Linkages and Mechanisms(1)

Position Analysis of the four-bar mechanism

The simplest and the basic kinematic chain is a four bar chain or quadric cycle chain as shown in fig. It consists of four links which each of them forms a turning pair at A, B, C and D. The four links may be of different lengths.

In a four bar chain, one of the links, in particular, the shortest link, will make the complete revolution relative to other three links. If it satisfies the Grashof’s law, such a link is known as a crank or driver. In fig, AD i.e. link 4 is a crank. The link BC link 2 which makes a partial rotation or oscillates is known to lever, rocker or follower and the link CD (link 3) which connects the crank and lever is called connecting rod or coupler. The fixed link AB link 1 is known as frame of the mechanism.

fig: four bar chain

Grashoff’s law

For a four bar mechanism the sum of shortest and longest link lengths should not be greater than the sum of remaining two link of length if there is to be continuous relative motion between the two links.

S + L≤ P+Q

where,

L = length of longest link.

P = length of particular remaining link,

Q =length of next remaining link.

if L+S>P+Qthen the linkage is aNon-GrashofType

if L+S<P+Qthen the linkages a Grashof type

Linkage position analysis

Loop Closure Equation

For simple mechanisms the solution of the loop closure equations reduces to the solution of a triangular relation also. The triangular relations can be solved analytically. In this approach rather than obtaining an explicit function between input variable and other position variables. Our aim will be to obtain a set of equations which when solved in steps will yield value of all unknown variables. Such a solution is a closed form solution and furthermore it is most suitable for a numerical solution using a computer, programmable calculator or even on a simple calculator.

As a first example consider a problem in which we want to determine the position of all the links of an off-set slider-crank mechanism which is shown below for different crank angles θ₁₂. The link lengths denoted by a₁, a₂, a₃ which are known.

The vector loop equation is

A_OA= A_OB +BA

In rectangular form, these vectors can be written as

A_OA = a₂ (cosθ₁₂ i + sinθ₁₂ j)

A_OB = xi + a1j

BA= a₃ (cosθ₁₃ i + sinθ₁₃ j)

Figure: Loop Closure Equation

Equating x and y components separately, the loop closure equation will yield two scalar equations:

a₂cosθ₁₂ = s₁₄ + a₃cos θ₁₃ (1)

a₂sin θ₁₂ = a₁ + a₃sinθ₁₃ (2)

Rewriting these equations:

Sinθ₁₃ = 1/a₃ ( a₂sinθ₁₂ – a₁ ) (3)

S₁₄ = a₂cosθ₁₂ – a₃cosθ₁₃ (4)

For a given value of the input variable, θ₁₂, one can solve for θ₁₃ from (3) and substitute the values of θ₁₃ and θ₁₂ into equation (4) to obtain the corresponding value of s₁₄. The variable s₁₄ can be solved only after θ₁₃ is solved from (c). If we are to determine the co-ordinates of a point C (x_C, y_C) we can write

X_C = s₁₄ + b₃cos (θ₁₃ -g₃) (5)

Y_C = a₁ + b₃sin (θ₁₃ -g₃) (6)

Again equations (5) and (6) can solved only after equations (3) and (4) are solved. Since the scalar equations obtained from loop closure equations are non-linear. The method of solution will differ from one mechanism to the other. The method used for the analysis of a four-bar is known as the Raven's Method.

Inversions of Four Bar Chain

Crank and lever mechanism

A part of the mechanism of beam engine known crank and lever mechanism which consists of four links as shown in fig. In this mechanism when the crank rotates about fixed centre A. The lever oscillates about a fixed center D. The end E of the lever CDE is connected to the piston rod that reciprocates due to the rotation of the crank. In other words the purpose of this mechanism is convert the rotary motion into the reciprocating motion.

fig:Crank and lever mechanism

Double crank mechanism

The mechanism of coupling rod of a locomotive known as the double crank mechanism which consists of four links is shown in fig.

Fig:Double crank mechanism

In this mechanism, the links AD and BC having equal length act as cranks and are connected to their own respective wheels. The link of CD acts as a coupling rod and the link AB is fixed in order to maintain a fixed centre to centre distance between them. This mechanism is meant for transmitting rotary motion from one wheel to the other wheel.

Double lever mechanism

A Watt’s indicator mechanism also known as Double lever mechanism that consists of four links, is shown in fig. The four links are: fixed link at A, link AC with link CE and link BFD. It is noted that BF and FD form one link because these two parts have no relative motion among them. The link of portion CE and BFD act as lever. The displacement vector of the link BFD is directly proportional to the pressure of gas that acts on the indicator plunger. On any small displacement of the mechanism, the tracing point of E at end of link CE trace out approximately straight line. The initial position of the mechanism is shown in fig by full lines where the dotted line indicate the position of the mechanism when the gas acts on the indicator plunger.

Fig:Double lever mechanism

Introduction to different mechanism

Single Slider Crank Chain Mechanism

The single slider crank chain is one of modification of basic four bar chain. It consistof one sliding pair and three of turning pairs. It is, mostly found in the reciprocating steam engine mechanism. This type of mechanism converts rotary motion to reciprocat motion and vice versa respectively. In a single slider crank chain which is shown in fig that the links 1 and 2 with links 2 and 3 and the links 3 and 4 form three pairs i.e turning while the links 4 and 1 form a sliding pair.

Fig: Single slider crank chain.

The link 1 is same to the frame of the engine, which is fixed. The link 2 is sameto the crank ; link 3 is sameto the connecting rod and link 4 is sameto cross-head. As the crank rotates the cross-head reciprocates in the guides and so that the piston reciprocate in cylinder.

References:
1. H.H. Mabie and C. F. Reinholtz, “Mechanism and Dynamics of Machinery”, Wiley.
2. J.S. Rao & R.V. Dukkipati Mechanisms and Machine Theory, New Age International (P) Limited..
3. J.E. Shigley and J.J. Uicker, Jr., “ Theory of Machines and Mechanisms”, McGraw Hill.
4. B. Paul, “Kinematics and Dynamics of Planar Machinery”, Prentice Hall.
5. C. E. Wilson, J.P. Sadler and W.J. Michels, “Kinematics and Dynamics of Machinery”, Harper Row.