Nothing is simpler than a pendulum: a bob suspended from a rope,

but the back-and-forth oscillations of the pendulum bob can be very helpful.

As it swings, the pendulum follows a reciprocating sine wave movement, which can be calculated approximately with the following formula: , where L is the length between the axis and the center of gravity of the bob.

Thus, if L = 1 m (3.2 ft.), the period will be 2 seconds.

If the oscillation amplitude increases, the period then slightly varies as follows:, where A is expressed in radians.

Thus, if L = 1m and the oscillation amplitude makes a 45° angle compared to the vertical position, the period is 2.08 seconds and 2.14 seconds for an amplitude of 60°, which is a slight difference.

The benefit of the pendulum is that it acts as a weight amplifier.

In the resting position, the tension on the rope supporting the bob equals the force exerted by the pendulum bob. For a mass M of 1 ton, the tension is T = 9.82 kN.

But the mass swinging movement creates additional tension on the rope, has a sine wave shape, and is a function of the angle at a given point in time and of the angle of the maximum amplitude A.

Thus, with an oscillation amplitude of 90°, the tension force in the rope is multiplied by 3 when the pendulum bob swings low.

On the other hand, when it swings high, the tension force is null.

This tension force exerts on the rope and triggers a reaction in the axis supporting the rope that equals the tension force.

This reaction can be decomposed into two projections: a horizontal projection and a vertical one.

Applied to the pyramid, the horizontal component is the most useful since it can be used as thrust.

This component also exhibits a sinusoidal pattern illustrated below for a motion amplitude of 45°, and another one of 60°.

It can be noticed that for an amplitude of 60°, the maximum horizontal projection is close to 85 % of the force produced by the bob’s mass and is 60 %for an amplitude of 45°.

This graph only shows one period of the motion; for the following one, the same pattern is reproduced but in reverse.

This means that the pendulum’s horizontal force component is zero on average.

If we want to take advantage of the force produced during a period to create motion, the motion of the pendulum for the next period must be blocked naturally; if not, the pendulum has a reciprocating motion, which may be interesting for some uses, like moving a saw, for instance.

If we aim to use a pendulum to shift a stone over a path, the stone must be blocked to avoid moving back; the stone is jerk-moved: it moves forward during one period, stops during the other period, and then moves forward again, and so on.

It is crucial to understand that the horizontal force exerted by the pendulum axis begins at the core of matter.

IT NEEDS NO ANCHOR POINT TO EXERT ITSELF.

For example, if a load had to be pushed down on an icy slope, there would be no need for studded tires or caterpillar tracks to exert forward motion, but the backward movement would have to be blocked.

Energy management:

As long as the pendulum axis is fixed, the pendulum motion consumes little energy—only the friction of the air and the friction in the rope. The force exerted moves along the axis; it works by consuming the energy stored in the mass of the pendulum bob. If let alone, the amplitude of the oscillations will quickly decrease, and motion will stop.

So, the energy produced by the pendulum bob has to be given back to it by giving potential energy to the system. For instance, workers could climb on the mass when in a high position and leave when in a low position. At these two points, vertical speed equals zero, so all the potential energy of the workers is transferred to the pendulum.

But another solution can be more efficient: workers stay on the pendulum bob and ask them to do an up-and-down movement in rhythm with the bob motion with their legs. Anyone who has ever been on a swing understands that!

As a matter of fact, this is the generic, basic solution to put to use that I suggest in my study.

Now the following question remains: how can a single worker doing swings produce a maximum power of 200 W worthy of a trained professional athlete?

The answer is simple, but it depends on the worker’s morphology.

For example, let’s take an 80 kg worker; both their femur and tibia are 0.4 m (1.31 ft) long. Between the sitting position on their heels and the standing position, their center of gravity shifted. To simplify, their body can be divided into three different masses: the feet and tibiae, representing, say, 10 %; the thighs, 30 %; and what is left of the body, 60 %.

Between the sitting position and the standing one, the center of gravity of the tibiae did not move, while that of the things did shift by 40 cm (15.74 in.) and that of the rest of the body by 80 cm (31.49 in.). Everything happens as if 75 % of their weight shifted by 0.8 m (2.62 ft.), so they produce a workforce of 470 joules at each elevation. A period of 470/200 = 2.35s is needed to produce a 200-w power, so the length of the rope had to measure 1.4 m (4.59 ft.) long, which is a rather short length for a big, strong man. And if the worker were ballasted with 100kg (220.46 lbs.), they would produce 590 kJ at each elevation, the period would be 2.95 s, and the length of the rope would be 2.17 m (7.11 ft.).

This example teaches us two things: it is easy to adapt the length of the rope and the weight of the ballast in accordance with the morphology to benefit the required force from it. Several workers could stand on the same pendulum, but they had to have compatible morphologies.

The swinging platforms would be 2 m (6.56 ft.) high and have a 4 m (13.12 ft.) long base to prevent them from tipping over under the torque of the horizontal component, and the width depends on the number of workers placed on the pendulum. These frames could range between 1.5 m (4.92 ft.) wide for two workers, 2.3 m (7.54 ft.) wide for four, 3 m (9.84 ft) wide for six, and 4 m (13.12 ft.) wide for eight.

We can ask ourselves: Why use a pendulum?

The answer is simple and obvious: the pendulum exerts a force proportional to its mass. To produce great force, a heavy pendulum bob, a strong rope to support it, and a sturdy frame are needed.

So considerable force, both horizontal and vertical, could be produced and concentrated.

It can be used, for example, in a device whose “course pusher” function requires little power and little movement, but that could produce high and calibrated force.

Let’s take, for the sake of the example, a 6-t cladding stone ready to be embedded in mortar; it has a force of resistance between the order of 20 and 30 kN to shift on its base. A pendulum made of copper, between 3 and 4 t, operated by a single worker could easily do it, but more importantly, with ACCURACY.