**1. Introduction**

According to Canada's government, about 40% of the world population does not have a pleasing way of getting sanitary water, which is hugely affected by the developing countries where up to 80% of illness in such areas is caused by inadequate water and sanitation [1]. Furthermore, Ref. [2] shows that more than a billion people in developing countries have insufficient clean water due to deprivation, change in climate, and bad governance. This leads to several issues such as under supply of drinking water, deficient structure to get water supply, swamp, droughts, and contamination of rivers as well as large dams [2]. In addition, people entail good water for livelihood, essential care, farming or agriculture, manufacturing, and trade. According to the 2019 UN World Development report, stated about four billion people, which is virtually two-thirds of the world population, encounter severe water scarcity at least one month in a year [2].

With the long-term problem of electricity and potable water in most developing and underdeveloped countries, especially in rural areas, more research needs to be done in areas that can positively affect their lives. One such area is the availability and accessibility of water for domestic, agricultural, and even industrial uses. There is a need to modify the existing water pump, such as the conventional hand water pumps. These rural dwellers can access good water even without electricity for the pumping (for some types of pumps). A pendulum can be used to provide the initial force for the pumping process in a reciprocating water pump [3,4]. With this principle, more water liters can be pumped with little

**Citation:** Yakubu, G.; Olejnik, P.; Awrejcewicz, J. Modeling, Simulation, and Analysis of a Variable-Length Pendulum Water Pump. *Energies* **2021**, *14*, 8064. https://doi.org/ 10.3390/en14238064

Academic Editor: Helena M. Ramos

Received: 29 September 2021 Accepted: 26 November 2021 Published: 2 December 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

effort since more energy can be overcome by small or little effort on the pendulum, thereby increasing the whole system's efficiency [5].

The importance of pendulum water pumps is that they significantly reduce to a minimum the human strain, making the water pumps easy to operate [6,7]. The pendulum is occasionally pushed with little effort, even with the fingers. It keeps the pumping continuous, unlike an ordinary hand pump, which always requires a load with significant steps [8]. A normal person can only make use of the hand pump for a few minutes. When pressed, water stops pumping. The situation is different from the pendulum water pump, i.e., a little pressure is needed to push the pendulum to keep changing and keep it oscillating for several hours without getting tired.

In [9], authors presented a three-dimensional model and a fabricated water hand pump with a pendulum. The pendulum's energy was analyzed based on the pendulum's kinetic and potential energy without further algorithms. In addition, 1200 L per hour of water discharge were observed using the pump with a pendulum. We noted that system dynamics need to be studied deeper to improve system efficiency and control for better performance and other research purposes.

In [8], a design, as well as the development of a pendulum-operated water pump, are presented. The mode of operation of the pendulum-operated water pump is based on defining the functions of all parts separately, but a mathematical model is not presented.

A fabricated set-up of an analytical design of a pendulum hand pump using Creo is covered by [6]. The design calculations were performed manually. The analyses show that the set-up has 70% efficiency at the initial angle of the pendulum, 39% efficiency when the pendulum is at 60◦ with the lever, and 25.5% efficiency when the pendulum angle is at 0 or 90◦ with the lever. Similarly, experimental statistics from a test rig, including validation of the dual-medium pressurizer's energy transmission strategy, are considered in [10]. An onshore pendulum's wave energy converter test rig was built for validation. It uses a hydraulic cylinder as a replacement for the wave that deploys a force on the pendulum. The overall result of the simulation shows a similar response to the experimental results.

In [11], authors designed and fabricated a pendulum hand pump. Parts of the functions were stated, the advantages and disadvantages of the pendulum pumps, the working principle, and its applications. The equations of motion need to be derived and solved numerically to allow further analysis to achieve better system performance. A kinematic approach in the theoretical analysis of a pendulum hand pump dynamics is presented in [5] where a nonlinear pendulum model is used to power the lever and the piston model for the applied excitation force to the pendulum. It is observed that satisfactory results are obtained where the frequency of excitation is greater than the pendulum's natural frequency of the model. In [5], we find the equation of the pendulum model as follows:

$$
\ddot{\varphi} + \frac{\mathcal{g}}{l} \sin \varphi = f(t),
\tag{1}
$$

and the equation of the lever and piston as below

$$
\ddot{\mathbf{x}} = \frac{3}{m} \left( f\_1(t) \frac{l\_1}{l\_2} - b\dot{\mathbf{x}} - k\mathbf{x} \right),
\tag{2}
$$

where: *ϕ*—pendulum's angular displacement, *g*—acceleration due to gravity, *l*—length of the pendulum, *f*(*t*)—input force from the pendulum model given as *f*(*t*) = *f*<sup>0</sup> cos (*ωt*); *f*0 forcing excitation amplitude, *ω*—excitation frequency, *m*—the mass of the lever and piston model, *k*—spring constant, *b*—viscous damping constant, *l*1—distance from the input force and the lever point of pivot, *P*, *l*2—distance from point *k* and *P*, and *f*1(*t*) = sin *ϕ*. Integrating numerically, Equation (7) yields the piston displacement, *x*. Using the same parameters presented in [5] (*l* = 3 m, *ω* = 5 rad/s, *f*<sup>0</sup> = 3 m, *l*<sup>1</sup> = 0.4 m, *l*<sup>2</sup> = 0.6 m, *l*<sup>3</sup> = <sup>1</sup> <sup>3</sup> *<sup>l</sup>*2, *<sup>k</sup>* = 5 N·m<sup>−</sup>1, *<sup>b</sup>* = 1 N·s·m<sup>−</sup>1, *<sup>m</sup>* = <sup>4</sup> kg), we use the proposed numerical approach of our modified system to solve Equations (1) and (2), and we have the numerical solution as shown in Figures 1 and 2, which is very similar to the ones presented in [5].

**Figure 1.** Angular displacement of the Pendulum in Equation (1).

**Figure 2.** Linear displacement of the Piston in Equation (2).

The study needs a pivotal and in-depth analysis of fluid mechanics, and the pendulum's length can also affect the system's performance.

To further reduce the human effort, we proposed to use a pendulum with a variablelength model instead of the conventional pendulum. This modification would give a faster and longer oscillation, which will, in turn, result in more rapid pumping of fluid since the variable-length pendulum can undergo a quicker and longer oscillation, as presented by the following authors: Ref. [12] derived the differential equations of dynamics for both the first and the second mutation from the sum of kinetic and potential energy for a rigid pendulum's two and three degrees of freedom. It was observed that a successive expansion in the forms of representation of the energy is introduced. The largest Lyapunov exponent was used to classify the system based on computational analysis. The phase planes and the Poincaré maps show some homogeneous dynamic patterns, such as quasiperiodic and chaotic motions. Refs. [13,14], the Euler–Lagrange equation, and the Rayleigh dissipation function are used to derive the equation for a three-degree of freedom pendulum system. The numerical results reveal that a variable-length spring pendulum hung from the occasionally forced slider can demonstrate quasi-periodicity and chaotic motions in a resonance condition. Furthermore, near the resonance, linking bodies on the system dynamics could lead to unforeseeable dynamical comportment.

Krasilnkov presents in [15] the variable-length pendulum harmonic oscillations, which depend on the length of the pendulum. Lyapunov exponents, bifurcation diagrams, and the Poincaré maps situated on phase plane diagrams were used to inspect the system behavior. It was concluded that the system exhibited chaotic properties in the domain of higher-level stability. A control scheme for a vertically excited parametric pendulum with variable length is presented in [16]. It offers two energy sources: a vibrating machine and sea waves simulated by a stochastic process [16]. For the pendulum to be controlled, a telescopic

adjustment of the pendulum length [16] is used during the motion. The numerical results show favorable terms for energy harvesting. Steady revolutions can be attained irrespective of the forcing factors and for all established initial conditions. It is concluded that it is hard to attain stable rotation due to the high reliance of the dynamical system on parameters that causes the forcing and the initial conditions. However, the controlled pendulum can reach stable rotations if the threshold velocity is sufficiently selected to modulate the control operation.

In this work, we present the following:

• The mathematical model and simulation results of the variable-length pendulum water pump were performed. The equations were solved using the Runge–Kutta method with 5th order adaptive step size.

A vertically excited parametric pendulum with variable length [16] is used instead of the conventional pendulum with constant length, minimizing human effort with increasing output of water or liquid from the pump outlet.

• With the variable-length pendulum, more and richer dynamics can be achieved flexibly, giving more and faster oscillations and providing long-lasting energy for the fluid pumping—thus drastically reducing the human effort required for pumping and saving time.

The presented work is practical and valuable because it shows the most responsible mechanism for obtaining the minimum effort to swing the pendulum as well as provides a methodology that guarantees relatively fast and long-lasting oscillations, as stated in the problem definition below. First, the three-dimensional model is introduced. Next, the components are listed, stating the functions of each part of the system. Finally, the system's working principle is supported by mathematical modeling, results of numerical simulations covered by essential conclusions.

### **2. Problem Definition and Modeling**

The rural area dwellers can use the pendulum water pump for farming and irrigation, water-wells, and can also be used for fire extinguishing both in rural areas and in cities. It can also be used for drainage to control liquid (water) levels in a protected area. Other areas of application include: sewage, chemical industries, medical fields, steel mills, etc. [8]. It is useful and practical for older people and children who can operate it easily, since it only requires minimum effort to swing the pendulum. Furthermore, the oscillating nature of the pendulum and maintenance do not require special training or skills to perform the task with hand or agility. Below, based on mathematical analysis and numerical simulation, we show that an initial force could be only required and then maintained for pumping the water. Based on the existing literature (to the best of our knowledge), none have used the pendulum with variable length. In our work, a vertically excited parametric pendulum with variable length [16] is used instead of the conventional pendulum, giving richer swinging dynamics in the entire mechanical coupling.
