*2.1. A Three-Dimensional Model*

The proposed system comprises three main parts: the pendulum, the lever, and the reciprocating pump assembly containing the piston and the pushrod. Other parts include bracket foundation link, pivot point, frame, support at the frame, delivery pipe, and suction pipe. Figure 3 shows the three-dimensional physical model with the various parts of the system. The pendulum's angular motion is transmitted into the to-and-from motion of the piston through the lever and the pushrod [2].

**Figure 3.** Configuration of the three-dimensional model of the investigated system.

The system input is from the pendulum where an initial force is applied. The lever and the spring act as a transmitter that transmits the energy to the piston (system output), where the pumping of the liquid takes place.

#### *2.2. Components of the System*

We modify the existing pendulum water pump to achieve a maximum effect by using a vertically excited parametric pendulum with variable length instead of the conventional pendulum. Each component is briefly described below:

Frame: the rigid part serving as a support where the whole system assembly is mounted. The parametric pendulum with variable length: Ref. [16], the required energy source

for commencing the action of pumping by oscillating. (Detailed in Section 3.1, case II). Bracket: a connection between the pendulum and lever and also between the lever

and the piston.

Pivot point: also called the fulcrum, it is the part where the lever turns. It plays a central role in the lever system, and the lever's power is supplied between the pivot point and the pendulum.

Bearing: to reduce rotational friction and support axial loads.

The rod: held up by double support bearing one on each side of the lever that forms the lever's fulcrum. The coupling of the lever crested on the bearing lever rotates with the use of thrashing. A different support bearing is utilized at the pendulum's bracket, allowing for the pendulum's motion.

Delivery pipe: it connects the pump's cylinder with the exit. The liquid is dispatched to the preferred exit point along this delivery pipe.

Suction pipe: it connects the origin of liquid to the reciprocating pump's cylinder. This pipe sucks the liquid from the source to the cylinder.

Reciprocating pump assembling: convert the mechanical energy into hydraulic energy by sucking the liquid into a cylinder. A piston is reciprocating, which uses thrust on the liquid and increases its hydraulic energy [17].

#### *2.3. The Working Principle*

The system free energy is based on the phenomenon of an oscillating pendulum-lever system. The pendulum pump's purpose is the oscillation of the body pendulum controlled with a small pressure hand. Change of the inertial forces causes the fluctuation of the lever attached to the pump piston connected to a spring and a damper. The oscillations of the pendulum serve as the model's input [18]. These oscillations bring about the swinging to and fro of the lever about its turning position; the lever is attached at one end to the pump's rod and brings about the lever movement. For water to flow from the pump, the pendulum does not need to be balanced. Instead, the piston starts oscillating based on gravitational potential, and water begins to come out through the delivery pipe continuously. The pendulum is to be pushed occasionally in other to maintain the continuous flow of water. The pendulum water pump works efficiently with 90◦ amplitude regardless of the pendulum's size [6]. The foot valve opens at the piston's upstroke, and suction brings water into the pump's upper part (head) from the suction pump. The piston's valve opens up and permits water to spout upward above the piston on the piston's following downstroke. On the subsequent piston's upstroke, water is propelled over the exit [19]. The system does not require fuel or electricity for its operation. Therefore, it is user-friendly and cannot cause global warming.

#### **3. Mathematical Modeling**

The complete system is made up of three parts: the pendulum, lever, and piston. These are modeled separately and connected through the bracket, where forces are transferred from one part to another. The motion of one makes the subsequent part move. As the pendulum is set to motion, it transfers the force to actuate the lever. The lever motion is imparted to the piston. The motion equation is established from Newton's second law of motion, i.e., *F* = *ma* since a force initiates the motion.
