Table of Contents

## What is Pendulum

Pendulum is the most basic mechanical oscillators. It has been the focus of scientists since the beginning of time including Galileo as well as Huygens. The main purpose of pendulum is to determine the duration of time. For tiny swing amplitude ph0, the duration T is dependent upon the size of pendulum that can be expressed in the following manner:

The pendulum is a basic design concept, but can display extremely complex dynamics. It has also been extensively used as a construction block in complicated non-linear systems. In the event that the pendulum exposed to stimulation with an angular frequency that is twice that of its natural frequency of, a parametric resonance of 2:1 is observed. The phenomenon of a parametric resonance in pendulum has prompted scientists to look into the possible use of pendulum to harvest energy. The majority of these ideas are using waves from the sea as sources of excitation. Pendulum systems, positioned on a floating structure, rotates or oscillates transforms electric energy from mechanical. The issue of a rotary solution for the pendulum that is parametrically excited is discussed in, while the optimization of the extraction of energy from such the system is described in. Energy extraction through the rotary motion of pendulums subjected to stochastic waves is discussed in. Wave flume laboratory experiments are carried out in, and where researchers test a 1:45 scale model of an energy harvesting device. Prepare a an analytical model that is simple and then conduct an usability study using oceanographic data of the Italian coastline. If you are using waves from the sea as an source of excitation for the pendulum system you must construct a long pendulums to achieve a the resonance of 2:1. For example , a period of 3 [s] sea waves roughly nine meters of pendulum (ordinary gravity waves have periods ranging between 1 [s] and thirty [s]). The challenge of obtaining a low natural frequencies of pendulums is dealt with in the article, where the authors propose a method consisting of n equally spaced pendula which form a ring, with an common pivot point that eliminates the direct dependency of pendulum’s length on its natural duration. The idea has been refined in. The alternative approach used in a variety of works is the vertical axis pendulum. This is where centrally pivoted pendulum floating on a structure is rotated around vertical the axis. The first patent for a device that utilized this concept was granted in the year 1966. Today, patents for contemporary designs are held to Neptune Wave Power LLC and Wello Oy. Papers explore the modeling and optimization of the vertical pendulum wave generators with axis. Harvesting energy from the angle perspective of absorption of vibration is described in.

We have proposed a simple method to capture energy from oscillating environments, based on pendulum oscillations that occur in 1:1 parametric resonance. The mechanical energy generated by pendulum is converted by a power generators to electrical power.

## Electric pendulum

The principles of motion for every mechanical device can be derived by the universal law of least action , also known as Hamilton’s principle. This is a statement that states that every mechanical system can be described as an equation for any continuous trajectory that is possible. This is referred to as the Lagrangian and leads to the Euler-Lagrange motion equations.

We study a simple pendulum limited to one degree of freedom that has variable!. In this case, l0 is an unchanging length that is measured from the lower end of the pendulum up to another vertically fixed piece. Two balls have mass m, m0 and charges q and q0. The system we developed at this research and also for the mass m that is considered to be the dynamic body we identify the Lagrangian function as follows:

in which the first element mx^2/ 2 is the kinetic energy , and U(x) represents the energy potential for the entire system. Although movement of the system is carried out in a bi-dimensional space, in reality, the motion is one-dimensional in accordance with its degrees of freedom.

The term U(x) is expressed as:

where

means the gravitational potential energy with g=9.8m/s^2 the acceleration due to Earth’s gravity, we have taken the y axis vertical downwards. The other term:

refers to that of the electrical potential energy refers to the electric permittivity in the vacuum. s represents an indication of the positive distance between charge. It is evident that i={0,1} is a repulsive or attractive interaction, respectively. Additionally, it is important to note that the nature of electric interaction is dependent on the shape of the charges qq0. However, we have taken them to be positive in magnitude and left the repulsive or attractive interactions solely upon the type of charge.

We can set s in terms of ɸ by means of cosine’s law:

We replace above expressions to obtain the total potential for the electric pendulum:

notice that the potential function holds U(ɸ)=U(-ɸ), then the system is governed by an even symmetry as it is known for the simple pendulum.

For i=1, we see an appealing electric potential that is very similar to a simple pendulum that has a stable points at ɸ=0. It is important to remember that in the event that i=0 is within an i-value range, as we’ll observe, the gravitational potential could be much greater in comparison to the electrical repulsive effect so the shape of potential appears like a simple pendulum.

For the considered system, the Lagrangian function L(ɸ,ɸ), corresponds to:

here we apply the formal theory of Euler-Lagrange equations, which are given by:

for the one-dimensional case, thus:

and:

so the equation of motion for the system is given by:

observe that all dynamic information for the electric pendulum is contained in the equation above. It is worth noting that if one of the charges q, q0=0, we recover the typical simple pendulum motion equation.

It is important to note that all details of the electric pendulum can be found within the above equation. It is important to note that when you take one charge q, q0=0, then we can recover the basic Pendulum motion equation.

## Electrostatic pendulum

The electrostatic pendulum is made up of two conductors made of metallic plates(Al foil with a dimension of 2cm x 3cm, and 0.1mm thick) connected through connecting rods(copper wire ~0.5cm in diameter and 4cm long). The rods don’t make any electrical contacts with each the other, and they serve to hang the pendulum vertically by using two rods.

tiny rings. Another conducting plate stationary that is large enough to cover both plates is put before the pendulum. This plate acts as a magnet for pendulum plates and provide the required acceleration to the plates. By connecting the rods,, high voltage (>400V DC) is applied to the plates. The pendulum can move around like a traditional pendulum. But due to its light weight and sporting broad plates, it is not likely to show oscillations when subjected to gravity acceleration due to the ease of damping. The oscillations are mostly controlled via electrostatic interaction between the pendulum plate and stationary plate. In this way, gravity acceleration is not considered in spite of the vertical mount.

The oscillatory nature of the pendulum is explained in the following manner. Because a conductor charged with charge is able to attract another conductor that is not charged, it begins that the higher plate(also known as positive plate) that is positively charged, comes into contact with the stationary plate. The stationary plate is able to reach exactly the same voltage as higher plate w.r.t the lower plate(also known as grounded plate) which we define to be in zero potential or just grounded. The stationary plate is positively charged, so that the its capacitance, when compared to the grounded plate creates the potential of a difference that is similar to that of the pendulum plates, that is, that it has a higher voltage source. When it is identical as the positive plate, there is no electric field that can cause attraction between the two plates. In contrast, the stationary plate draws its grounded plate. This causes the pendulum to change into its second phase, which is the movement of the grounded plate toward the stationary plate. As the ground plate comes into contact with the stationary plate, it disengages and then returns it to its ground potential. When the stationary plate has reached the same potential as the lower plate, and the attraction that they share increases. However, the attraction between upper plate and stationary plate is able to take over. The entire process repeats for a long time. Pendulums can be constructed using common materials such as aluminum foil, copper-based wire for commercial use (0.5mm diameter) and high-voltage plastic for structural needs. But the high-voltage source must be designed using an appropriate electronic circuit(Millman and Halkais as well as the references in) As the power required to maintain oscillations of the pendulum is minimal and therefore a low-power high voltage generator is adequate. The force of attraction is determined by the potential difference between Pendulum’s plates as well as the fixed plate. Actually, it is directly proportional to the voltage. Doubling the potential difference will increase the attraction force by two times at all times during the pendulum’s movement resulting in double the acceleration. Additionally, a charge transfer takes place at every contact between pendulum plates and the plate that is stationary. It is basically a discrete current flow between pendulum plates by the stationary plate that acts as a temporary reservoir for charge. The next section briefly discusses upon the quantitative aspects of electrical/mechanical energy in the pendulum.

Reference:

http://perlikowski.kdm.p.lodz.pl/papers/2017/Energy2017.pdf

https://www.researchgate.net/publication/334149454_Dynamics_for_an_electric_pendulum#pf2

https://arxiv.org/pdf/1201.3092.pdf