Derive an expression for the magnitude of the couple M required to maintain the equilibrium of the linkage shown.

Derive an expression for the magnitude of the couple M required to maintain the equilibrium of the linkage shown.

Let us consider the virtual displacement of the linkage as shown in figure:

From the principle of virtual work,

𝛿U = 0

or, -MδΘ + PδY_E + PδYF = 0-------(1)      [ M and δΘ are in opposite direction]

Keeping B as fixed point and measuring distance from it, we have,

Y_E = 3asinΘ                    YF = 4asinΘ

Differentiating we get,

δY_E = 3acosΘδΘ            δYF = 4acosΘδΘ

Then, from eqn 1 we get,

Or, -MδΘ + Px3acosΘδΘ+ Px4acosΘδΘ = 0

Or, -M + Px3acosΘ+ Px4acosΘ = 0

∴ M = 7aPcosΘ