Experimental Study on Porpoising of a High-Speed Planing Trimaran

Abstract

Porpoising is defined as unstable coupled heave and pitch motions. In this study, seven test conditions with various longitudinal locations of the center of gravity and the moment of inertia were designed to study the porpoising phenomenon of a tunnel-type planing trimaran. After comparing the lower speed limit for porpoising, this study revealed that moving the center of gravity forward could increase the lower speed limit of porpoising, but it could also increase the amplitude of heave and pitch. Furthermore, the reduction of the moment of inertia of the planing hulls could effectively reduce the oscillation amplitude of planing trimaran porpoising and improve the longitudinal stability of planing trimarans. In this test, condition A5 had a 15% reduction in the inertia moment compared to A3, but the amplitude of the pitch oscillation in porpoising decreased by 60%. Unlike conventional trimaran towing tests, this experiment studied the effect of speed on porpoising after exceeding the lower speed limit for porpoising and found that with an increase in speed, the amplitude of porpoising first increased, then decreased, and then increased again. This study defines the speed at which the amplitude of porpoising increases for the second time as the second critical speed of porpoising. The porpoising generated by a planing trimaran after reaching the second critical speed is defined as the second porpoising. Finally, the limitations of the conventional longitudinal stability limit curve method to predict the porpoising of a planing trimaran are discussed.

1. Introduction

A planing trimaran is a composite ship design that combines the advantages of conventional planing mono-hull, multi-hull, and aerodynamic booster ships [1]. In the high-speed planing state, the ventilation tunnel enclosed by the demi-hull and the main hulls provides considerable aerodynamic lift, which may not only improve the resistance performance of the hull, but also play a role in reducing oscillations and buffering to a certain extent [2,3,4]. The hull lift and trim at high-speed planing reduce the hull tail wetted length, which results in hydro- and aerodynamic lifts and moves back the center of buoyancy. Therefore, hull longitudinal motion stability is disturbed, and motion instability can occur. Porpoising is one of the most common and concerning longitudinal unstable motions for high-speed planing ships, and it is caused by the breaking of the lift and momentum balance acting on the hull. Porpoising is a hull’s heave and pitch, coupling oscillation with sustained or increasing amplitudes [5]. During porpoising, the vessel jumps out of the water because of dynamic lift and subsequently slams into the water because of dynamic lift reduction. This drastically vertical and rotating motion can severely damage the hull structure. Therefore, longitudinal stability, especially porpoising, is one of the primary limiting factors for determining the maximum speed of operation in addition to resistance performance and is a key concern in the design of a planing trimaran [6]. Therefore, considering porpoising in the design stage is necessary.

To date, studies on the inhibition of porpoising have focused on porpoising-occurrence criteria [7], high-speed region motion behavior and mechanisms [8,9], limited speed prediction for longitudinal stability [10,11], and the extreme speed prediction of numerous porpoising suppression methods, such as moving forward the center of gravity [12,13], which effectively increases the gravitational bow-burying moment and reduces the sailing trim value of the high-speed section. Thus, longitudinal motion stability is improved. Extensive studies have shown that appendages can improve longitudinal stability [14]. Alternatively, stern pressure can be increased by setting the tail pressure wave plate [15,16,17] or the tail intercepting plate [18,19,20,21,22], the hydrofoil in the tunnel, and other appendages [13], which suppresses porpoising. Furthermore, appendages combining automation control have been verified both in rough and calm water and can be used to dynamically adjust the trim angle in real time to obtain superior comfort and resistance performance. The dynamic longitudinal stability and extreme speed increase considerably [23,24]. Porpoising prediction is performed using the theoretical method in which the motion response eigenvalue of the hull equation of motion is utilized [25,26]. The porpoising boundary line method is used for regression with model towing test results [7,10,27].

Within the design speed range, the resistance curve of a planing hull typically exhibits two peaks. The resistance of a planing hull increases with an increase in speed before the first resistance peak due to the increasing speed of the synchronous trim angle and wetted surface, whereas the resistance decreases with the hull lifted out of the water and a reduction in the trim angle after the first peak value. The navigation attitude changes considerably with the increase in speed [28]. This phenomenon reveals that the navigation attitude of a planing hull changes frequently during speed improvement. Thus, frequently changing the hull’s attitude is a result of the dynamic adjusting process for the center of gravity, the center of buoyancy, and the metacenter balance. Similarly, during the porpoising process, maintaining the three-center dynamic balance becomes difficult because of the pitch–heave coupling motion. Under such a circumstance, the navigation attitude, interfering with the balance considerably, is critical in hull navigation stability performance. However, the relationship between the navigation attitude (pitch and heave) of the planing hull and the speed is not a simple positive correlation; it typically reverses after the resistance peak. Therefore, whether the planing hull will maintain the dynamic balance depends on the three-center dynamic balance and navigation attitude matching the increase in speed. Similar phenomena have been observed by Kayatama [29], Hongjie Ling [30], and Yuan [14]. However, in most studies on porpoising, the speed at which porpoising occurs has been considered to be extreme. Limited studies have focused on the oscillation trend and related mechanisms of the hull motion response of a planing trimaran after porpoising at extreme speeds.

The purpose of this experiment was not to study the extreme speed at which a planing trimaran experiences porpoising but rather to investigate the movement characteristics of porpoising. Towing tests were conducted at a higher speed interval after the speed at which porpoising occurred. Not only the resistance performance, but also the longitudinal stability of a planing trimaran at various positions of the center of gravity and moment of inertia were investigated. The experimental data of the full-speed-range motion response of a planing trimaran revealed secondary porpoising of the planing trimaran. The effects of the design parameters, such as the longitudinal location of the center of gravity, the moment of inertia on the navigation attitude, and the longitudinal motion stability of the planing trimaran were investigated.

2. Towing Tank Tests

2.1. Physical Description of the Model

Figure 1 illustrates the proposed planing trimaran model. The step of the planing trimaran was set at a distance of 0.28 times the length of the boat. The specific model design parameters are presented in

Table 1. Model parameters.

Main Size	Parameter Notation	Parameter Value
Length overall	LOA (mm)	1300.00
Breadth moulded	BOA (mm)	470.00
Main-hull beam	B (mm)	230.00
Deadrise angle	β (deg)	17.00
Tunnel height	HT (mm)	34.00
Tunnel beam	BT (mm)	93.21
Tunnel length	LT (mm)	370.00
Demi-hull length	LDOA (mm)	320.00
Demi-hull height	HD (mm)	22.80
Demi-hull beam	BD (mm)	8.80
Step height	HS (mm)	5.00
Model scale	λ	10

.

2.2. Experimental Setup and Data Processing

A towing test was conducted according to the ITTC recommendations for the towing tests of high-speed marine vehicles. The towing tank of the China Special Vehicle Research Institute (i.e., No. 605 Subsidiary Research Institute, Aviation Industry of China Group), located in Jingmen, China, was used for the test.

The length, width, and depth of the towing tank were 510, 6.5, and 6.8 m, respectively. The towing tank corollary carriage towing system can run at a steady speed ranging from a minimum of 0 m/s to a maximum of 25 m/s, limiting the speed error to under 0.1%. The other main parameters, equipment, sensors, and calibration information of the towing tank are presented in

Table 2. General parameters of the towing tank.

Main Feature	Unit Type	Value (Range)	Accuracy
Length of the tank (m)	--	510	--
Width of the tank (m)	--	6.5	--
Water depth of the tank (m)	--	5	--
Density of towing tank water (kg/m3)	--	12.07	--
Kinematic viscosity (10−7 m2/s)	--	999.38	--
Temperature of water (°C)	--	13℃	--
Carriage system (m/s)	non-standard	0~25	0.1%
Dynamometry sensor	U3B1-50K-B	50 kg	0.01 kg
Electronic angle sensor	02111102-000	±60°	0.02°
Cable-extension displacement sensor	CLMD2-AJ1A8P01500	500 mm	0.1%
Inertial measurement units (IMU)	IMV610H	Pith: ±60° Roll: ±180°	dynamic accuracy <0.3°
Electronic hoist scale	DR150	0~150 kg	0.02 kg
Rotational inertia measurement instrument	ZGT-2000	<500 kg	<1%

.

In the current study, lightweight special foam material with a geometrical scale factor of λ = 10 was used to manufacture the planing trimaran model. The surface was covered with fiberglass and sanded and painted to smoothen the surface line. The demi-hull was pre-embedded with wood on both sides to strengthen the towing point and the installation position of the navigation groove. The hull was covered with light waterproof paper and treated with a waterproofing treatment. The free towing method was used in the towing tests. In the tests, the model was attached to the carriage system with two degrees of freedom (pitch and heave). To assure the accuracy of the towing point layout, the height in the vertical direction and length in the longitudinal direction were accurately measured, which rendered the towing points on both sides collinear with the gravity center of the hull. Sensors and devices measuring the heave amplitude, resistance, and pitch angle were mounted to the hull. The experimental setup is displayed in Figure 2b. The position of the center of gravity and the initial coordinate system are illustrated in Figure 2c.

The moment of inertia investigated in this study refers to the moment of inertia around the transverse axis at the center of gravity, specifically Iyy, with the reference axis being the y-axis shown in Figure 2d. When changing the working conditions, the counterweight was moved longitudinally without changing the height, in order to modify the moment of inertia without changing the center of gravity position. During the adjustment process, repeated measurements of the position of the center of gravity and the moment of inertia were necessary for ensuring that the center of gravity remained unchanged while changing the moment of inertia. The center of gravity and the moment of inertia were both measured using a mass, center of mass, and inertia measurement system, a model of which is presented in Table 2.

First, the resistance and navigation attitude of the hull with the center of gravity in various positions were tested. In the second stage, the counterweight position was adjusted to change the moment of inertia, and the center of gravity was maintained. Furthermore, the relationship between the moment of inertia and the longitudinal motion stability of the planing trimaran was tested. The seven test cases are listed in

Table 3. Test conditions.

Cases	Displacement	Gravity Center (GX, GY, GZ)	Moment of Inertia (Iyy)
A1	10.04 kg	(400, 0, 90) mm	1.200 kg·m2
A2		(400, 0, 90) mm	0.987 kg·m2
A3		(350, 0, 90) mm	1.064 kg·m2
A4		(350, 0, 90) mm	1.198 kg·m2
A5		(350, 0, 90) mm	0.903 kg·m2
A6		(300, 0, 90) mm	1.020 kg·m2
A7		(300, 0, 90) mm	1.190 kg·m2

.

For a high-speed planing trimaran, the length of the waterline changes dynamically with the speed because of the change in the attitude during navigation. Therefore, the Froude number of the ship’s width or volume are typically used as the similarity criteria, and their forms are $F{r}_B=V/\sqrt{gB}$ and $F{r}_{\nabla }=V/\sqrt{g{(\nabla )}^{1/3}}$, respectively. For the stepped planing trimaran discussed in this paper, when the step was in the skidding state, the actual width of the sliding surface was less than the width of the planing trimaran, so the volume Froude number was used as the similarity criterion of the stepped planing trimaran in this paper.

Figure 3a displays the pulsation monitoring and acquisition time–history curve of each physical parameter of the data collection system in the trailer acceleration section, the stable towing section, and the trailer deceleration sections when the model was under condition A4 with a towing speed of $F{r}_{\nabla }=8.25$. In model towing tests, the model should fully accelerate to the preset speed, and wave-making should be fully developed to collect and record the experimental results of resistance and navigation attitude. The model speed was fully stabilized in the second half of the towing of the trailer. The time–history curves of the monitored physical parameters in this navigation segment were effective data, which included predictions of the resistance and the motion attitude response of the model at corresponding speeds.

Figure 3 reveals that porpoising occurred in the smooth operation stage of the hull model for the planing trimaran, and the time–history curves of the related physical parameters of resistance and motion attitude exhibited considerable periodic vibrations. For the processing of the experimental data, the results generally represent the average using the time-domain mean of significant values for the data collection time–history curve. Notably, the resistance and navigation attitude amplitude after stabilization in the model rapidity test were stable, as displayed in Figure 3b. Therefore, in this study, the time-domain means in several stable oscillation periods were the data-averaging results.

In this study, after the planing hull navigation attitude became stable, the instantaneous value of the physical parameter data collected by the test monitoring at any time t can be expressed as follows:

$$\xi (v,t)=\overline{\xi }(v)+{\xi }^{’}(v,t)$$

(1)

where $\overline{\xi }$ is the time-domain mean of the monitored physical parameters, and the pulsation value ${\xi }^{’}(v,t)$ is a function of the speed and time.

The averaging of the time-domain and pulsation zero mean values of the physical parameter data collected in the test was performed according to the equation. The mean term $\overline{\xi }$ after the time-domain averaging process represents the variation trend of the resistance and the model motion attitude with the towing speed. Similarly, the pulsation term ${\xi }^{’}(v,t)$ after processing the zero mean represents the time–history oscillation intensity and frequency of the monitored physical parameters at a fixed speed.

Figure 4 displays the trim angle variation of a planing trimaran under various speeds in A4. As displayed in Figure 4a, the trim exhibits porpoising characteristics with increases in the towing speed. Moreover, Figure 4b presents the temporal variation in the trim angle at various speeds where porpoising occurs, providing an intuitive comparison of the vibration characteristics and trends of porpoising at different speeds.

3. Experimental Results and Analysis

In this paper, to discuss the model’s oscillation motion-related parameters and indicators (e.g., the oscillation period and amplitude, frequency) expediently, a naming method is applied. M_XYZ is defined as follows: M presents the physical quantity of the vibration period. The data range of the symbol M contains T and A, where T indicates the time of the vibration period and A indicates the oscillation amplitude. Here, X represents the exact kind of towing test result considered. When the X value is P, the vibration parameter is the statistical data on pitch; H denotes heave; and R denotes resistance. Furthermore, Y represents the towing case number, which corresponds with the towing scheme number. Furthermore, Z is the towing speed. For example, $T_{PA311}$ indicates the time of the vibration period of the pitch in towing scheme A3 at a speed of 15 m/s. The prefix and subscript value conventions are presented in

Table 4. Naming scheme for parameters related to hull oscillatory motion.

Prefix and Subscript	Value Definition	Representational Meaning
M	T	Time
	A	Amplitude
	f	Frequency
X	P	Pitch
	H	Heave
	R	Resistance
Y	A1, A2, …, A7	Cases no.
Z	01, 02, …, 15	Speed (m/s)

.

3.1. Hull Motion Behavior Evolution Process with Increasing Speed

Figure 5 displays the navigation attitude and wave-making form of the planing trimaran at various speeds under test condition A3. At low speeds, the planing trimaran was in the drainage navigation attitude. Although the trim of the vessel was small, the draft was deep and submerged the tunnel in water. The hull generated almost no splash, and the slipstream was flat.

When the towing speed reached $F{r}_{\nabla }=2.75$, the hull transitioned from the draining and semi-planing state to the planing state. With the increase in speed, obvious water splashes appeared near the demi-hull and tunnel, and the trim increased sharply. Behind the stern seal plate, a cocktail flow and a Kelvin wave with a larger angle were generated.

At a towing speed of $F{r}_{\nabla }=4.13$, the hull was lifted out of the water due to the hydrodynamic lift provided by the bottom tunnel. With the increase in speed, the hull entered the planing state, causing the Kelvin angle to gradually decrease.

At $F{r}_{\nabla }=6.19$, the waves and splashes generated by the main hull were absorbed by the tunnel, and only a small number of splashes were observed on both sides of the hull. After the wave-making in the tunnel was blocked by the inner wall, the wave-making was separated from the hull at the stern of the tunnel and extended backward at a small angle.

The results show that as the towing speed increased, the planing trimaran underwent a transition from a drainage navigation attitude to a planing state, generating considerable water splashes and causing the Kelvin angle to gradually decrease. Additionally, the use of the bottom tunnel provided hydrodynamic lift, which enabled the hull to lift out of the water and reduced the wave-making.

Notably, with the increase in speed, the model exhibited considerable periodic porpoising within the $7.57\le F{r}_{\nabla }\le 8.25$ speed range. In this case, the periodic oscillation of the hull provoked a strong slam splatter. As displayed in Figure 6, when the model was pitched into the water, a small splash was generated at the initial stage and absorbed by the tunnel. Next, with the intensification of the splash in the water, the splash could not be completely intercepted and absorbed by the tunnel, which resulted in the appearance of the splash on the outside of the demi-hull. Finally, the hull was gradually lifted, and the demi-hull sputtering phenomenon and attacking wave-making gradually weakened and disappeared, and porpoising started in the next period.

With 8.94 ≤ $F{r}_{\nabla }$ ≤ 9.63, the hull regained stability. In this case, the stagnation line of the main-hull waterplane moved forward as the trim angle decreased, and a small number of whisker-like splashes from the bottom of the boat exceeded the masking range of the demi-hull. With the increase in the towing speed, at the towing speed point $F{r}_{\nabla }=10.32$, a discernible slight oscillation occurred again. Similar phenomena occurred in test conditions A3 and A5. The speed range of the porpoising varied depending on the test parameters, but the amplitude of the porpoising oscillations did not increase or diverge with the increase in the speed and gradually decreased or even restabilized.

3.2. Effects of the Position of the Center of Gravity on the Porpoising of the Planing Trimaran

A comparison of the resistance and pitch angle test results for A1, A4, and A7 is shown in Figure 7. The resistance of the trimaran planing boat was greatly affected by the longitudinal position of its center of gravity. When the center of gravity moved backward, the resistance decreased, and the trim angle increased.

Figure 8 presents a comparison of the pitch and heave amplitudes and periods of the planing trimaran at various centers of gravity. The results indicate a significant correlation between the longitudinal position of the center of gravity and the porpoising motion of the trimaran. According to the sailing posture of the trimaran in Figure 7, a forward shift in the center of gravity led to a decrease in the average pitch angle and heave value during this speed range, which increased the critical speed at which porpoising occurred. For example, the longitudinal instability range of A7 was 6.19 ≤$F{r}_{\nabla }$≤ 8.94, the heave motion pulsation peak remained at a speed point of $F{r}_{\nabla }=7.57$, and the average heave amplitude reached 6.377 mm. The oscillation peak of the pitch pulsation in the range occurred at a speed point of $F{r}_{\nabla }=8.25$. At this stage, the average amplitude of the pitch oscillation reached 3.323°. However, the longitudinal instability range of A4 was reduced to 6.19 ≤ $F{r}_{\nabla }$ ≤ 8.94, and the heave motion pulsation peak remained at the speed point of $F{r}_{\nabla }=7.57$. At this stage, the average heave amplitude reduced to 4.923 mm, the pitch pulsation oscillation peak in the range was moved to a speed point of $F{r}_{\nabla }=8.94$, and the average oscillation amplitude increased to 4.822°.

As shown in Figure 9, the vertical force acting on the hull in front of the center of gravity is defined as $N_{FG}$, and the vertical force acting on the hull behind the center of gravity is defined as $N_{AG}$. The moment generated by $N_{FG}$ on the transverse axis of the center of gravity is $M_{FG}$, and the moment generated by $N_{AG}$ on the transverse axis of the center of gravity is $M_{AG}$. When the planing trimaran was in stable navigation, the sum of $N_{AG}$, $N_{FG}$, and gravity was zero, and the sum of $M_{FG}$ and $M_{AG}$ was also zero. When porpoising occurred, the hull underwent periodic oscillations under the influence of hydrodynamic and aerodynamic loads. As the center of gravity gradually moved forward, the moment arm $L_{AG}$ between the force $N_{AG}$ and the gravitational force increased gradually, causing an increase in $M_{AG}$, which in turn exacerbated the pitching oscillation. On the contrary, the trim decreased as the center of gravity moved rearward. This resulted in a larger vertical component of the hydrodynamic and aerodynamic force, and a corresponding increase in $N_{AG}$ and $N_{FG}$, resulting in a larger amplitude of heave motion.

3.3. Effects of the Moment of Inertia on the Porpoising of the Planing Trimaran

In order to study the effects of the moment of inertia on the porpoising of the planing trimaran, the test kept the displacement and center of gravity position of the planing trimaran constant under A4, and three different moments of inertia were obtained by adjusting the counterweight position, which were $I_3$= 1.064, $I_4$= 1.198, and $I_5$= 0.903. As shown in Figure 10, the time averages of the trim and resistance for the three test conditions were compared, and it was found that the moment of inertia had very little effect on the time averages of the trim and resistance of the planing trimaran.

To analyze the effects of the moment of inertia on the porpoising of the planing trimaran, the effects of the moment of inertia on the unstable motion speed range of the planing hull and its oscillation amplitude in this range were tested, and the average oscillation amplitude and oscillation period of multiple stable oscillation periods in the A3, A4, and A5 solutions were calculated (Figure 11).

As displayed in Figure 11, the moment of inertia exhibited a significant correlation with the longitudinal unstable motion speed range of the hull and its oscillation amplitude in this range. The longitudinal unstable motion speed range moved backward and expanded with the increase in the moment of inertia. Furthermore, with the increase in the moment of inertia, the motion attitude of the hull and the oscillation of resistance intensified, its oscillation amplitude peak increased rapidly, and the speed point of the amplitude pulsation peak moved slightly backward. For example, A5 only exhibited a pitching motion with an amplitude of A_PA511 = 0.717° at a speed point of $F{r}_{\nabla }=7.57$, and then its oscillation amplitude decreased rapidly, and the longitudinal stability motion was restored. With the increase in the speed, the longitudinal instability occurred first in the $F{r}_{\nabla }=9.63$ range again, which resulted in second porpoising, and the pitch oscillation amplitude increased to A_PA514 = 1.583°. The first pitching instability range of A3 and A4 gradually moved backward and expanded to $7.57\le F{r}_{\nabla }\le 8.25$ and $7.57\le F{r}_{\nabla }\le 8.94$, respectively. The speed point of the pitch amplitude peak moved backward to $F{r}_{\nabla }=8.25$ and $F{r}_{\nabla }=8.94$. At this stage, severe pitching oscillations with amplitudes of A_PA312 = 3.751° and A_PA413 = 4.822° were observed. With a further increase in speed, condition A3 lagged behind A5, and a second porpoising with an amplitude of 1.593° occurred at $F{r}_{\nabla }=10.32$.

The smaller the moment of inertia of a planing trimaran, the less moment is required to change its pitch state, and its motion state is also easier to change. Therefore, when the time mean of the navigation attitude was similar, the planing trimaran with a low moment of inertia caused porpoising at a lower speed (less external force) and recovered a stable navigation attitude quickly with a recovery force (moment). By contrast, the hull with a larger moment of inertia exhibited the opposite tendency, and the oscillation amplitude increased. The moment of inertia considerably influenced the longitudinal stability of the planing trimaran. Because the longitudinal instability range and longitudinal unstable motion amplitude determined the degree of damage to the navigation, a reasonable reduction in the moment of inertia enabled the planing trimaran to pass the vibration peak of the first longitudinal motion smoothly, which improved the longitudinal stability of the planing hull.

4. Analysis and Prediction of Porpoising of Planing Trimaran

4.1. Analysis of Secondary Porpoising of Planing Trimaran

Studies have revealed that porpoising is a type of coupling motion of pitch and heave [29]. When porpoising occurs, cyclical pitch and heave motions are both generated. Furthermore, because of the characteristics of the planing hull, the pitch motion was more observable than the heave. Day and Haag [27] used the actual criterion for porpoising in the model test in which the amplitude of the pitch motion was greater than 1° and accompanied by a heave motion with a certain amplitude. In the current study, the Day and Haag criterion was followed, and only the pitch motion was used as the criterion, whereas the heave motion was considered as a porpoising severity comparison scale. This criterion was used to define porpoising in this study.

To compare the variation trend of the motion characteristics with speed after the planing trimaran entered porpoising, the data collection results of the trim and heave time–history in the range of $6.88\le F{r}_{\nabla }\le 10.32$ in test plan A3 were zero-averaged. The pulsation terms ${\xi }^{’}(v,t)$ of the corresponding monitored physical parameters were obtained. After the phase difference elimination process, the pulsation values was plotted in Figure 12. In the test condition A3, with the increase in speed, the model exhibited bounded pitch vibrations with amplitudes of A_PA311 ≈ 1.98° and A_PA312 ≈ 2.89° at speed points of $F{r}_{\nabla }=7.57$ and $F{r}_{\nabla }=8.25$, respectively. The pitch period was reduced from T_PA311 ≈ 0.31 s to T_PA311 ≈ 0.29 s. Next, a small vibration of amplitude occurred at the towing speed points of $F{r}_{\nabla }=8.94$ and $F{r}_{\nabla }=9.63$, and the amplitudes were similar with values of 0.45°. The increase in speed did not cause continuous pitch vibrations. The model recovered longitudinal stability, until the pitching motion with an amplitude of A_PA315 ≈ 1.48° occurred again at a speed of $F{r}_{\nabla }=10.32$, and the period continuously decreased to T_PA315 ≈ 0.25 s.

In test condition A3, a considerable bounded period heave motion occurred at the two speed points of $F{r}_{\nabla }=7.57 , 8.25$, and the heave amplitude periods were A_HA311 ≈ 4.22 mm and T_HA311 ≈ 0.31 s, and A_HA312 ≈ 3.89 mm, and T_HA312 ≈ 0.29 s. After the heave occurred, the amplitude rapidly increased to 4.22 mm at $F{r}_{\nabla }=7.57$, and the amplitude gradually decreased and eased. However, no significant heave motion occurred at the drag speed point of $F{r}_{\nabla }=10.32$.

Under test condition A3, porpoising was observed at the speed points of $F{r}_{\nabla }=7.57 , 8.25$. As the stable periodic bounded vibration occurred, the pitch and heave vibration periods were the same and gradually decreased as the speed increased. Therefore, the vibration frequency increased. When $F{r}_{\nabla }\ge 8.94$, the pitch and heave amplitude decreased gradually with the increase in speed and returned to the amplitude level at $F{r}_{\nabla }=6.88$. However, secondary porpoising occurred at a speed point of $F{r}_{\nabla }=10.32$, and at this time, the vibration period had a certain recovery compared to the towing speed point of $F{r}_{\nabla }=10.32$. In terms of the experimental phenomenon, the porpoising evaluation using the Day and Haag criterion was consistent with the intuitive observation results of the experimental phenomenon. When the model was excited during sailing, longitudinal unstable motion occurred, and the oscillation amplitude gradually increased with the increase in the speed but decreased after porpoising, and the hull motion became stable. Next, longitudinal instability occurred again at $F{r}_{\nabla }=10.32$, and secondary porpoising occurred. Additionally, the occurrence of porpoising under each towing condition was determined, and the relationship between each speed and the porpoising of a series of towing conditions was obtained, as presented in

Table 5. Variation in speed longitudinal motion stability performance of planing trimaran under different towing conditions.

Speed (m/s)	Fr∇	A1	A2	A5	A3	A4	A6	A7
8	5.47	S	S	S	S	S	S	S
9	6.15	S	S	S	S	S	S	S
10	6.83	S	S	S	S	S	U	U
11	7.52	S	S	Semi-U	U	U	U	U
12	8.20	S	S	S	U	U	U	U
13	8.88	U	S	S	S	U	U	U
14	9.57	U	S	U	S	S	S	S
15	10.25	U	Semi-U	U	U	S	S	S

.

As presented in Figure 5, U denotes unsteady, where the hull had longitudinal motion instability, whereas S indicates steady, that is, the hull did not exhibit porpoising. Semi-U represents the slight vibration of hull. At this stage, the vibration was not obvious from the direct observation of the test model. To intuitively observe the relationship between the gradual increase in the moment of inertia of the conditions A3, A4, and A5 and the porpoising speed range, the results are arranged in the order of A5, A3, and A4 in the table. In Table 5, A2 exhibits a trim amplitude of A_PA215 ≈ 0.715° in the hull at the speed point of $F{r}_{\nabla }=10.32$. And A5 corresponds to a slight pitch vibration with an amplitude of A_PA511 ≈ 0.717° in the hull at the speed point of $F{r}_{\nabla }=7.57$. Therefore, this phenomenon is represented by Semi-U. As observed, the planing trimaran exhibited multiple stable states under the A3–A7 towing conditions; the hull was exposed to secondary porpoising under test conditions A3 and A5. Additionally, assuming that the towing test speeds were increased, the planing trimaran hull configuration in the test produced similar secondary longitudinal motion instability in conditions A4, A6, and A7.

4.2. Limit Curve Method for Predicting Porpoising of the Planing Trimaran

To determine the relationship between the porpoising and speed of the planing trimaran, the Clement method was introduced based on the stability test data of the TMB Series 62 [31]. As displayed in Figure 13, the exponential relationship curve between the planing trimaran hull dynamic load coefficient and the speed is drawn, and the curve function form is expressed as follows [32]:

$${\left[\frac{C_{LB}}{x_g/B}\right]}_{critical}=mF{r}_{\nabla }{}^n$$

(2)

where $m$ and $n$ are undetermined coefficients, and $C_{LB}=\Delta /0.5\rho V^2{B}^2$ is the dynamic load coefficient that represents the dynamic lift level of the hull.

A larger width B of the planing trimaran ensures that the coefficient $C_{LB}/\left(x_g/B\right)$ can exceed the stability critical line even at higher speeds, so the planing trimaran can have superior longitudinal motion stability.

According to the experimental results, the dimensionless calculation results of the dynamic load coefficient and the position of the center of gravity for the upper limit of the longitudinal motion stability speed and the lower limit of the motion instability speed are given.

In the

Table 6. Extreme instability speeds of the longitudinal motion of the planing trimaran under various test conditions.

Cases	B(m)	xg	$F{r}_{\nabla stb}$	$F{r}_{\nabla ust}$	CLBstb	CLBust	$\raisebox{1ex}{${\mathit{C}}_{\mathit{L}\mathit{B}\mathit{s}\mathit{t}\mathit{b}}$}\!\left/ \!\raisebox{-1ex}{${\mathit{x}}_{\mathit{g}}/\mathit{B}$}\right.$	$\raisebox{1ex}{${\mathit{C}}_{\mathit{L}\mathit{B}\mathit{u}\mathit{s}\mathit{t}}$}\!\left/ \!\raisebox{-1ex}{${\mathit{x}}_{\mathit{g}}/\mathit{B}$}\right.$
A2	0.23	0.40	8.88	9.57	0.022461	0.019367	0.012915	0.011136
A3	0.23	0.35	6.83	7.52	0.037958	0.031371	0.024944	0.020615
A6	0.23	0.30	6.15	6.83	0.046862	0.037958	0.035927	0.029101

, $F{r}_{\nabla stb}$ represents the upper limit of the speed at which the planing trimaran maintained a stable state of navigation, and $F{r}_{\nabla ust}$ represents the lower limit of the speed at which porpoising occurred. The least square method was used to fit each towing condition hull stability critical curve in the test. The longitudinal stability critical curve fitting result was obtained using formula (3). The shaded area above the curve corresponds to the longitudinal motion instability range as follows:

$${\left[\frac{C_{LB}}{x_g/B}\right]}_{critical}=\frac{4.8473}{F{r}_{\nabla }{}^{2.694}}$$

(3)

To verify the accuracy of the margin of the stability curve on the prediction of the longitudinal motion stability of the experimental hull configuration, Figure 13 displays the results of the inspection of the curve function values of towing condition A3 at various speeds.

Because of the conventional margin of the stability curve, only the speed points near the longitudinal margin of stability, that is, the maximum stable speed ($F{r}_{\nabla }stb$) and the minimum unstable speed ($F{r}_{\nabla }ust$), were collected for curve fitting. Therefore, after unstable longitudinal motion occurred at $F{r}_{\nabla }=7.57$ in towing condition A3, the calibration curves were in the instability range above the margin of the stability line.

However, when the planing trimaran was traveling at speeds of $F{r}_{\nabla }=8.94$ and $F{r}_{\nabla }=9.63$, the pitch motion amplitudes were 0.438° and 0.288°, respectively, and the heave motion amplitudes were 0.41 and 0.241 mm, respectively. According to the judgment criteria set by Day and Hagg [27], the planing was in a stable straight running state.

However, when $F{r}_{\nabla }$ is 8.94, $\frac{C_{LB}}{x_g/B}=0.015>{\left[\frac{C_{LB}}{x_g/B}\right]}_{critical}=0.013$.

When $F{r}_{\nabla }$ is 9.63,$\frac{C_{LB}}{x_g/B}=0.013>{\left[\frac{C_{LB}}{x_g/B}\right]}_{critical}=0.011$.

Based on the stability limit line chart, both speeds were in the unstable range. Therefore, a discrepancy exists between the experimental observations and the predicted results.

Therefore, the conventional margin of stability cannot provide an accurate forecast for the secondary stability (i.e., secondary porpoising) of the planing trimaran because of the reduction in the hull vibration. Thus, certain limitations exist. This study revealed the limitations of Clement’s conventional method of predicting the longitudinal stability of a planing boat to determine the porpoising of a trimaran. The model can predict the first porpoising but not the secondary stability (i.e., secondary porpoising) accurately.

5. Conclusions

The oscillation characteristics and mechanisms of porpoising in planing trimarans were analyzed in this study. This study investigated the effects of the center of gravity location, the moment of inertia, and speed on the oscillation characteristics of porpoising in planing trimarans. Additionally, the study observed secondary porpoising during experiments, leading to the following conclusions:

(1)

Previous studies have indicated that moving the center of gravity forward increases the lower speed limit for porpoising in planing trimarans. However, this maneuver may also cause increased resistance [5,33]. In this experiment, it was found that moving the center of gravity forward not only increased the resistance of the planing trimaran, but also increased the amplitude of the porpoising. The position of the center of gravity of A4 moved forward by 16% compared to A7 and the lower speed limit for porpoising increased from $F{r}_{\nabla }=6.9$ to $F{r}_{\nabla }=7.59$, but the maximum pitch oscillation amplitude increased by 45%. Since the amplitude of the porpoising is one of the important factors in evaluating the safety performance of a planing trimaran, this finding is very important for the future design of planing trimarans. In the future research of porpoising and the design process of planing trimarans, attention needs to be paid to the increase in the oscillation amplitude caused by the center of gravity adjustment.

(2)

In this experiment, after comparing the changes in the trim and resistance of the planing trimaran at different moments of inertia, it was found that although the moment of inertia had little effect on the time-domain mean of parameters, such as the trim and resistance, it had a significant effect on porpoising. The moment of inertia of A5 was reduced by 15% compared to A3, and the amplitude of the maximum pitch oscillation during porpoising was reduced by 60%. With the increase in the moment of inertia, the hull oscillation was intensified, the oscillation amplitude peak increased rapidly, and the speed point of the amplitude pulsation peak also moved slightly backward. This finding provides valuable insights and references for researching porpoising in planing trimarans. Future studies can investigate the effects of the moment of inertia on porpoising to explore the underlying mechanisms of this phenomenon in planing trimarans, and ultimately drive innovation and advancement in planing trimaran design.

(3)

In the towing tests of the planing trimaran, the phenomena of secondary stability and secondary porpoising were observed. That is, when the planing trimaran experienced porpoising and the speed continued to increase, the planing trimaran gradually returned to stable motion, but it eventually experienced porpoising again. Currently, to avoid damage to the hull and passenger, it is usually necessary to reduce speed to eliminate porpoising when it occurs. The discovery of secondary stability and secondary porpoising provides new ideas and methods for improving the maximum speed of planing trimarans, which is of great significance to improve the porpoising stability of planing trimarans.

(4)

Planing hull porpoising is caused by the dynamic imbalance of force and momentum acting on the hull. To study the porpoising of planing trimarans, accurate time–history force analysis should be conducted in a future study. Therefore, the mechanism of porpoising of the planing trimaran base in the current and other towing test results using the CFD method was proposed to analyze the force change process to investigate the porpoising mechanism. Furthermore, numerical simulations can be used to evaluate the relationship between the parametrization design of the planing trimaran (e.g., the main-hull design parameters and the tunnel and demi-hull design parameters) and porpoising. These results may provide a design reference for planing trimarans.