Slider-crank Mechanism

A common mechanism that converts rotary motion into reciprocating motion or vice-versa is sometimes referred as a "Slider-crank mechanism", which is shown at the left. It has wide range of applications like pumps, engines, and compressors.

Generally, the mechanism consists of two rigid links and a piston that are connected by frictionless joints and constrained to move in a single plane. The link AC is called a "crank-shaft" and the link CB is called a "connecting rod". The slider-crank mechanism has only one degree of freedom since the location of both links can be specified by the single independent coordinate θ

In this case, the crank-shaft, AC, weighs 20 N and connecting rod, CB, weighs 35 N. The weight of the links pushes the piston towards right. To keep this system in equilibrium, a force, F, of 70 N pushes in the opposite direction. Determine angle θ for the equilibrium of the slider-crank mechanism.

Begin with a free-body diagram of the slider-crank mechanism as shown at the left. If the origin is established at the fixed support A, the location of F and center of gravity for each link can be specified by the position coordinates x_B, y_W1, and y_W2, respectively.

Let point B undergoes a virtual displacement δx_B in the negative x direction. The deflection is assumed to be small even though it is shown large in the diagram.

Because point A is fixed, the reaction forces at A perform no work and B_y does not move in the direction of the force so no work is performed by B_y.

Expressing the position coordinates of point B and center of gravity of links in terms of the independent coordinate θ and taking the derivatives to find virtual displacements yields,

     x_B = 0.5 cosθ + 0.5 cosθ = 1 cosθ
     δx_B = -sinθ δθ

     y_W1 = 0.25 sinθ
     δy_W1 = 0.25 cosθ δθ

     y_W2 = 0.25 sinθ
     δy_W2 = 0.25 cosθ δθ

As shown in the free-body diagram, an increase in θ (i.e. δθ) causes a decrease in x_B and an increase in y_W1 and y_W2

The total virtual work performed during the virtual displacement is the sum of the virtual work performed by the force F, and the virtual work performed by the weight of the connecting rod and crank-shaft,

     δU = 0
     -20 δy_W1 - 35 δy_W2 - F δx_B = 0

Putting results obtained earlier in the above equation gives,

     20 (0.25) cosθ δθ + 35 (0.25) cosθ δθ
          + 70 (-sinθ δθ = 0

     (13.75 cosθ - 70 sinθ) δθ = 0

Noting that δθ can not be zero,

     θ = tan^-1(13.75 / 70) = 11.11^o

Equilibrium of crank-shaft mechanism is obtained when crank-shaft is rotated by 11.11^o in the anti-clockwise direction.