**3.3.1 Experiment design**

Fig.13 gives the experiment principle. We use one NEC 2301 stain gage as the force sensor and use NEC AC AMPLIFIER AS 1302 to amplify the output signal from strain gage, which are shown in Fig.14.

Fig. 13. Experiment Design for Identification

Fig. 14. Strain Gage and Amplifier

Firstly, we give a brief illustration for the strain gage measurement. Let *Fd* be deformation force, and *ε* be the deformation of aluminium lever used in the experiment. Therefore, from the theory of mechanics of materials, we can get the relation of *Fd* and *ε* as:

$$F\_d = \frac{ZE}{X}e\tag{16}$$

where, *Z* is second moment of area, *E* is the Young's modulus, *X* is the distance from acting point of force to stain gage.

### **3.3.2 Experiments and analysis**

In the experiment, there are four variables we need to consider, the equivalent cross-section of propeller, flow velocity, incoming angle and control voltage. What we are interested in is the variation of propulsive force in different incoming angles and different control voltage.

### 3.3.2.1 Equivalent cross-section variation of propellers

10 Will-be-set-by-IN-TECH

Flow velocities Depth Incoming angles Control voltages 0.1*m*/*<sup>s</sup>* <sup>80</sup>*cm* <sup>0</sup> <sup>−</sup> *<sup>π</sup>* <sup>3</sup>*<sup>V</sup>* <sup>−</sup> <sup>7</sup>*<sup>V</sup>* (DC) 0.2*m*/*<sup>s</sup>*

In this part, we try to identify the dynamics model of the water-jet propeller by experiment. What we are interested in is the relation of flow incoming angles, flow velocities and

Fig.13 gives the experiment principle. We use one NEC 2301 stain gage as the force sensor and use NEC AC AMPLIFIER AS 1302 to amplify the output signal from strain gage, which are

Firstly, we give a brief illustration for the strain gage measurement. Let *Fd* be deformation force, and *ε* be the deformation of aluminium lever used in the experiment. Therefore, from

where, *Z* is second moment of area, *E* is the Young's modulus, *X* is the distance from acting

In the experiment, there are four variables we need to consider, the equivalent cross-section of propeller, flow velocity, incoming angle and control voltage. What we are interested in is the variation of propulsive force in different incoming angles and different control voltage.

*<sup>X</sup> <sup>ε</sup>* (16)

*Fd* <sup>=</sup> *ZE*

the theory of mechanics of materials, we can get the relation of *Fd* and *ε* as:

Table 1. Experiment Condition

**3.3.1 Experiment design**

shown in Fig.14.

**3.3 Experiments for the dynamics modeling**

Fig. 13. Experiment Design for Identification

Fig. 14. Strain Gage and Amplifier

point of force to stain gage.

**3.3.2 Experiments and analysis**

propulsive forces. Experiment condition is listed in Table 1.

As a vectored water-jet-based propulsion system, it should be noted that both the propulsive force and its direction can be changed. Therefore, when the propeller changes its direction, actually, the incoming angle of flow is also changing, and the equivalent cross-section of the propeller is changing. From Equation 11 we know the propulsive force will change if cross-section *A* changes. Fig.15 gives a demonstration of this case. When the propeller rotate from position I to II, the equivalent cross-section will change from cross-section I to cross-section II. So we try to find an equation to describe this variation.

Fig. 15. Variation of Equivalent Cross-section of Propellers

Considering that the measured force from stain gage is actually a resultant force of propulsive force and fluid force. And we also know that the fluid force acted on the propeller depends on the equivalent cross-section.

So the first experiment is measurement of the equivalent cross-section variation. The propeller is submerged in the flow which has a speed of 0.2*m*/*s*, propeller is powered off. And we only change its orientation in *X* − *Y* plane. Because of experiment limits, we can not change the flow direction, so in the experiment, the incoming angle equals to the orientation angle of the propeller. Fig.15 gives a demonstration of the equivalent cross-section. We give some special angles, 0, *π*/6, *π*/3, 2*π*/3, 5*π*/6, *π*, for this experiment. Fig.16 gives the experiment data of equivalent cross-section. You may notice that, we did not adopt the orientation angle of *π*/2. Because, when the propeller rotate to *π*/2, which means that the measure surface of the strain gage is parallel to the flow direction, the strain gage can not measure the flow force.

Fig. 16. Variation of Equivalent Cross-section

We substitute Equation 19 into 18 we get:

*Ft* = *Fd* + *<sup>ρ</sup>V*<sup>2</sup>

are results at the flow velocity of *Vf* = 0.1*m*/*s* and *Vf* = 0.2*m*/*s*, respectively.

So now, we can calculate the real propulsive force by using equivalent cross-section, deformation force and incoming angles. The results is shown in Fig.18. Fig.18(a) and Fig.18(b)

Development of a Vectored Water-Jet-Based Spherical Underwater Vehicle 15

(a) *Vf* = 0.1*m*/*s*

(b) *Vf* = 0.2*m*/*s*

From the results of 3.3.2.3, we can obtain the relation of control voltage and propulsive force. First, we give the experiment data, in Fig.19. From the diagram, we can see that the relation

Because of the symmetrical shape of the hull, it is obvious that motion characteristics of surge, sway and heave should be similar. However, from another point of view, surge and sway are motions in *X* − *Y* plane while heave is a motion that its motion surface perpendicular to *X* − *Y* plane. Therefore, we carry out experiments for horizontal motion surface and vertical motion surface respectively. Besides, for the experimental prototype vehicle, we only consider one

of control voltage and propulsive force can be described using linear equation.

Fig. 18. Propulsive forces with Different Incoming Angles

3.3.2.4 Control voltage and propulsive force

**4. Underwater experiments for basic motions**

rotational DOF in Z axis, so the third experiment is yaw motion.

*<sup>c</sup> Ae*(*φ*)*cosγ* (20)

From Fig.16 we can see, the data curve is similar with a sinusoid, so we use a sine function to fit this experiment data:

$$A\_{\varepsilon}(\phi) = \lambda\_1 \sin(\lambda\_2 \phi + \lambda\_3) \tag{17}$$

where *φ* is incoming angle, *λ*1, *λ*2, *λ*<sup>3</sup> are coefficients.

3.3.2.2 Incoming angle and deformation force

In this case, the flow velocity is seen as constant. Two groups of experiment are carried out at flow velocity of 0.1*m*/*s* and 0.2*m*/*s*. The control voltage to thruster is from 3*V* to 7*V* every 1*V*. From the data shown in Fig.17, we can see that the deformation force does not simply increase

Fig. 17. Deformation Forces with Different Incoming Angles

with the increasing of incoming angle, the maximum deformation of the lever happens at about 60 degree of the incoming angles. They are not a linear relation. Then, how about the real propulsive force?

3.3.2.3 Incoming angle and propulsive force

As we mentioned, what we measured by stain gage is actually a resultant force of propulsive force and fluid force. Therefore, the real propulsive force *Ft* should be calculated using deformation force *Fd* and the equivalent cross-section *Ae*.

$$F\_d = F\_t - F\_f \cos \gamma \tag{18}$$

If we consider about the fluid force of the flow, we can refer to Equation 11 and 17, then write the fluid force as:

$$F\_f = \rho V\_\mathbb{C}^2 A\_\ell(\phi) \tag{19}$$

We substitute Equation 19 into 18 we get:

12 Will-be-set-by-IN-TECH

From Fig.16 we can see, the data curve is similar with a sinusoid, so we use a sine function to

In this case, the flow velocity is seen as constant. Two groups of experiment are carried out at flow velocity of 0.1*m*/*s* and 0.2*m*/*s*. The control voltage to thruster is from 3*V* to 7*V* every 1*V*. From the data shown in Fig.17, we can see that the deformation force does not simply increase

(a) *Vf* = 0.1*m*/*s*

(b) *Vf* = 0.2*m*/*s*

with the increasing of incoming angle, the maximum deformation of the lever happens at about 60 degree of the incoming angles. They are not a linear relation. Then, how about the

As we mentioned, what we measured by stain gage is actually a resultant force of propulsive force and fluid force. Therefore, the real propulsive force *Ft* should be calculated using

If we consider about the fluid force of the flow, we can refer to Equation 11 and 17, then write

*Ff* = *ρV*<sup>2</sup>

*Fd* = *Ft* − *Ff cosγ* (18)

*<sup>c</sup> Ae*(*φ*) (19)

Fig. 17. Deformation Forces with Different Incoming Angles

deformation force *Fd* and the equivalent cross-section *Ae*.

3.3.2.3 Incoming angle and propulsive force

real propulsive force?

the fluid force as:

*Ae*(*φ*) = *λ*1*sin*(*λ*2*φ* + *λ*3) (17)

fit this experiment data:

where *φ* is incoming angle, *λ*1, *λ*2, *λ*<sup>3</sup> are coefficients.

3.3.2.2 Incoming angle and deformation force

$$F\_l = F\_d + \rho V\_c^2 A\_e(\phi) \cos \gamma \tag{20}$$

So now, we can calculate the real propulsive force by using equivalent cross-section, deformation force and incoming angles. The results is shown in Fig.18. Fig.18(a) and Fig.18(b) are results at the flow velocity of *Vf* = 0.1*m*/*s* and *Vf* = 0.2*m*/*s*, respectively.

Fig. 18. Propulsive forces with Different Incoming Angles

#### 3.3.2.4 Control voltage and propulsive force

From the results of 3.3.2.3, we can obtain the relation of control voltage and propulsive force. First, we give the experiment data, in Fig.19. From the diagram, we can see that the relation of control voltage and propulsive force can be described using linear equation.

#### **4. Underwater experiments for basic motions**

Because of the symmetrical shape of the hull, it is obvious that motion characteristics of surge, sway and heave should be similar. However, from another point of view, surge and sway are motions in *X* − *Y* plane while heave is a motion that its motion surface perpendicular to *X* − *Y* plane. Therefore, we carry out experiments for horizontal motion surface and vertical motion surface respectively. Besides, for the experimental prototype vehicle, we only consider one rotational DOF in Z axis, so the third experiment is yaw motion.

(a) The trajectory of case 1 (b) Steering angle of case 1

Development of a Vectored Water-Jet-Based Spherical Underwater Vehicle 17

(c) The trajectory of case 2 (d) Steering angle of case 2

(e) Surge and Brake of case 3

the experimental results fit well with simulation results in surge stage, but when the vehicle rotating, errors become large. The reason of this is because the simulation experiment only considered linear damping force and quadratic damping force, but in reality, there are other

Even though we design the working depth of the vehicle to 10*m*, because the depth of experimental pool is only 1.2*m*, we can only make the experiments in shallow water. So we set the vertical motion time in a relatively small range. We also carried out two experiments:

Fig. 20. Experimental Results of Horizontal Motion

step 1. Set the top point of the spherical hull as the start point;

step 1. Set the top point of the spherical hull as the start point;

step 2. Move downward in Z axis for about 7*s* ;

hydrodynamic forces act on the vehicle.

**4.2 Experiment of vertical motion**

step 3. Float up to the surface.

**Case 1**:

**Case 2**:

Fig. 19. Propulsive forces with Different Control Voltages
