank mechanism (Slider is above the crank axis) transmission angle, minimum transmission angle and maximum transmission angle. To calculate minimum and maximum transmission angles, just enter crank length L2, coupler length L3 and offset e in the below calculator.

Inline Slider Crank mechanism | Transmission angle calculator

 Inline Slider Crank  mechanism | Transmission angle calculator

In the below picture shown are equations for an Inline slider crank mechanism transmission angle, minimum transmission angle and maximum transmission angle. To calculate minimum and maximum transmission angles, just enter crank length L2 and coupler length L3  in the below calculator. For offset enter zero, since slider is inline with the crank axis.

Drag link quick return mechanism calculator

 Drag link quick return mechanism calculator

In the below picture shown is a Drag link quick return mechanism, In which forward stroke time of the slider is greater than return stroke time. The equations for theta 1, theta 2, forward stroke angle alpha, return stroke angle beta and time ratio are given. In the below calculator tool, enter link lengths L1, L2, L3 and L4 to find theta 1, theta 2, forward stroke angle alpha, return stroke angle beta and time ratio Q.

Degrees of freedom calculator for linkages

 Degrees of freedom calculator for linkages

Crank and Slotted lever - Quick return mechanism calculator

 Crank and Slotted lever - Quick return mechanism

Time ratio and Stroke calculator:

In the below pictures shown a Crank and Slotted lever quick return mechanism. In which forward stroke angle is greater than return stroke angle.Forward stroke angle is theta 1, reverse stroke angle theta 2, equations for forward stroke angle theta 1, reverse stroke angle theta 2, time ratio Q and length of stroke are given below. In the below calculator enter link lengths l1, l2 and L4 to find theta 1, theta 2, time ratio Q and length of stroke S. 

Whitworth Quick return mechanism - Updated

 

 Whitworth Quick return mechanism

Time ratio Q = Theta / Beta

Calculator

Whitworth Quick return mechanism - Calculator

 Whitworth Quick return mechanism

Time ratio Q = Theta / Beta

Calculator

Quick return mechanisms – A short note

 

Quick return mechanisms – a short note

 The quick return mechanisms are used on machine tools to give a slower cutting/forward stroke and a quick return stroke for a given constant angular velocity of the driving crank.

While designing quick return mechanism, the ratio of the crank angle for the cutting stroke to that of the return stroke is very important parameter and it is called as time ratio. To produce a quick return of the cutting tool, this ratio must be greater than unity and as large as possible. 

 

Different types of quick return mechanisms discussed below.

 1. Offset slider crank mechanism:

 The slider crank mechanism can be designed with an offset y as shown in below fig., so that the path of the slider does not intersect with the crank axis. Which will give a quick return motion. However, the amount of quick return/ time ratio is very small and this mechanism will only be used where space was limited and the mechanism had to be simple.

2. Drag link mechanism:

 In the mechanism shown below links 1,2,3 and 4 comprise a drag link mechanism, in which the shortest link is fixed. If the driving crank rotates at constant speed, the other crank (driven crank) will rotate in the same direction but at a varying speed which will allow the slider (link 6) to make a slow stroke to the left and a quick stroke to the right. The time ratio is equal to (theta 1/theta 2)

3. Crank shaper mechanism:

 The below picture is a crank shaper mechanism, in which link 2 rotates completely whereas link 4 oscillates. If the driver link (link 2) rotates counter clock wise at constant velocity, whereas slider 6 will have a slow stroke to the left and a fast return stroke to the right. The time ratio is equal to (theta 1/theta 2)

4. Whitworth mechanism: 

This mechanism is obtained by making the distance O2O4 less than the crank length O2B of the crank shaper mechanism shown in above fig. Both links 2 and 4 rotate completely. If the driver crank 2 rotates counter clockwise with constant angular velocity, slider 6 will move from D' to D" with a slow motion while 2 rotates through angle theta 1, then as 2 rotates through the smaller angle theta 2, slider 6 will have a quick return motion from D" to D'. The time ratio is theta 1/theta 2.

Drag link/ Double crank mechanism

 

Drag link/ Double crank mechanism:

In the below picture shown a four bar linkage in which the shortest link is fixed. Such a linkage is called as a drag link mechanism. Both links 2 and 4 make complete rotations about centers O₂ and O₄ respectively. If one crank rotates at constant speed, the other crank will rotate in the same direction at a varying speed. The following conditions must exist for this mechanism to work.

                                                             BC > O₂O₄ + O₄C – O₂B

                                                            BC < O₄C – O₂O₄ + O₂B

 These relations can be derived from triangles O2B'C' and O2B"C".

Crank and rocker mechanism

 

Crank and rocker mechanism:

In the below picture shown a four bar linkage in which link adjacent to shortest link is fixed. Crank 2 (shortest link) rotates completely about pivot O2 and by means of coupler 3 causes crank 4 to oscillate about O4. Hence the mechanism transforms motion of rotation into oscillating motion. The following conditions must exist for this mechanism to work.

                                                  O₂B + BC + O₄C > O₂O₄

                                                            O₂B + O₂O₄ + O₄C > BC

                                                  O₂B + BC – O₄C < O₂O₄

                                                  BC – O₂B + O₄C > O₂O₄

 Either link 2 or 4 can be the driving crank. If link 2 drives, the mechanism will always operate. If link 4 is the driver, a flywheel or some other aid will be required to carry the mechanism beyond the dead points B' and B". The dead points exist where the line of action BC of the driving force is in line with O₂B.

Inconsistencies of Gruebler's equation

 

Inconsistencies of Gruebler's equation:

In some cases, Gruebler's equation appears to give incorrect results, for example

1. The mechanism has a lower pair which could be replaced by a higher pair, without influencing output motion.

Figure 1 depicts a mechanism with three links and three sliding pairs. According to Gruebler's theory, this combination of links has a degree of freedom of zero.

In this case

N =3 , L = 3 and H = 0

so d.o.f = 3(3-1)-2 x 3 = 0

But by inspection, it is clear that the links have a constrained motion, because as the link 2 is pushed to the left, link 3 is lifted due to wedge action. A little consideration shows that the sliding pair between links 2 and 3 can be replaced by a slip rolling pair as shown in fig. 2, ensuring constrained motion.

In this case N = 3, L = 2 and H = 1

                    so d.o.f. = 3(3-1) – 2 x 2 – 1 = 1

2. The mechanism has a kinematically redundant pair,

 The cam and follower mechanism shown in fig. Has 4 links, 3 turning pairs and a rolling pair, giving d.o.f. as 2.

                                            d.o.f = 3(4-1)-2(3)-1 = 2

 However, a close inspection reveals that as a function generator, oscillatory motion of the follower is a unique function of cam rotation. In other words, degrees of freedom of the cam and follower mechanism is only 1. In this mechanism, the function of roller is to minimize friction, it does not in any way influence the motion of the follower. If the roller is made an integral part of the follower, the motion of the follower will not be affected. Thus the kinematic pair between links 2 and 3 is redundant, if this pair is eliminated, then N = 3, L = 2 and H = 1, which gives degrees of freedom is equal to 1.

                                                d.o,f = 3(3-1)-2(2)-1 = 1

3. There is a link with redundant degree of freedom.

If a link can be moved without producing any movement in the remaining links of mechanism, the link is said to have redundant degree of freedom. Link 3 in the mechanism of the below fig., can slide and rotate without causing any movement in links 2 and 4. Since the Gruebler's equation gives d.o.f. as 1, the loss due to redundant d.o.f. of link 3 implies effective d.o.f. as zero. Fig. 1 represents a locked system. However, if link 3 is bent, as shown in fig. 2, the link 3 ceases to have redundant d.o.f. and constrained motion results for the mechanism. 

Below fig. Shows a mechanism in which one of the two parallel links AB and PQ is a redundant link, as none of them produces additional constraint. By removing any of the two links, motion remains the same. It is logical therefore to consider only one of the two links in calculating degrees of freedom.

4. No consideration was given to the lengths of the links or other dimensional properties.

Fig. 1 represents a structure and that the criterion properly predicts d.o.f. = 0. However, if link 5 is arranged as in Fig. 2, the result is a double-parallelogram linkage with d.o.f. = 1, even though the below equation indicates that it is a structure. The actual d.o.f. = 1 results only if the parallelogram geometry is achieved.

                                                    d.o.f. = 3(5-1)-2(6)-0 = 0

In the development of the Kutzbach criterion, no consideration was given to the lengths of the links or other dimensional properties. Therefore, it should not be surprising that exceptions to the criterion are found for particular cases with equal link lengths, parallel links, or other special geometric features.

 Although there are exceptions, the Kutzbach criterion remains useful, because it is so easily applied during mechanism design. To avoid exceptions, it would be necessary to include all the dimensional properties of the mechanism. The resulting criterion would be very complex and would be useless at the early stages of design when dimensions may not be known.

Gruebler’s Paradox

 Gruebler’s Paradox

The below figure shows a five bar linkage arranged in parallelogram form. There are five links and six pin joints so that,

Gruebler’s equation predicts that the mechanism has zero DOF and cannot move, but our intuition would seem to indicate that it can move. This phenomenon is known as Gruebler’s Paradox and reinforces the notion that we should always accompany our DOF calculations with skepticism and intuition.

Index of merit for a four-bar linkage

 

Index of merit for a four-bar linkage.

 The ability to transmit torque or force effectively and efficiently is one of the major criteria to be considered for any mechanism. Four bar linkages/mechanisms are widely being used in practice, so there needs to be a criteria to judge the quality of such a mechanism for its intended application.

Some mechanisms, such as a gear train, transmits a constant torque ratio from input to output shaft. Apparently, this is possible because there is a constant speed ratio between input and output shaft. But in case of a four bar mechanism this is not possible because torque ratio is a function of geometric parameters which generally change during the course of the mechanism's notion. Generally Mechanical advantage and transmission angle are the two common parameters used as index of merit in designing a four bar linkage.

Mechanical advantage:

The mechanical advantage of a mechanism is defined as the ratio of the force or torque exerted by the driven/output link to the necessary force or torque required at the driver/Input link.

 Therefore, MA = Output torque/Input torque = output force/Input force

If friction and inertia neglected,

                              Input power = Output power

Note that this is directly proportional to the sine of the angle γ between the coupler and the follower, and is inversely proportional to the sine of angle β between the coupler and the driver. Of course, both these angles, and therefore the mechanical advantage, are continuously changing as the linkage moves.

 When the sine of angle β becomes zero, the mechanical advantage becomes infinite; thus, at such a posture, only a small input torque is necessary to produce a very large output torque load. This is the case when the driver AB is directly in line with the coupler BC as shown in Fig. it occurs when the crank is in posture AB1 and again when the crank is in posture AB4. Note that these also define the extreme postures of travel of the rocker DC1 and DC4. When the four-bar linkage is in either of these postures, the mechanical advantage is infinite—that is, β =0◦ or β =180◦—and the linkage is said to be in a toggle (or limit) posture.

Transmission angle:

The angle γ between the coupler and the follower is called the transmission angle.

The above equation indicates that the mechanical advantage diminishes when the transmission angle is much less than a right angle. If the transmission angle becomes too small, the mechanical advantage becomes small, and even a small amount of friction may cause a mechanism to lock or jam. To avoid this, a common rule of thumb is that a four-bar linkage should not be used in a region where the transmission angle is less than, say, 45◦ or 50◦. The better four-bar linkage, based on the quality of its force transmission, has a transmission angle that deviates from 90◦ by the smaller amount.

Because of the ease with which it can be visually inspected, the transmission angle has become a commonly accepted measure of the quality of a design of the four-bar linkage.

 A double-rocker four-bar linkage has a dead-center posture when links 3 and 4 lie along a straight line. In a dead-center posture, the transmission angle is γ = 0◦ or γ = 180◦, and the linkage is locked. The mechanism must be designed in such a way that, either the dead center posture be avoided or provide an external force, such as a spring, to unlock the linkage.