*3.3. Piston Model*

According to [20,24], piston has a reverse effect on the lever, and it damps oscillations of the pump. Damping of the lever motion causes damping of the pendulum, but the force damping the pendulum is less than the work of force that dampens the lever. Figure 8 shows a schematic diagram of the piston. An analysis leads to the equations of motion, as it is shown below.

**Figure 8.** Schematic diagram of the piston.

We introduce the following denotations for the masses (kg): *m*1—buoy mass, *m*2 piston model mass (total), *m*3—mass of the rod, *m*4—mass of the piston, *m*5—mass of the pumped fluid, together with the two relations *m*<sup>2</sup> = *m*<sup>3</sup> + *m*<sup>4</sup> + *m*<sup>5</sup> in the upstroke and *m*<sup>2</sup> = *m*<sup>3</sup> + *m*<sup>4</sup> in the down stream. Therefore, the mass *m*<sup>2</sup> changes in time *t* as it moves upward and downward. In addition, the mass *m*<sup>2</sup> is different in the upstroke due to the pumping processes of the fluid. Ref. [25] delivers a mathematical representation of the dynamical motion of the piston given in a state-space representation as it follows:

$$
\dot{q} = Aq + f, \quad \text{for } q(0) = q\_0. \tag{21}
$$

The state vector is presented as:

$$q = \begin{bmatrix} \mathbf{x}\_{\text{bc}} \ \mathbf{x}\_{\text{bc}} \ \mathbf{x}\_{\text{pc}} \ \hat{\mathbf{x}}\_{\text{pc}} \ p\_{\text{ur}} \ p\_{\text{lr}} \end{bmatrix}^T,\tag{22}$$

where: *xbc*—position of the buoy's center of mass (m), *<sup>x</sup>*˙*bc*—buoy velocity (m·s−1), *xpc* position of the piston center (m), *<sup>x</sup>*˙ *pc*—piston velocity (m·s−1), *pur*—upper reservoir pressure (Pa), *plr*—lower reservoir pressure (Pa).

The governing equations are given in Equations (23) and (24):

$$k\_1 \ddot{x}\_{bc} + c \left(\dot{x}\_{bc} - \dot{x}\_{pc}\right) + k \left(\mathbf{x}\_{bc} - \mathbf{x}\_{pc} - I\_R\right) = F\_{bc} - m\_1 \mathbf{g}\_\prime \tag{23}$$

where: *lR*—length of the rod (m), *Fbc*—force of the buoy's center of mass (N).

We get the second-order differential equation of dynamics of the piston:

$$m\_2 \ddot{\mathbf{x}}\_{\rm pc} + c \left( \dot{\mathbf{x}}\_{\rm pc} - \dot{\mathbf{x}}\_{\rm bc} \right) + k \left( \mathbf{x}\_{\rm pc} - \mathbf{x}\_{\rm bc} - l\_R \right) = -A\_c (p\_{\rm ur} - p\_{\rm lr}) - m\_2 \mathbf{g} - F\_{\rm f},\tag{24}$$

where:

$$\alpha\_{\rm fc} = \frac{L\_r + 0.5H\_b - \left(m\_1 + m\_4 + \rho A\_c (L\_c + H\_u)\right)}{\left(S\_b \rho\_{sw} - \frac{H\_u}{2}\right)},\tag{25}$$

$$\propto\_{pc} = \frac{0.5H\_b - \left(m\_1 + m\_4 + \rho A\_c \left(L\_c + H\_{\text{fl}}\right)\right)}{\left(S\_b \rho\_{sw} - \frac{H\_{\text{fl}}}{2}\right)}.\tag{26}$$

Continuing, the pumping force is found in the following form:

$$F\_p = -A\_c p\_{Ir} + \rho (l\_c + L\_u) A\_{ur} \left(\ddot{z}\_p + \text{g}\right) + \rho A\_c \dot{z}\_{p\prime}^2 \tag{27}$$

where area of the piston *Ac* = *πRp* <sup>2</sup> (m2), *zp*—piston displacement about a zero mean, *Aur*—area of the upper reservoir (m2), *Ff*—initial approximation of the friction in the contact between the piston and the cylinder given as <sup>−</sup>*Bx*˙ *pc*, *<sup>B</sup>* <sup>=</sup> *<sup>μ</sup> Sp* 2*πRpHp*—cylinder damping coefficient (N·s·m−1), *Sp*—separation of piston and cylinder (m), *Rp*—radius of the piston (m), and *Hp*—height of the piston (m).

The amount of water pumped by the piston in every upstroke is the water in the cylinder, and the water inside the upper reservoir. Because of this, the fluid mass has to be modified as follows: 

$$m\_5 = \rho \left( l\_c + \frac{p\_{ur}}{\rho \mathcal{g}} \right) A\_{c\prime} \tag{28}$$

where: *<sup>ρ</sup>*—density of the fluid (kg·m−3), *<sup>g</sup>*—gravitational acceleration (m·s−2), and *lc* length of the cylinder (m).

The buoyancy force depending on the buoy *Xbc* and the position of the wave *xw* is given in the form [25]:

$$F\_{\rm bc} = \left(\varkappa\_{\rm w} - \varkappa\_{\rm bc} + \frac{1}{2}H\_{\rm b}\right) S\_{\rm b} \rho\_{\rm sw} \mathfrak{g}\_{\prime} \tag{29}$$

where: *Hb*—height of the buoy (m), *Sb*—surface of the buoy, *ρsw*—reservoir fluid density (kg·m<sup>−</sup>3) and *xw*—definition of wave used for the simulation is given:

$$\chi\_w = L\_r + \frac{H\_w}{2}\sin\left(\frac{2\pi}{T\_w}t - \frac{\pi}{2}\right),\tag{30}$$

where: *Hw*—wave height (m), *Tw*—wave period (s).

#### **4. Numerical Results and Discussion**

The numerical solutions of the system governing equations are solved using the Runge–Kutta method with adaptive step-size, with a simulation time step size of 0.003 s for case I and 0.005 s for case II. The initial condition for the pendulum angle *ϕ*(0) is 0.5*π* radians for both case I and case II, with a time scale of 30 s and 50 s for the case I and case II, respectively.

Case I: Water pump pendulum with constant length. We start with the pendulum displacement, then the lever displacement, and finish at the piston's displacement. Finally, various parameters that determine the pendulum pump's output discharge are analyzed, and the numerical results are presented in Figures 9–11 for the pendulum, lever, and piston displacement, respectively. Parameters analysis includes the pendulum's mass, angle of

suspension, and the pendulum's length. Therefore, we present only the results with the parameters that show good system responses.

For the pendulum displacement, the computations are performed using the below stated parameter values: *l* = 0.5 m, *M* = 0.38 kg, *m* = 0.095 kg, *f*<sup>0</sup> = 0.005 N, *<sup>ω</sup>* = 5 rad·s<sup>−</sup>1, *<sup>g</sup>* = 9.81 m·s−2, *<sup>b</sup>* = 0.003 N·s·m−1. The period of rotational motion depends on the length of the pendulum. The length is varied at a point in time until the desired periods are obtained. As shown in Figure 9, amplitude of the total energy decreases with time due to damped oscillations. Therefore, the pendulum has to be push occasionally for a continuous fluid flow.

**Figure 9.** Angular displacement *<sup>ϕ</sup>*(*t*) of the pendulum for *<sup>l</sup>* <sup>=</sup> 0.5 m, *<sup>b</sup>* <sup>=</sup> 0.003 N·s·m<sup>−</sup>1.

Simulation of the lever displacement is shown in Figure 10, using the following parameter values: *ml* <sup>=</sup> 2.0 kg *<sup>c</sup>* <sup>=</sup> 1.0 N·s·m−1, *<sup>k</sup>* <sup>=</sup> 5.0 N·m−1, *<sup>l</sup>*<sup>1</sup> <sup>=</sup> 0.4 m, *<sup>l</sup>*<sup>2</sup> <sup>=</sup> 0.6 m, *<sup>l</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup> <sup>3</sup> *l*2, *f*1(*t*) = sin *ϕ*, where *ϕ*—the pendulum angular displacement (rad) is the pendulum system input. The results show that the total energy gradually decreases with time.

**Figure 10.** Linear displacement *<sup>x</sup>*(*t*) of the lever for *<sup>l</sup>* <sup>=</sup> 0.5 m, *<sup>b</sup>* <sup>=</sup> 0.003 N·s·m<sup>−</sup>1.

An analysis of the piston is presented. Some studies of fluid mechanics are also included in the piston analysis for optimal pump performance. The linear displacement of the piston is shown in Figure 11 with the values of the parameters as *m*<sup>1</sup> = 10 kg, *<sup>g</sup>* = 9.81 m·s−2, *<sup>c</sup>* = 1.0 N·s·m−1, *<sup>k</sup>* = 50.0 N·m−1, *lR* = 4 m, *<sup>g</sup>* = 9.81 m·s−2, *<sup>B</sup>* = 1.25915 N·s·m−1, *<sup>m</sup>*<sup>2</sup> = 6.0242 kg, *<sup>ρ</sup>* = 1000 kg·m−2, *<sup>ρ</sup>sw* = 1030 kg·m−<sup>2</sup> *Aur* = 4 m2, *Alr* = 4 m2, *sb* = 2 m, *Hb* = 2 m, *Tw* = 10 s, *Hw* = 4 m, *Lc* = 10 m, *zp* = 0 m, *Hp* = 2 m, *Lr* = 2 m *zp* = 0 m, *Rp* = 0.05 m2. The same values for the spring constant, *k*, and the viscous damping constant, *c*, was used because of the same connection. It can be observed that the response of the piston is similar to the lever displacement. However, the displacement is not as much as that of the lever because of higher piston mass and other considered factors, overall pump parameters, and fluid analysis.

**Figure 11.** Linear displacement *xpc* of the piston *<sup>l</sup>* <sup>=</sup> 0.5 m, *<sup>b</sup>* <sup>=</sup> 0.003 N·s·m<sup>−</sup>1.

The analysis for different lengths is carried out for a deeper understanding of the relation between pendulum length and output of the system, which was not addressed in [5]. Changing the right-hand side pendulum length alone, i.e., without changing initial parameters of other parts, affects performance of the whole system. Therefore, the efficiency of the pumping depends on the length of the right-hand side pendulum. As can be seen in Figures 12–14, the system becomes more stable, but with a reduced number of oscillation periods for the whole system when the parameter value *l* is changed from 0.5 to 1.2 m.

**Figure 12.** Linear displacement *<sup>ϕ</sup>*(*t*) of the pendulum (continued) for *<sup>l</sup>* <sup>=</sup> 1.2 m, *<sup>b</sup>* <sup>=</sup> 0.003 N·s·m<sup>−</sup>1.

**Figure 13.** Linear displacement *<sup>x</sup>*(*t*) of the lever (continued) for *<sup>l</sup>* <sup>=</sup> 1.2 m, *<sup>b</sup>* <sup>=</sup> 0.003 N·s·m<sup>−</sup>1.

**Figure 14.** Linear displacement *xpc* of the piston (continued) for *<sup>b</sup>* <sup>=</sup> 0.003 N·s·m<sup>−</sup>1.

The simulation results shown in Figures 9–14 have been compared with the result delivered by [5]. It has proved that our model follows a more realistic trend because it includes friction and the effect of excitation force on the response of the investigated system. Furthermore, the presented system in case I is more stable than one in [5], as shown in Figures 1, 2 and 12–14. The obtained results clearly show how the pendulum's length plays a critical role in the overall system stability.

In addition, further analysis of Case I of the pendulum model is carried out by increasing the viscous damping with other parameters left unchanged. When the viscous damping *<sup>b</sup>* is increased to 0.3 N·s·m−<sup>1</sup> with the same length *<sup>l</sup>* = 1.2 m, the time response of the system decreases and vanishes within a few seconds, as shown in Figures 15–17. Providing some technical recommendations, the value of *b* should be minimum for the system to be more stable and oscillate for a more extended period. Therefore, the value of the pendulum length, *l*, and the viscous damping, *b*, are to be selected with care as they have more effect on the system performance.

**Figure 15.** Linear displacement *<sup>ϕ</sup>*(*t*) of the pendulum (continued) for *<sup>l</sup>* <sup>=</sup> 1.2 m, *<sup>b</sup>* <sup>=</sup> 0.3 N·s·m<sup>−</sup>1.

**Figure 16.** Linear displacement *<sup>x</sup>*(*t*) of the lever (continued) for *<sup>l</sup>* <sup>=</sup> 1.2 m, *<sup>b</sup>* <sup>=</sup> 0.3 N·s·m<sup>−</sup>1.

**Figure 17.** Linear displacement *xpc* of the piston (continued) for *<sup>l</sup>* <sup>=</sup> 1.2 m, *<sup>b</sup>* <sup>=</sup> 0.3 N·s·m<sup>−</sup>1.

Case II: Water pump pendulum with a vertically excited parametric pendulum with variable length [16] is used instead of the conventional pendulum. Figures 18–20 show the simulation results for the whole system when this type of pendulum with variable length is used with the following parameter values: *l*<sup>01</sup> = *l*<sup>02</sup> = 4 m, *m* = 5 kg, *xe* = 1.5 m, *<sup>f</sup>*<sup>0</sup> = 10 N, *<sup>ω</sup>* = 0.5 rad·s−1, *<sup>g</sup>* = 9.81 m·s−2, *cp* = 0.1 N·s·m−1. The parameters' values of the lever and the piston remain unchanged. With *cp* = 1 N·s·m−<sup>1</sup> and *<sup>ω</sup>* = 0.4 rad·s−1, Figures 21–23 are obtained, which shows more stable oscillation of the variable-length pendulum with more stability of the lever and the piston model as a result of the effect of increasing damping.

**Figure 18.** Angular displacement *ϕ*(*t*) of the pendulum (Case II—a variable length concept of the pendulum pump) for *cp* <sup>=</sup> 0.1 N·s·m<sup>−</sup>1, *<sup>ω</sup>* <sup>=</sup> 0.5 rad·s<sup>−</sup>1.

**Figure 19.** Linear displacement *x*(*t*) of the lever (Case II—a variable length concept of the pendulum pump) for *cp* <sup>=</sup> 0.1 N·s·m<sup>−</sup>1, *<sup>ω</sup>* <sup>=</sup> 0.5 rad·s<sup>−</sup>1.

**Figure 20.** Linear displacement *xpc*(*t*) of the piston (Case II—a variable length concept of the pendulum pump) for *cp* <sup>=</sup> 0.1 N·s·m<sup>−</sup>1, *<sup>ω</sup>* <sup>=</sup> 0.5 rad·s<sup>−</sup>1.

**Figure 21.** Angular displacement *ϕ*(*t*) of the pendulum (Case II—a variable length concept of the pendulum pump) for *cp* <sup>=</sup> 1 N·s·m<sup>−</sup>1, *<sup>ω</sup>* <sup>=</sup> 0.4 rad·s<sup>−</sup>1.

**Figure 22.** Linear displacement *x*(*t*) of the lever (Case II—a variable length concept of the pendulum pump) for *cp* <sup>=</sup> 1 N·s·m<sup>−</sup>1, *<sup>ω</sup>* <sup>=</sup> 0.4 rad·s<sup>−</sup>1.

**Figure 23.** Linear displacement *xpc*(*t*) of the piston (Case II—a variable length concept of the pendulum pump) for *cp* <sup>=</sup> 1 N·s·m<sup>−</sup>1, *<sup>ω</sup>* <sup>=</sup> 0.4 rad·s<sup>−</sup>1.

Figures 18–23 show the time histories of the pendulum, lever, and piston, respectively. Some irregularity and quasi-periodicity are reported, and the behavior does not settle even after 50 s of simulation time. The pattern of recurrence does not lend to precise measurement. However, the oscillations can occur regularly when a regular external forcing forces them. In other words, the quasi-periodicity behavior can be compensated when a regular external forcing is applied to the system.
