5 bar linkage (3R & 2P)
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The five bar linkage consists of two prismatic pairs and three
revolute pairs.Although the degree of freedom of 3R and 2P linkage
is 2 {3(5-1)-2*5=2} but here one degree of freedom is lost because
of the restricted motion of link 2(because the other end of link 2
is restricted to move in a circular path).The 2 prismatic pairs are
connected by a spring so that while closing the umbrella we store
potential energy in the spring which could be used to open the umbrella
automatically.Here we study the closing of the umbrella,what happens in
the openinig is just the reverse the only difference being that since
no external force is applied so there is acceleration of the umbrella parts.
To close the umbrella we apply force on the link 4.Initially the angle made by link 2 with horizontal is approximately Zero(slightly>0). So the force transmitted from link 4 to link 1 through link 2 is less because the component of force in downward direction is less, but as the umbrella is further closed this angle increases and so the componet of the applied force in the in the downward direction increases(Although the spring gets more compressed still the force required is less because the increase in component of force due to variation of angle is more than sufficient to balance the spring force).So finally when the umbrella is closed,the two prismatic pairs move a certain distance(fixed =D) from their other extreme positions(Infact they now come above the base shaft). This fixed distance is less than the total usable length of the shafts put together.Now suppose you fold the umbrella (without allowing relative motion between the two shafts),what you will find is that one of the prismatic pair has come on the base shaft this implies that D>the length of the hollow shaft and so you have to have a part of the base shaft to completely fold the umbrella. Now since the required force to fold the umbrella was decreasing as it was folded we know that the force required to hold umbrella in folded position is less.When you completely close the umbrella you have to apply a force for two purposes 1.To keep the umbrella in folded position. 2.The spring force between the two shafts.
In the folded position the umbrella is stable(force required to hold the umbrella in folded position is less) ,so if we leave umbrella in this position it would open slowly.What is done to open the umbrella quickly is that a relative velocity is imparted to the prismatic pairs w.r.t the hollow shaft in the upward direction,this is done by the spring between the hollow and base shafts.When you open the umbrella(from completely closed position)the spring between the shafts pushes the hollow shaft upward and thus imparts velocity to both the prismatic pair and the hollow shaft,but the movement of the hollow shaft is limited by a catch and so it stops suddenly at a point but the prismatic pairs keep on moving and thus a relative velocity is imparted to prismatic pairs w.r.t the hollw shaft and the umbrella opens.Above mechanism also gives an account of why the umbrella first extends and then opens up.
4 bar linkage
This is a parallelogram linkage consisting of four revolute pairs. The function of this linkage is to achieve the final and initial configuration of the umbrella i.e it acts as a guide way for the cloth from opened condition to the closed one.
The link 1 moves in a circular motion about the fixed point O.
Since the link 3 is parallel to link 1 it's motion is exactly
goverened by the motion of link 1. In the opened position the angle
made by the link 1 with the vertical axis is more than in the closed
one therefore the link 3 makes more angle with the vetical axis in the
opened position and the enhances the spreading of the clothe,while in
closed position it makes less angle and so enhances folding of the
cloth.
Another pecularity of the four bar linkage is that the joint between the link 2 and link 3 is made smoother as visible in the photograph. It is done so that the cloth tied to this linkage does not tear off if some rougness is present.
start
mechanism
springs shafts catches