Mechanical Advantage

There are two main aspects of mechanical advantage that compound each other in the context of this mechanism. The first is the advantage provided by the ratio of radii from the turn-crank to the interior spool which unfolds the umbrella. The other aspect is the mechanical advantage inherent in the multi-bar mechanism which takes the input force from the slider motion of h₂ and applies that to the surrounding links comprising the "roof" of the umbrella (link 3 in our case).

In order to calculate the mechanical advantage of this part of the mechanism, we simply look at the power relationship between the input (crank) and output (h₂ slider) motion. Assuming for the moment that we have 100% efficiency in the mechanism we have the following.

This is the same expression that we would expect to get when dealing with a pair of interlocked gears, which shows that the same concept of mechanical advantage applies in the case of tightened string on a spool, similar to a fishing reel. Plugging in the values that we have for the radii we get a mechanical advantage of 5.17, which means that the input velocity is reduced by a factor of 0.19, but the output force is amplified by a factor of 5.17.

The second aspect of mechanical advantage regards the relationship between the force applied to the slider (F_out in the last calculation) and the force applied by the mechanism to lift the bars of the umbrella. Since the link L₆ in general has a low velocity during the opening motion of the umbrella, it will be the link considered the base link for this mechanical advantage calculation. Once again we can make a power equivalence calculation, this time based on the fulcrum between links 6 and 3.

All of the terms in this equation are either parameters of the mechanism geometry or velocities that we have already solved for in the Velocity Analysis section. As such we now a general term for the mechanical advantage

The following shows the mechanical advantage profile moving the value of h₂ from it's maximum value (where the umbrella is closed) to its minimum value (where the umbrella is fully open).

As we can see from the plot, the mechanism is designed with mechanical advantage in mind. Logically, this makes sense since the umbrella itself is quite large and heavy and the product needs to be operated by any selection of consumer regardless of their physical strength. The mechanical advantage seems to peak at a point and tend towards infinity. Examining the data, this point was found to be h₂ = 0.711 m which corresponds to the following configuration of the mechanism.

Point of Maximum Mechanical Advantage

Mathematically, this is the point in the opening motion in which ω₃ = ω₆. Returning to the plot shown in the Velocity Analysis section, this can be clearly seen occurring at the point h₂ = 0.71

Examining Velocity in the Scope of Mechanical Advantage