Inversions of Four-Bar Mechanism

In this tutorial, you will learn about the types of Four-Bar Mechanisms, the laws which define them, their different variations. Also, we will study the various devices which work on these inversions and their applications.

Contents:

  1. Inversions in Mechanisms
  2. Grashof’s Law
  3. Four-Bar Chain
  4. Single Slider Crank Chain
  5. Double Slider Crank Chain
  6. Whitworth Quick Return Mechanism
  7. Crank and Slotted Lever Mechanism
  8. Uses of different Four-Bar Mechanisms

Inversions in Mechanisms

The process of generating different mechanisms by fixing different links in the same mechanism is known as Inversion of Mechanism.

  • The relative motions in between the links does not change irrespective of the inversions. However, the absolute motion can change to a great extent.
  • For a given number of links in a mechanism, the possible number of inversions is given by the following equation:
    Number of Inversions ≤ L
    where N = number of links in a mechanism.

    A simple inversion consisting of 4 kinematic links is shown below:

Inversion of a Four-Bar Mechanism

The above diagram describes about a four-bar mechanism. In the first figure, link 4 is fixed whereas in the second figure, link 1 is fixed. As we can see in both diagrams, the relative motion between the links in both the mechanism do not change.

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Grashof’s Law

Grashof’s law is a mechanism used to identify the nature of each mechanism. The statement of the law is as follows:

“For a 4-bar mechanism, the sum of the shortest and longest link in a planar quadrilateral linkage must be less than or equal to the sum of the remaining two links, then the shortest link can rotate fully with respect to a neighboring link.”

S + L P + Q

where,
S = Shortest link
L = Longest link
P & Q = Adjacent links to the shortest and longest link.

The following natures of mechanism are observed in Grashof’s law:

  • Double Crank
    This type of mechanism is obtained when the sum of the shortest and the longest links in a mechanism is less than the sum of the other two links according to the equation below:
    S + L < P + Q

    The below diagram describes a Double Crank Mechanism:

    Double Crank Mechanism

    The above figure shows a Double Crank Mechanism. Both the cranks rotate in the same
    direction connected by a common link.

  • Crank Rocker
    S + L < P + Q

    The sum of the shortest and the longest link must be less than the sum of the other two links as indicated by the equation above. Also, the adjacent to the shortest link must be fixed. If these conditions are satisfied, then the mechanism obtained is known as a Crank Rocker Mechanism.
  • Double Rocker
    S + L > P + Q

    The sum of the shortest and the longest link must be greater than the sum of the other two links indicated by the equation above. Also, the shortest link must be a coupler and the shortest to the opposite link should be fixed. If these conditions are satisfied, then the mechanism obtained is known as Double Rocker Mechanism.

    A Crank Rocker and a Double Rocker Mechanism are shown below:

    Crank Rocker and Double Rocker Mechanism

    The above figure shows a Crank Rocker and a Double Rocker Mechanism. While the former has only one rocker connected to the crank via a connecting rod, the latter has two rockers oscillating about the dotted curve as shown in the figure.

  • Parallelogram Linkage
    This type of linkage is obtained when opposite links in a Four-Bar Mechanism are of the same lengths.
    S + L = P + Q

    The above equation is obeyed by a parallelogram linkage. If any of the shortest or longest link is fixed, then the linkage is bound to follow Double Crank Mechanism.
  • Deltoid Linkage
    Deltoid Linkage is obtained when two adjacent links are either shortest or longest in a mechanism.

    A parallelogram and a deltoid linkage are shown below:

    Parallelogram and Deltoid Linkages

    The following diagram shows a parallelogram and a deltoid linkage. In case of a deltoid linkage, if the shortest link is fixed, then the mechanism obtained is Double Crank and if the longest link is fixed, then the mechanism obtained is Crank Rocker.

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Four-Bar Chain

The simplest chains or mechanisms generally consists of four kinematic pairs either having a sliding or a turning pair. They are primarily divided into 3 categories amongst which Four-Bar Chain is one of the simplest kinematic chains.

  • It consists of 4 rigid links connected by four binary joints. As a result, there are 4 turning pair in a 4-bar chain.
  • The fixed link which does not perform any motion is called as the Frame.
  • The link which makes a complete revolution about a fixed axis is called a Crank.
  • The link which oscillates and does not perform a complete revolution is known as Rocker.
  • By fixing different links of the Four-Bar Mechanism, we generate three different mechanisms or inversions as described below:

  • Crank and Rocker Mechanism (Beam Engine)
    This inversion consists of one complete rotational and an oscillatory motion made up of 4 kinematic links. Beam Engine is an example of this mechanism.

    A diagram of Beam Engine is shown below:

    Beam Engine

    The above figure describes a Beam Engine. It consists of a crank which performs rotatory motion and transmits the motion to the beam lever of the engine (the rocker in this mechanism) which is then transferred into the piston (reciprocating motion). The purpose of this mechanism is to transfer rotatory motion into reciprocating type.

  • Double Crank Mechanism (Coupled Rod of a Locomotive)
    This inversion consists of two cranks which complete 3600 rotations. Coupled rod of a locomotive is an example of Double Crank Mechanism as shown below:

    Coupling Rod of a Locomotive

    The above figure shows a coupling rod in a locomotive. It consists of two wheels rotating in the same direction connected with one crank on each wheel and coupled by a rigid coupling rod. The purpose of this mechanism is to transmit rotatory motion from one wheel to another with minimal losses.

  • Double Rocker Mechanism (Watt’s Indicator Mechanism)
    The Double Rocker Mechanism consists of two rockers or levers which perform oscillatory motion. It consists of four links out of which link 4 is fixed, the two links which function as a lever, 1-4 & 1-5, and the fourth link 6-2-3 which is connected to the indicator cylinder.

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    It is to be noted that the link 6-3-2 is a single link and not two individual links as there is no relative motion between link 6-3 and 3-2. The entire mechanism is pivoted about the point 4 which is a hinge joint.

    Watt’s Indicator Mechanism

    The above figure describes a Watt’s Indicator Mechanism. As the plunger travels upwards, the point 5 travels upward almost trace a straight path.

Single Slider Crank Chain

The second category of kinematic chain is the Single Slider Crank Chain. It consists of four kinematic pairs out of which one is a sliding pair and the rest three are turning pairs. If different links of the Single Slider Crank Chain are fixed at different time, four different inversions can be generated.

  • First Inversion
    When the frame is fixed, one link is made a crank and one link is made the slider, then we obtain the first inversion which is the sole mechanism of reciprocating IC Engines and Compressors.
  • Second Inversion
    When the crank of the mechanism is fixed, the second inversion is obtained. This inversion is applied in Rotary Engine and Whiteworth Quick Return Mechanism.
    A figure of Rotary Engine is shown below:

    Rotary Engine

    The above figure describes a rotary engine. As we can conclude from the diagram, there are 8 cylinders connected to a common fixed crank by a connecting rod instead of one single cylinder.

  • Third Inversion
    When the connecting rod of the mechanism is fixed, we obtain the third inversion. This inversion is applied to oscillating cylinder mechanism and crank and slotted-lever mechanism.
    A figure of an oscillating cylinder engine is shown below:

    Oscillating Cylinder Engine

    The above figure describes about an Oscillating Cylinder Engine. Instead of the connecting rod, the cylinder functions as a rocker.

Double Slider Crank Chain

The third category of basic four kinematic chain is known as Double Slider Crank Chain. It comprises of two sliding pairs and two turning pairs. By fixing different links at times, we obtain various inversions of this mechanism.

  • First Inversion
    When the frame of the Double Slider Crank Chain is fixed, we obtain the first inversion. This inversion is applied in Elliptical Trammels, a device used to draw ellipses.
    A diagram of Elliptical Trammels is drawn below:

    Elliptical Trammels

    The figure above shows an Elliptical Trammel. The slider A reciprocates from point 1 to 2 and the slider D reciprocates from 3 to 4.

  • Second Inversion
    When one of the sliders is fixed, we obtain the second inversion. Scotch Yoke Mechanism is based on the second inversion of Double Slider Crank Chain. A figure describing the mechanism is shown below:

    Scotch-Yoke Mechanism

    The above figure describes Scotch-Yoke Mechanism with one of the sliders fixed. The other slider reciprocates on the rotation of the crank. Here, it is to be noted that the piston is not the second slider. The fixed link is the second slider in this case.

  • Third Inversion
    When the link connecting the slider is fixed, we obtain the third inversion. Oldham Coupling uses this inversion.
    A figure of Oldham’s Coupling is given below:

    Oldham’s Coupling

    The above figure describes an Oldham’s Coupling. The two shafts have flanges connected by an intermediate flange/shaft.

Whitworth Quick Return Mechanism

Whitworth Quick Return Mechanism is the second inversion of Single Slider Crank Chain. This mechanism is used in mechanical workshops to cut metals.

A basic diagram of the Whitworth Quick Return Mechanism is given below:

Whitworth Quick Return Mechanism

The above figure describes a Whitworth Quick Return Mechanism. The tool connected to the ram performs two actions- forward stroke in which the metal is removed, and an idle stroke.

  • The angle α is known as the cutting angle whereas the angle β is known as the return angle.
  • While operation the idle/return stroke is of no use. Hence, the angle β is deliberately kept as small as possible to increase productivity.
  • The forward to idle ratio is of the ratio 2:1.
  • Since the time taken for the forward and the return stroke in the mechanism is proportional to the angle α and β respectively. Therefore, we generate a relation as given below:

    \(\frac{Time \,of \,cutting \,stroke }{Time \,of \,return \,}\) = \(\frac{α }{β }\) =\(\frac{α }{ (360^0 – α) }\)

Crank and Slotted Lever Mechanism

Crank and Slotted Lever Mechanism is the third inversion of Single Slider Crank Chain. Here, the connecting rod is fixed and a slider reciprocates in the slotted lever connected to a crank as shown below:

Crank and Slotted Lever Mechanism

The figure above describes a Crank and Slotted Lever Mechanism. There are two strokes performed by this mechanism- forward and return similar to Whitworth quick return mechanism.

  • The crank CO connected to the slotted lever rotates about point O. The rotational motion of the crank is converted to oscillatory motion with the help of the slider.
  • The oscillatory motion of the slider is then converted to reciprocating motion by the connecting rod attached to the Ram.
  • The time taken by the forward stroke and the time taken by the return stroke is denoted by a ratio known as Quick Return Ratio.Quick Return Ratio = Time taken by cutting stroke/ Time taken by return stroke

    = \(\frac{β}{α}\) 

  • Quick Return Ratio is always greater than 1.
  • The distance travelled by the Ram from its mean position to final point is called Stroke Length.
    Stroke Length = (2 × L × r) / l
    where,
    L = Length of the Slotted Lever PX
    R = Radius of the crank CO
    L = Length of Connecting Rod OX

Uses of different Four-Bar Mechanisms

The mechanism analyzed above forms the basis of all the working mechanical equipment and devices in the world. Some of them are listed below:

  • Mechanisms like single slider crank chain is the backbone of all the reciprocating IC Engines available in the market.
  • Rotary engines, an inversion of Single Slider Crank Chain is still used in some old-age aircrafts.
  • Whitworth quick return mechanism and Crank and Slotted lever mechanisms form the basis of cutting tools like shaper used in the manufacturing industry.
  • Oldham’s Coupling is used to connect two non-collinear shafts.
  • Coupling Rod of a Locomotive was the most important part used in Steam Engines.
  • Pantograph, a lower pair mechanism, is used in copying devices which is an inversion of Four-Bar Chain.
  • Tailor’s machine used extensively in the fabric and sewing industry, is also an inversion of Four-Bar Chain.
  • Beam engine, another inversion of Four-Bar Chain is used in extracting water out of mines and in water pumps.

Key Points to Remember

Here is the list of key points we need to remember about “Inversion of Four-Bar Chain”.

  • When performing inversions, the relative motions of the links in a mechanism does not change.
  • If a linkage follows Grashof’s Law, then it is necessary to have at least one crank.
  • Double-Rocker Mechanism does not follow Grashof’s Law.
  • A mechanism in which none of the links perform revolving motion is not a useful one.
  • If the length of anyone of the links is greater than the sum of the lengths of the other three links in a 4-Bar Chain, then a 4-Bar Mechanism is not possible.
  • If the slider in a Single Slider Crank Chain is fixed, then we can also obtain another mechanism and this mechanism is used in Hand Pumps used in rural areas.
  • Watt’s Indicator Mechanism is also known as Straight Line Mechanism.

If you find any mistake above, kindly email to [email protected]

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Manish Bhojasia - Founder & CTO at Sanfoundry
Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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