Experimental Calculation of Added Masses for the Accurate Construction of Airship Flight Models

Abstract

In recent years, interest in airships for cargo transport and stratospheric platforms has increased, necessitating accurate dynamic modeling for stability analysis, autopilot design, and mission planning, specifically through the calculation of stability derivatives, like added mass and inertia. Despite the several CFD methods and analytical solutions available to calculate added masses, experimental validation remains essential. This study introduces a novel methodology to measure these in a wind tunnel, comparing the results with prior studies that utilized towing tanks. The approach involves designing the test model and a crank-slider mechanism to generate motion within the wind tunnel, considering load cell sensitivity, precision, frequency range, and Reynolds numbers. A revolution ellipsoid model, made from extruded polystyrene, was used to validate analytical solutions. The test model, measuring 1 m in length with an aspect ratio of 6, weighing 482 g, was moved along rails by the crank-slider system. By increasing the motion frequency, structural vibrations affecting load cell measurements were minimized. Proper signal processing, including high-pass filtering and second-order Fourier series fitting, enabled successful virtual mass calculation, showing only a 2.1% deviation from theoretical values, significantly improving on previous studies with higher relative errors.

1. Introduction to Flight Models for Airships: From Equations of Motion to Stability Derivative Coefficients

Interest in airships has received renewed impulse during the last decade prompted by two great potential applications: one as surveillance and communications platforms with very long persistence in the stratosphere (known as HAPS or High Altitude Pseudo-Satellites, e.g., [1]), and another as an alternative, more efficient platform for air transport of heavy payloads [2].

The development of these vehicles requires provision of an autonomous flight capability that avoids the need for people on-board. This makes it important to have a highly precise flight model of the airship that helps to ensure adequate controllability by the autopilot. However, airship flight dynamics notably differ from those of airplanes. As a highly representative example, the effect of added masses or moments of inertia is usually irrelevant for airplanes, while they have a significant impact on airships. In the following sections, airship flight modeling is reviewed. We highlight the importance of the dynamic terms within the stability derivatives of such models and discuss how the added masses could be measured.

1.1. Equations of Motion

The definition of a flight dynamics model for an airship will typically be carried out by considering it as a rigid body with three translational and three rotational degrees of freedom. Therefore, the dynamic model is composed of six differential equations that describe the dynamics of translation (Equations (1)–(3)) and rotation (Equations (4)–(6))–known as equations of motion. In this case, they are written in body axes and the mass and inertia are considered as constant (adapted from [3])

$$\sum F_x=m\left(\dot{u}+wQ-vR\right),$$

(1)

$$\sum F_y=m\left(\dot{v}+uR-wP\right),$$

(2)

$$\sum F_z=m\left(\dot{w}+vP-uQ\right),$$

(3)

$$\sum M_x=\dot{P}{I}_{xx}+\left(I_{zz}-I_{yy}\right)QR-\left(P^2-R^2\right)I_{yz},$$

(4)

$$\sum M_y=\dot{Q}{I}_{yy}+\left(I_{xx}-I_{yy}\right)PR-\left(\dot{R}+PR\right)I_{yz},$$

(5)

$$\sum M_z=\dot{R}{I}_{zz}+\left(I_{yy}-I_{xx}\right)PQ-\left(\dot{Q}-QP\right)I_{yz},$$

(6)

where $m$ is the mass of the airship, $I_{ii}$ are the elements of its inertia tensor and, according to the sign convention and the reference system defined for the airship depicted in Figure 1, $P$, $Q$ and $R$ are angular velocities for the roll, pitch and yaw. Furthermore, the airship is considered to be axisymmetric, thus $I_{xz}=I_{xy}=0$.

The integration of the equations of motion, knowing the external forces ($\sum F_x$, $\sum F_y$, $\sum F_z$) and moments ($\sum M_x$, $\sum M_y$, $\sum M_z$) that are applied to it, allows determination of the new state of the body at a later instant. However, this integration process requires the ability to estimate, for each state of the airship, the magnitude of such aerodynamic forces and moments. Thus, a mechanism to obtain forces and moments from the state of the aircraft must be defined.

For this, one of the most common procedures is to linearize the equations of motion, which can be carried out under the assumption that the movement of the aircraft is restricted to small perturbations over an equilibrium flight condition. Linear dynamic models have been widely applied to study the behavior of airships in flight, as in [4], as well as for control design, as in the work presented in [5] for a high-altitude airship. These linear models contain the derivatives of the external force or moment with respect to the corresponding states of the aircraft, resulting in the terms usually called aircraft stability coefficients. These are depicted in Equations (7)–(12).

$$\sum F_x=F_{x0}+F_x^u·u+F_x^{\dot{u}}·\dot{u},$$

(7)

$$\sum F_y=F_{y0}+F_y^v·v+F_y^{\dot{v}}·\dot{v},$$

(8)

$$\sum F_z=F_{z0}+F_z^w·w+F_z^{\dot{w}}·\dot{w},$$

(9)

$$\sum M_x=M_{x0}+M_x^{\varphi }·∆\varphi +M_x^{\dot{\varphi }}·∆\dot{\varphi }+M_x^P·P+M_x^{\dot{P}}·\dot{P},$$

(10)

$$\sum M_y=M_{y0}+M_y^{\alpha }·∆\alpha +M_y^{\dot{\alpha }}·∆\dot{\alpha }+M_y^Q·Q+M_y^{\dot{Q}}·\dot{Q},$$

(11)

$$\sum M_z=M_{z0}+M_z^{\psi }·∆\psi +M_z^{\dot{\psi }}·∆\dot{\psi }+M_z^R·R+M_z^{\dot{R}}·\dot{R}.$$

(12)

Those stability coefficients include both static and dynamic terms. Static aerodynamic coefficients have been extensively studied in the literature for a wide range of airship shapes and tail configurations. Thus, they are not discussed in this work. Some relevant conclusions are summarized in [6]. Regarding the dynamic examples, these represent the forces and moments produced by the velocity and acceleration of the body, instead of its position or orientation. They are more difficult to predict, as they involve transient situations that are harder to reproduce and measure.

1.2. Airship Stability Derivatives and Theory of Virtual or Added Mass

Among the dynamic terms for the stability derivatives, there are those referring to the virtual masses and moments of inertia, also known as added masses and moments ($F_x^{\dot{u}}, F_y^{\dot{v}}, F_z^{\dot{w}}$ and $M_x^{\dot{P}}, M_y^{\dot{Q}}, M_z^{\dot{R}}$). Within this work, we will focus only on the force terms and linear displacements; torques and angular velocities will be out of the scope of this document, and could be addressed in future work.

The aerodynamic effect of virtual masses and moments usually generates confusion and is mostly misunderstood (a detailed explanation of the theoretical foundations can be found, among other sources, in the works of Imlay [7] and Lamb [8]). However, it is of special relevance for the dynamics of an airship. When a body accelerates, decelerates, or changes its direction while moving in a fluid, it behaves as if it had more mass than it really does, in such a way that, for the forces that are apparently being applied to it, the resulting accelerations are lower than expected.

The apparent increase in mass, as well as the distribution of this added mass, vary with the nature of the movement. This explains why it is necessary to study and characterize it precisely to construct the dynamic model of the vehicle. Although the added mass affects the motion of any object that accelerates or decelerates in a surrounding fluid, its effects are only significant when the mass of the object is similar to that of the displaced fluid, a situation typical of vehicles such as airships, submarines or parachutes. That is the reason why its study is not considered when performing the dynamic analysis of conventional fixed wing or rotary wing aircraft.

The added mass derivatives are usually expressed as proportionality factors $k_i$, which define the amount of added mass as a fraction of the displaced mass of the surrounding fluid, as in Equations (13)–(15).

$$F_x^{\dot{u}}={\rho }_{\mathrm{fluid}}·\mathrm{Vol}·k_1,$$

(13)

$$F_y^{\dot{v}}={\rho }_{\mathrm{fluid}}·\mathrm{Vol}·k_2,$$

(14)

$$F_z^{\dot{w}}={\rho }_{\mathrm{fluid}}·\mathrm{Vol}·k_3,$$

(15)

where ${\rho }_{\mathrm{fluid}}$ is the density of the surrounding fluid and $\mathrm{Vol}$ is the volume of the displaced fluid, equal to the volume of the airship body.

Thus, to characterize the virtual mass of a given body in terms of the proportionality factors $k_i$, the following methods will be considered:

• the analysis of flight test data [9], a very expensive and quite complex procedure;
• wind tunnel tests [10], which require complex movement mechanisms and measurement equipment;
• engineering models, a less precise technique [1]; and
• numerical aerodynamics methods [11,12], commonly known as CFD (Computational Fluid Dynamics), a widely used method that requires a validation process for the numerical model.

Given the increase in computing power experienced in the last decades, numerical methods have gained preponderance and are currently by far the most common techniques for estimating the coefficients of stability derivatives. As an alternative to more complex CFD codes, potential solvers have also been used in recent times, taking advantage of very fast computation times and good enough accuracy when viscous effects in flow configuration are minor―for example, when there is no flow detachment. In [13] a panel method is used to calculate the velocity potential function of the flow around the airship. Then, the added masses can be derived from the sum of the kinetic energy of the displaced fluid. In [14], a similar but even simpler approach is used, as the geometry definition is limited to elliptic cones and no tail planes can be added. Furthermore, a quite recent and comprehensive review is provided in [15]. Although the study is applied to underwater vehicles, the methodology can also be used for airships and is based on Analytical and Semi-Empirical (ASE) methods, Computational Fluid Dynamics, or captive and free model testing. These analyze all the vehicle’s derivatives, including inertial and damping coefficients, and the review incorporates some novel methods based on System Identification and machine learning algorithms.

However, in most cases, the scientific community has left aside the lab testing of scale models as a source of precise data for validating the numerical results for the added mass, probably due to the intrinsic difficulty of the required tests and measurement methods.

One of the very few experimental works with models in recent years is that of Lee et al. [16]. They ran a pure heaving motion test at a towing tank for a spheroid-type Unmanned Underwater Vehicle, comparing the results with theoretical calculation and CFD analysis. The differences between the test measurements and the theoretical calculation of added mass ranges from 18 to 37% in excess, depending on the frequency of the movement employed for the analysis. This test, as with most of those performed in the past, such as in [17] or [18], has been made by submerging the model under water. Running the test in water instead of air dramatically increases the magnitude of the added masses to be measured (water density is a thousand times that of air density), at the expense of more complex facilities and measurement equipment. Furthermore, some old test data [17] present an important limitation. The experiments have been conducted in water tanks, so the velocity of the fluid is null. As a result, the angle of attack is ±90 deg while the model is oscillating up and down. Flow around the spheroid is totally detached in these circumstances and, consequently, added mass estimation is expected to differ from potential flow calculations and flight conditions [7]―where small angles of attack are usually considered.

As far as the authors know, the measurement of airship added masses on wind tunnels has not been documented. Only the additional moment of inertia was quantified in the past by swinging models as compound pendulums in a vacuum tank under different air density conditions [19].

This work will present a methodology to obtain the virtual mass of airship bodies thanks to wind tunnel tests. The movement mechanism, the instruments and the model manufacturing will be presented, while the accuracy that can be expected for this method will be analyzed. The implementation of the experimental set-up will face several challenges due to the small magnitude of the virtual mass (compared to airship model mass) and the cross coupling of inertia forces, aerodynamic effects and structural vibrations of the system. If proved successful, the proposed methodology will make it possible to provide a dataset for virtual masses, obtained for different vehicle geometries, that can help to alleviate the lack of precise experimental data in order to accomplish the validation of simpler-to-use numerical methods.

2. Experimental Design

To appreciate the contribution of virtual masses, not only in experiments but also in numerical simulations, it is necessary to pay attention to two main considerations that essentially define the methodology (Figure 2): first, having a movement configuration capable of providing the model with the necessary accelerations for dynamic components to appear and be captured by the force sensor (defined as β, the ratio between the virtual mass force and the sensor signal noise); and, secondly, ensuring that these dynamic components represent a significant magnitude compared to the aerodynamic and inertia force terms (defined as α, the ratio between the inertia force and the virtual mass force). This is important to properly identify the different force components within the measured force signal.

Regarding the movement configuration of the model, the simplest pattern of motion is the displacement caused by a pure sinusoidal signal, which is commonly used in CFD studies. However, the mechanical configuration required to achieve this pattern for lateral displacement is not easy to implement and the few experiments that exist to date in this regard use a crank-slider type of movement. As an example, Lee et al. [16] and Yavuz [20] employed the combination of a crank and a connecting rod to perform the lateral displacement of the model. In [17], a scotch-yoke mechanism is used, allowing a pure sinusoidal movement for heaving or pitching. Here, the crank-slider approximation has been adopted for the lateral displacement of the model, using a long connecting rod to achieve a similar performance between outward and return movements (see

Table 1. Design parameters for the experiment.

Variable	Value	Units
Model characteristic length ($2a$)	1	m
Model aspect ratio ($a/b=a/c$)	6	-
Model weight	482	g
Lateral displacement-crank ($r$)	10	cm
Connecting rod-slider ($l$)	105	cm
Angular velocity $\left(\mathsf{\Omega }\right)$	6.12	rad/s2
Max. expected net force	2	N

).

Thus, the following equations are used to reproduce the model position $y$, velocity $v$ and acceleration $\dot{v}$ [21] for Equation (8):

$$y=r\cos \left(\mathsf{\Omega }t\right)+\sqrt{l^2-r^2{\sin }^2\left(\mathsf{\Omega }t\right)}-\sqrt{l^2-r^2},$$

(16)

$$v=-r\omega \sin \left(\mathsf{\Omega }t\right)\left(1+{\displaystyle \frac{r\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\Omega }t\right)}{\sqrt{l^2-r^2{\sin }^2\left(\mathsf{\Omega }t\right)}}}\right) ,$$

(17)

$$\dot{v}=-r{\mathsf{\Omega }}^2\left(\cos \left(\mathsf{\Omega }t\right)+{\displaystyle \frac{r\left[l^2\cos \left(2\mathsf{\Omega }t\right)+r^2\mathrm{s}\mathrm{i}{\mathrm{n}}^4\left(\mathsf{\Omega }t\right)\right]}{{{(l}^2-r^2{\sin }^2\left(\mathsf{\Omega }t\right))}^{\frac{3}{2}} }}\right),$$

(18)

in which $r$ is the lateral displacement (i.e., radius of the crank), $l$ is the longitude of the connecting rod and $\mathsf{\Omega }$ is the angular velocity of the crank.

Once the movement mechanism has been defined, it is possible to analyze the different forces (dynamic, aerodynamic and inertia terms) that will be produced by the test model. However, to do that, the characteristics of the test model need to be defined. In many studies of the literature for virtual mass estimations, not only for airships but also for underwater vehicles, the shape of interest tends to be approximated as acceptable to a prolate spheroid or an ellipsoid of revolution. This model has an analytical solution that allows the authors to perform a comparative analysis against their own models, experiments or numerical simulations. Additionally, the effect of other parts, like rudders or elevators, could be added later following the literature recommendation, which is essentially semi-empirical. The theoretical contribution of a thin flat plate within a perfect fluid can be estimated and then corrected experimentally, as has been proved in the work of Gracey [19]. In this study, we propose the use of an ellipsoid of revolution (aspect ratio 6:1), while the effect of additional parts is left for future works.

For a body with this shape, the $k_2$ coefficient from Equation (14) has a theoretical constant value that is defined from the semiaxes of the ellipsoid of revolution $a>b=c$ [8]:

$$k_2={\displaystyle \frac{{\beta }_0}{2-{\beta }_0}},$$

(19)

$${\beta }_0={\displaystyle \frac{1}{e^2}}-{\displaystyle \frac{1-e^2}{2e^3}}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{1+e}{1-e}}\right),$$

(20)

where $e$ is the eccentricity of the ellipsoid of revolution:

$$e=\sqrt{1-{\displaystyle \frac{c^2}{a^2}} }.$$

(21)

The aerodynamic term in Equation (8) $F_y^v·v$ is the product of Equation (17) and the stability derivative of the force $F_y^v$. This example has been calculated from the dimensionless stability derivative of force ${\overline{F}}_y^v$ in [11], using Equation (22) to represent the proper dimensions.

$${\overline{F}}_y^v={\displaystyle \frac{F_y^v}{\frac{1}{2}\rho v_{ref}\mathrm{Vol}}}.$$

(22)

Finally, $F_{z0}$ is related to the inertia force, which is easily estimated by multiplying the test model weight by the acceleration (18).

Once the different force terms have been properly modeled for the test model (the ellipsoid of revolution), it is possible to study the ideal combination of parameters that ensure the two aforementioned considerations. Inertia and virtual mass forces are in the same phase, so they need to be in good proportion to capture the effect produced by the added masses. As a result, a large model―displacing a large volume of surrounding fluid―with very low mass―producing the lowest possible inertia―is preferrable. Thus, the values displayed in Table 1 constitute the most appropriate combination to ensure the analytical solution plot in Figure 3.

The values of these parameters $F_y^v$ and $F_y^{\dot{v}}$ are, respectively, −0.07663 and −0.01467, resulting in a maximum aerodynamic force of −0.047 N and a maximum virtual mass of −0.060 N, while the maximum net force is −2.03 N. This means that the maximum net force is 33 times the maximum virtual mass.

Therefore, a test model has been implemented that consists of a mock-up of an ellipsoid of revolution made of extruded polystyrene and a combination of elements that allows a displacement of 10 cm at the desired frequency. Figure 4 is presented for a better understanding of the hardware list and distribution: a computer for motor control and data recording, a data acquisition system with a force sensor, whose housing is detailed in Figure 5, a stepper motor, and a crank-slider system in combination with two guide rails.

2.1. Mock-Up Model

Aiming to minimize the mass of the test model while maintaining a good stiffness, the ellipsoid of revolution has been manufactured with extruded polystyrene. This material is a very light foam that is extruded with a foaming gas, similar to the expanded polyethylene. The main advantage of the extruded polystyrene is its smooth finish with a low value for bulk density of around 30 kg/m³. The model is 1 m long with an aspect ratio of 6, resulting in 0.0145 m³ of volume and 0.436 kg of estimated weight. After the internal emptying to house the sensor, the mock-up measured weight, including some connecting pieces made of aluminum, is 0.482 kg (see Table 1). The inner housing allocates the force sensor and cables, avoiding the measurement of the drag forces related to the supporting structure and the cables themselves (Figure 5).

Figure 5. Detail of the load cell housing and the mock-up production.

Figure 5. Detail of the load cell housing and the mock-up production.

2.2. Force Sensor

A piezoelectric force transducer with constant current line drive technology has been chosen for the experiment, since they are specially indicated for dynamic force measurements, specifically the 8230 model from Brüel and Kjaer. with a range of +44/−44 N and a resolution of 0.00062 N. It is inserted directly between the mock-up model and the structure that holds and moves it. The maximum of the virtual mass force is up to 0.060 N (3.5% of the maximum expected net force); therefore, the effect of its contribution is captured by the instrument.

The data are collected using a universal measuring amplifier system, model QuantumX MX44B (Hottinger Brüel & Kjaer GmbH, Darmstadt, Germany), with a sample rate of 40 kHz and a 24-bit analog-to-digital converter. Thus, it is capable of capturing <10⁻⁵ N variations within the range of the sensor, so it does not generate noise louder than the noise of the sensor signal itself.

2.3. Mobile Device

To move the entire assembly, a mobile device has been implemented that can perform lateral displacements according to the requirements of Table 1. For this purpose, a stepper motor uses a crank-slider system with 105 and 10 cm rods, respectively, to move the mock-up along two rails aligned perpendicularly to the incident wind flow (see Figure 6). The motor can be programmed to move at different frequencies, achieving various lateral displacement speeds and accelerations in addition to those of the design specification.

The whole device is placed inside the test chamber of a low-speed wind tunnel with a cross-sectional area of 1 m². The mock-up model is aligned so that the center of the lateral displacement is in the middle of the chamber, and it displaces 10 cm left and right perpendicular to the free stream direction, with a minimum separation of 81.67 cm between the mock-up surface and the wind tunnel wall, which implies a 2.18% cross-sectional blocking percentage of the free stream.

3. Results

During January and February of 2024, a test campaign was conducted following the procedures detailed in the previous section. Different combinations of wind speeds in the tunnel (from 4.5 m/s to 7.5 m/s) and model motion frequencies (from 1.88 rad/s to 6.12 rad/s) were tested. These ranges were chosen to explore how different combinations affect the accuracy of the results, beyond the initial values proposed in Section 3. The data acquisition rate was set to 2400 Hz, and the data were collected for more than two minutes for each test, obtaining enough samples for subsequent signal post-processing.

Figure 7 illustrates the data post-processing methodology employed in this study. Raw data (step 1), representing the force measured by the load cell, underwent spectral analysis (step 2) to identify the optimal model motion frequency (step 3), minimizing measurement noise. Subsequently, a low-pass filter was applied (step 4), and the inertial contribution was removed (step 5) based on the known acceleration and mass of the model. This resulted in a force signal suitable for Fourier series fitting (step 6). The first cosine coefficient of the fitted series was then utilized to calculate the virtual mass (step 7). Detailed descriptions of each post-processing step are provided in the following paragraphs.

The raw load cell signal is subject to contamination from multiple noise sources, notably aerodynamic disturbances and structural vibrations. The overlapping frequency content of these sources makes it difficult to isolate or filter them individually, which could introduce measurement errors. Testing (step 1) across a broader range of wind speeds and frequencies allows for a more comprehensive understanding of these noise effects and their impact on the accuracy of the measurements.

Nevertheless, as vibrations (and their associated noise) are frequency-dependent, adjusting the model’s displacement frequency (while maintaining other test conditions) can minimize signal noise and enhance the accuracy of virtual mass estimation. Furthermore, additional improvements are achieved, since increasing the frequency raises the virtual force, thereby increasing the signal-to-noise ratio measured by the sensor.

Following the proposed methodology, as part of step 2, the spectral content of the measured signal was examined for several model displacement frequencies. Results are shown in Figure 8 for a Reynolds number of $4.1\times {10}^5$ (based on the airship length), although no notable differences were found for other Reynolds numbers.

The spectrum of the signal $e\left(\omega \right)$ is computed using the Fast Fourier Transform (FFT), following the standard procedure as detailed in [22]. To enhance the accuracy of this transformation and reduce the spectral leakage, the signal is windowed using the Hamming function before performing the FFT. Windowing with the Hamming function tapers the signal at the beginning and the end, smoothing discontinuities at the boundaries of the sampled data. This step is crucial for minimizing artifacts in the frequency spectrum. Given the frequency-dependent nature of the force amplitude, spectra were normalized by total signal energy to enable meaningful comparison across different frequencies.

Figure 8 illustrates how the first harmonic’s amplitude decreases with lower oscillation frequencies, resulting in broader peaks in the energy spectrum. It should be noted that the total area under the curve is normalized to 1, thus a higher peak amplitude implies a narrower pulse width. Test frequencies were determined based on the electric motor’s control pulse characteristics.

Thus, broader spectral peaks correlate with increased noise and vibrational contamination in the signal, compromising the precision of virtual mass estimations.

The two tested frequencies that minimize structural vibrations and other noise components are the higher frequencies, corresponding to $\mathsf{\Omega }$ = 6.12 rad/s and $\mathsf{\Omega }$ = 5.34 rad/s. Although the differences are not readily apparent on the logarithmic scale graph, the peak value obtained for $\mathsf{\Omega }$ = 6.12 rad/s is 0.48, which is 50% higher than the 0.32 calculated for the other value.

Thus, the highest tested frequency within the operational limits was identified as optimal for virtual mass extraction. This result strongly aligns with the initial design parameter estimates obtained in Section 3. Frequencies exceeding these limits were avoided to prevent load cell overload and to maintain small angles of attack, thereby minimizing the risk of flow separation, which can modify the virtual mass value [23].

Following the removal of noise-contaminated signals, pre-processing of the remaining data involves applying a low-pass filter (step 3) with a cutoff frequency of 5 Ω:

$$e\left(\omega >5\Omega \right)=0,$$

(23)

Next, the filtered signal is recovered by means of the inverse Fast Fourier Transform of $e\left(\omega \right)$. This filtering operation effectively attenuates high-frequency noise while preserving the amplitude of the initial harmonics, which are essential for the accurate estimation of the stability derivatives. Figure 9 provides a visual comparison of the original and filtered signals. Note that $T$ denotes the period of the model’s movement.

Following signal filtering, the next step (5) is the removal of the inertial force contribution ($F_{y0},$ a function solely of the model’s mass and the kinematic acceleration imparted by the connecting rod-crank mechanism). The resulting signal ($R_s=\sum F_y-F_{y0}$), comprising the superposition of aerodynamic and virtual forces (under the assumption of minimal vibrational contamination), is subsequently fitted with a Fourier series:

$$R_s=a_1\cos \left(\mathsf{\Omega }t\right)+b_1\sin \left(\mathsf{\Omega }t\right)+a_2\cos \left(2\mathsf{\Omega }t\right)+b_2\sin \left(2\mathsf{\Omega }t\right).$$

(24)

Figure 10 presents the results of the fitting process, which was performed using the nonlinear least squares method with the Trust–Region algorithm [24]. The coefficient $a_1$, extracted from the fitted signal, is directly related to the virtual mass as $F_y^{\dot{v}}=a_1/{\mathsf{\Omega }}^2$.

Experimental tests were repeated several times for the same Reynolds number across a range of wind speeds, obtaining the results presented in

Table 2. Results of Virtual Mass Estimation at Tested Reynolds Numbers.

Wind Speed (m/s)	Max. Angle of Attack (Deg)	Reynolds Number	Test Number	$\mathit{R}$	Virtual Mass, ${\mathit{F}}_{\mathit{y}}^{\dot{\mathit{v}}}$ (g)	${\mathit{k}}_2$
4.5	7.8	$2.8\times {10}^5$	1	0.97	1$5.65$	0.98
			2	0.98	14.38	0.90
5.5	6.4	$3.5\times {10}^5$	3	0.97	15.07	0.95
			4	0.98	14.23	0.89
			5	0.97	15.19	0.95
6.5	5.5	$4.1\times {10}^5$	6	0.97	14.02	0.88
			7	0.97	14.06	0.88
			8	0.97	14.53	0.91
7.5	4.7	$4.7\times {10}^5$	9	0.97	13.51	0.85
			10	0.97	13.32	0.84
			11	0.97	12.23	0.77
			Mean		14.20	0.89

. The expected virtual mass was determined to be 14.2 g, with a standard deviation of 0.9 g. All datasets exhibited strong correlation with the Fourier series model. This is demonstrated by the high Pearson correlation coefficient ($R$ > 0.97). No statistically significant correlation was found between virtual mass and wind speed, as long as flow conditions remained within the attached boundary layer regime, with a maximum angle of attack of less than 8 deg for each combination.

The theoretical value of the virtual mass corresponding to the geometry is calculated by using Equation (19), with the result being $k_2=0.91$, which represents a relative error of merely 2.1% compared to the experimental value. This result is noteworthy, as similar experimental studies have often yielded results that significantly deviate from theoretical predictions. For instance, Lee et al. [16] reported discrepancies ranging from 17.6% to 36%, highlighting the great accuracy of our findings.

4. Conclusions

In recent years, there has been a renewed interest in airships as cargo transport and stratospheric platforms. Proper modeling of the dynamics of these platforms, which is essential for stability and control analysis, autopilot design, and mission planning, requires calculation of their stability derivatives, specifically the calculation of virtual mass and inertia.

Despite the abundance of CFD methods and analytical solutions found in the literature, experimental studies remain necessary to validate the results obtained by these other models. Previous experimental tests are more abundant for submarines, with set-ups where the geometry to be studied is submerged in a water tank. By providing a relative motion to this geometry, the forces that arise are determined and, from the correct analysis, the stability derivatives are obtained.

This study presents, for the first time to the author’s knowledge, a methodology for measuring virtual masses in a wind tunnel, rather than in water. An analysis was conducted to dimension both the geometry to be chosen and the crank-rod mechanism necessary to impart motion to the model within the wind tunnel. Variables to consider include the sensitivity and precision of the load cell used, as well as the range of frequencies and Reynolds numbers to be employed.

To validate the analytical solutions, a model of a revolution ellipsoid was fabricated from extruded polystyrene, with a length of 1 m and an aspect ratio of 6. The model weighs 482 g. The crank-slider system, with rods measuring 105 cm and 10 cm, respectively, was used to move the model along two rails oriented perpendicularly to the incident wind flow. Given the low mass of the model and the rigidity of the crank-rod mechanism, the load cell measurements can be affected by structural vibrations during testing. These vibrations, however, can be minimized by increasing the frequency of the motion, provided the load cell is not overloaded. The processing of the measured force signal involved analyzing the Fourier spectrum and comparing the harmonics at various motion frequencies to ensure minimal vibration interference.

With proper signal processing, including the use of a high-pass filter, the removal of the inertial term, and fitting the result to a second-order Fourier series, the virtual mass value has been successfully calculated. The obtained result is particularly valuable when compared with previous studies. The results obtained through 11 different tests in this study differ by only 2.1% from the theoretical value, improving by about one order of magnitude the relative error obtained in other previous experimental studies. Some actions have been considered to improve the reliability of the results in the future, such as the construction of models of different sizes to test at the same Reynolds numbers in order to evaluate the effect of the scalability of the model, or to carry out tests at higher Reynolds numbers, which are closer to the more typical values of large airship-type platforms.