
As we have seen, simple hinged rod
mechanisms can solve both theoretical and practical problems. The simplest of these mechanisms is
the articulated quadrilateral. If you
consider the ABCD quadrilateral with a fixed side AD, it
is clear that the position of one of the rods determines
that of the remaining ones. This mechanism has only
one degree of freedom. The difficulty in predicting the type of movement depends also on the fact that, by varying the rods' length ratio, the system may function in a very different way. This is the fascinating, interesting and useful nature of the articulated quadrilateral. 

If the opposed rods are of equal
length, the system becomes an articulated
parallelogram, and the opposed rods always
remain parallel. This is the basis of numerous
applications of the mechanism which are under our eyes
every day. Drawing boards, scales, Venetian blinds, the
windscreen wipers of a bus (and of a cars, but in a less
obvious way)  all use this simple mechanism, which is
also found in some table lamps, in lifting machinery, in
bicycles gearboxes, in sewing baskets... 

The pantograph is
itself practically based on a double articulated
parallelogram (in the case of Figure 1, it is a rhombus).
If PC = CD = DE = EB = AE = AC, then the points P, A, B
are always on a straight line and the distance of B from
the fixed point P is always twice that of A, so that, if
A draws a curve, B draws a similar curve, augmented by a
factor of 2. Go back to the top of the page In the general case, as we have seen, the articulated quadrilateral works differently according to the ratios between the rods' lengths. This is not the time and place for a detailed analysis of the various types. We will simply try to show the great versatility of this mechanism. Often it is used as a force multiplier, in the cases when the movement is caused by a shaft rotating at nearconstant speed, while the force must be applied only on a short distance with small movements, as it happens in oilmills or in some cutting devices. Or, on the contrary, this mechanism may be used to transform alternating motion in circular motion, as in the pedals of old sewing machines, or with even more discreet functions, such as in rubbish bins. 
Also of interest is the use of the articulated quadrilateral in the project for a car's suspension (see figure) such that the wheels may absorb all the shocks due to the road's irregularities, without directly transmitting them to the passengers. In this case, one must ensure that both wheels maintain the same vertical angle. Even more interesting is the use of the articulated quadrilateral in car steering. The problem here is to allow the wheels of a steering vehicle to have axes that go through the same point, no matter the size of the steering radius. If this condition is not met, the vehicle will lose its stability in curves, and the tires will be under pressure from lateral forces and will not be very durable. The solution was found by the German engineer Ackermann in 1818, and applied to horsedrawn carriages. It was perfected by Jantaud and Panhard, on the basis of an articulated trapeze, and it is described in the diagram.  
If one considers a
joint plane with the rod CB, leaving AD fixed and
rotating the shorter rod, the points of this plane draw
very diverse curves, according to the position of the
point. To individuate the point tracking the plane joint
to the BC rod, it is sufficient to add to the mechanism
other two rods BE and CE that form a triangle with BC.
The latter opportunity is exploited e.g. in the film
pulling mechanism in motion picture projectors and
cameras (figure).
When B performs a full revolution around A, the point E
draws the dotted trajectory, causing the sudden
advancement of the film during its nearrectilinear part,
then frees itself and moves back on the other half of the
trajectory.
A crane's arms also takes advantage of this property. In this case the problem lies in ensuring that, varying the distance from the base, the height of the load remains basically unchanged. Obviously, more complex mechanisms can perform more complex functions, but the problem there is not different in principle from the simple cases which we have analysed: what one wants to obtain from a mechanism is to have a given point draw a given trajectory which describes its function, without needing this trajectory to exist physically. As we have seen, Kempe has demonstrated that there exists a connecting rod mechanism to trace any algebraic curve. However, in general such a mechanism is constituted by a very high number of rods. Having determined more and more simple and functional mechanisms to perform the same function, however approximately, has been one of the fundamentals of technological progress. 