Abstract
Intermediate Bulk Containers (IBCs) are among the most widely used packaging and transport systems worldwide for bulk goods such as liquids. When IBCs are only partially filled, liquid sloshing can occur during both intralogistics and external transport. This sloshing generates dynamic forces on the container walls, which are transferred to the supporting structure. In particular, the horizontal sloshing forces can significantly exceed the inertial force, potentially causing critical issues for the carrier when handling the IBC. This study focuses on intralogistics operations, for example in warehouses using Automated Guided Vehicles (AGVs) and Autonomous Mobile Robots (AMRs), which operate with limited accelerations and velocities. Since these vehicles provide only restricted torque capacity (typically handling pallet loads up to 1500 kg), the additional horizontal sloshing forces may impair their ability to drive and position accurately. To address this, this study provides a fundamental, experimental characterization of the sloshing dynamics that cause these disturbances. Specifically, it identifies the natural resonant frequencies, systematically characterizes the resulting horizontal forces across a range of fill levels and excitation energies, and determines the worst-case conditions where these disruptive forces are maximized for low-viscosity liquids such as water.
1 Introduction
An Intermediate Bulk Container (IBC) is a type of packaging designed for the transport of bulk goods and liquids in industrial applications. It typically consists of a plastic container enclosed in a metal cage, or less commonly, a fully metal container, both mounted on a pallet. IBCs are available in various sizes, with the 1000 L version being the most common (industry pallet size) (; ), followed by the 600 L version (Euro pallet size), etc.
In sectors such as the chemical industry, food production, and pharmaceuticals, vast quantities of liquids are transported each year in Intermediate Bulk Containers (IBCs). The global IBC market was valued at USD 15.3 billion in 2024 and is projected to reach USD 25.6 billion by 2034 (). This growth is primarily driven by the increasing demand for safe and efficient packaging solutions across these industries. Compared to traditional drums, IBCs provide a higher volume per unit area, for instance a 1000 L IBC offers more capacity per square meter than four 200 L drums. In addition, IBCs are easier to stack and handle, as they are pallet-mounted by design.
Transportation of these IBCs occurs through two primary modes: external logistics, referring to movements beyond the company premises, and intralogistics, referring to the handling and transport on the site itself. When an IBC is only partially filled during the transport, the liquid can move relative to the imposed motion of the container. This relative motion induces waves at the free liquid surface, a phenomenon known as sloshing. The intensity of sloshing depends on the frequency and amplitude of the external excitation: large accelerations or motions close to the resonance frequency of the liquid-container system significantly amplify wave formation. Highly viscous liquids are less prone to sloshing and tend to dissipate such motion more rapidly than low viscous liquids. The liquid waves exert dynamic forces on the container walls, which are subsequently transmitted to the carrier, this is the system responsible for moving the container. Under strong liquid motion, these forces can increase substantially, potentially leading to undesirable effects such as damage to the container or carrier, tipping of the container, extended braking distances, or reduced positioning accuracy. The study of sloshing dynamics is a well-established and critical field of research, historically driven by the aerospace industry to ensure the stability of space vehicles. Landmark studies, such as those summarized by NASA in “The Dynamic Behavior of Liquids in Moving Containers” (), underscore the complexity and importance of this phenomenon. Despite decades of research, sloshing continues to be an important challenge in the aerospace industry, particularly in rocket applications where it influences guidance and control systems ().
When IBCs are transported externally it is often by trucks that can withstand high forces in all directions. In addition, care is taken to ensure that the number of partially filled containers in this transport is limited. If, for example, several half-full IBCs are in the load and they slosh simultaneously, the truck may also be affected by these forces. For intralogistics transport, forklifts are still widely used, but in recent years there has been a strong rise in the application of Automated Guided Vehicles (AGVs), Autonomous Mobile Robots (AMRs), and fully automated (high-bay) warehouses. A comprehensive market study () projects the sector to expand from approximately USD 6 billion in 2024 to around USD 22 billion by 2030. This industry-wide shift is driven by the well-documented inefficiencies, safety concerns, and labor dependency inherent to traditional, manual warehouse operations (). When transporting IBCs with forklifts, sloshing is less of a concern, since the mass of the forklift is large compared to the sloshing forces, the vehicles are typically rated for loads well above 1500 kg, and the operator can actively intervene if problems arise due to liquid motion. For AGVs and AMRs, however, the situation is quite different. These vehicles are designed to be as compact as possible, with relatively low mass and limited payload capacities (typically 1500–2000 kg). In such cases, sloshing forces can have a significant impact on driving performance and positioning accuracy. However, precise positioning is a critical requirement in modern warehouse environments. In high-bay storage systems, for instance, it is fundamental to achieve high-density storage, requiring shuttles to pick and place loads with minimal tolerances.
Common approaches to mitigate liquid sloshing in containers include the use of horizontal baffle plates (), vertical baffle plates (), spring-mounted baffle plates (; ), flexible porous baffle plates (), floating foams () and baffle balls (). More advanced strategies have also been proposed, such as actively controlled baffles using deep reinforcement learning to suppress sloshing dynamics (; ). However, these solutions require mechanical modifications or the addition of materials inside the container. For IBCs, such modifications or internal components are not feasible, making sloshing a persistent challenge in their transport. Therefore, solutions must be sought within the vehicle control system. In this context, various command shaping and input shaping techniques have been proposed to reduce sloshing by appropriately designing the motion profiles of the system. Early work in this area demonstrated that residual vibrations in flexible systems can be effectively suppressed using input shaping techniques (; ), which were later extended to liquid sloshing problems (). These approaches aim to reduce excitation of the system’s natural modes by modifying the acceleration profiles.
In subsequent years, the application of command shaping and input shaping expanded toward multi-axis and multi-mode settings. For instance, (), proposed command-smoothing and combined input-shaping/smoothing architectures to suppress not only the fundamental mode but also higher sloshing modes, supported by experiments on moving containers. Similarly, input shaping has been integrated with robot trajectory planning to reduce sloshing during three-dimensional container transfer motions ().
Building on these developments, more recent research has focused on increasingly refined and application-oriented motion design strategies. Adjustable smooth command profiles can suppress three-dimensional sloshing in transported containers (), while robust multi-step input commands effectively limit fluid oscillations under varying conditions (). In addition, polynomial-based trajectory shaping methods () and classical command shaping approaches () demonstrate that carefully designed motion inputs can significantly reduce sloshing without requiring modifications to the container itself. Furthermore, anti-sloshing trajectory design methods combining input shaping, model-based inversion, and filtering techniques have been proposed and experimentally validated for high-acceleration motions in automatic machines ().
To use this kind of control strategies a deeper understanding of the sloshing problem in case of IBCs is necessary. While numerous theoretical and computational models have been developed to predict sloshing behavior (), these models are often validated against experiments on simplified, small-scale, and rigid laboratory setups (; ; ). However, a significant gap remains in the literature regarding the direct experimental characterization of sloshing in full-scale, deformable IBCs.
The primary contribution of this research is therefore to bridge this specific gap. Instead of relying on scaled models or pure simulation, this paper presents direct experimental force measurements on a 1000 L IBC to provide data that is immediately relevant to the design and control of real-world AGV and AMR systems. To assess the risks of sloshing in the intralogistics applications, it is essential to understand the order of magnitude of the fluid forces, to identify the situations in which sloshing becomes most severe, and to determine the fill levels that generate the largest sloshing forces. To address this, this study provides a fundamental, experimental characterization of the sloshing dynamics that cause these disturbances. Specifically, it identifies the natural resonant frequencies, systematically characterizes the resulting horizontal forces across a range of fill levels and excitation energies, and determines the worst-case conditions where these disruptive forces are maximized for low-viscosity liquids such as water.
2 Materials and methods
2.1 Test setup
In this study, a 1000 L industry-standard IBC was used, as this is the most common type in industrial applications. The IBC dimensions are shown in Figure 1b and are as follows: height = 1.00 m, width = 0.94–0.97 m, and length = 1.14–1.17 m. The reported dimensions correspond to the inner container (fluid vessel) only, excluding the outer cage and pallet. The inner container is made of HDPE, which is the standard material for such applications due to its chemical resistance. The outer cage is made of galvanized steel, providing structural support and protection to the inner container.
FIGURE 1
To subject the IBC to predefined motion profiles, a custom-built linear motion platform (Figure 1a), hereafter referred to as the Pallet Test Machine (PTM), was utilized. The PTM consists of a base frame and a moving carriage. The system is actuated by an AC servomotor (Beckhoff AM8062-0P20-0000) and a gearbox, controlled by a servo drive (Beckhoff AX5125-000-0202) and a PLC (Beckhoff CX5140-0155). The carriage, which offers a 1200 × 1200 mm loading surface suitable for both EUR and IND-pallets, travels along two linear guideways mounted on the base frame. A rack and pinion drive system provides a maximum travel stroke of 770 mm. For safety, the setup is enclosed by a frame to contain the payload in case of excessive tilting.
The force measurement system (Figure 1b) was designed to isolate and quantify the horizontal fluid forces (sloshing forces). To this end, the IBC is placed on two roller conveyors and clamped between four load cells (Forsentek FH8C-1t). The motion of the carriage is transmitted to the IBC through these load cells, allowing them to register the combined inertial forces of the IBC and the dynamic sloshing forces. The data from the four load cells is acquired using a Siemens Simatic S7-1500 Programmable Logic Controller (PLC). This central controller is interfaced with a distributed I/O system, a Siemens ET200SP station. To digitize the analog signals from the load cells, two high-speed strain gauge input modules (Siemens AI 2xSG 4-/6-wire HS) are integrated into this station. Each module is configured to process the signals from two load cells simultaneously. This setup facilitates a high sampling rate of up to 10 kHz per channel, which is more than sufficient to accurately capture the dynamic sloshing forces with high resolution and without the risk of aliasing.
The uncertainty of a single load cell was estimated from the manufacturer specifications by considering the specified nonlinearity (±0.02% RO), hysteresis (±0.02% RO) and nonrepeatability (±0.017% RO). Creep (±0.02% RO over 30 min) was neglected due to the short measurement duration and the use of a tare prior to each test. Temperature effects were neglected since the lab temperature remained within 20 °C ± 1 °C, corresponding to a potential drift of only ±0.002% RO. A conservative worst-case bound was obtained by linearly summing the remaining error limits, resulting in ±0.057% FS, i.e. ±5.59 N for a 1-t load cell. The total force is computed from four load cells; assuming full correlation yields a maximum possible bound of ±22.4 N. In practice these error contributions are not expected to be (fully) correlated across sensors, so the actual combined error is likely smaller than this conservative bound.
A key design challenge was the mitigation of measurement artifacts. The first artifact is the tilting moment generated during acceleration, which can introduce undesirable vertical force components onto the horizontally oriented load cells. This was addressed by clamping an IND-pallet between the load cells and placing the IBC on top of it. This configuration allows the IBC to pivot freely, decoupling the tilting moment from the force measurement.
A second, more complex artifact arises from vertical forces. Measurement errors are caused by both the static deflection of the roller conveyors under the fluid’s weight and, more importantly, the dynamic vertical forces generated as the fluid sloshes. Merely taring the load cells is insufficient as the vertical load distribution changes continuously during a test. To completely decouple the horizontal force measurement from these vertical influences, linear roller bearings were integrated between the load cells and the IND-pallet. These components permit free vertical displacement while rigidly transmitting the horizontal forces. The final experimental setup, shown in Figure 2, thereby ensures a pure, uncontaminated measurement of the horizontal sloshing forces.
FIGURE 2
The overall control architecture and communication pathways of the experimental setup are illustrated schematically in Figure 3.
FIGURE 3
2.1.1 Initialization and motion profile generation
Following the system’s initialization sequence, which includes safety checks and a homing procedure, the user inputs three key parameters via a Human-Machine Interface (HMI): the fluid level within the IBC, the desired number of oscillation cycles and the displacement amplitude. The fluid level parameter is used for automated file naming, while the number of cycles and the displacement amplitude are primary inputs for defining the motion profile.
Next, a “CalculateProfile” command executes an external Python script. Within this script, the user specifies the desired oscillation period. The script then dynamically generates a complete displacement profile as a set of position-time data points. This profile is automatically loaded into the CAM-table within the Beckhoff TwinCAT motion control software.
2.1.2 Test execution and data acquisition
The test sequence is initiated by activating the “StartShakeTest” command on the HMI. Data logging of the load cells on the Siemens PLC commences immediately (sample rate of 50 Hz), but the PTM remains stationary for the first second. This initial quiescent period is intentionally recorded to:
Establish a stable zero-force baseline, confirming the validity of the initial tare.
Ensure the fluid is entirely at rest before the motion begins.
At the precise moment the PTM initiates the motion profile, the Beckhoff PLC sends a digital trigger signal to the Siemens PLC. This signal serves as a synchronization marker, precisely aligning the force data with the motion data. The total data acquisition duration is fixed at 2 minutes. This duration is deliberately longer than the motion sequence itself, ensuring that the subsequent free decay of the fluid’s oscillation is fully captured after the forced motion has ceased.
2.1.3 Data storage
Upon completion of the 2-min acquisition period, the logging process terminates automatically. The collected data is then exported and saved as an Excel file, ready for post-processing and analysis.
2.2 Preliminary test
Following the calibration of the load cells, a validation test was performed to ensure the integrity and accuracy of the complete measurement system prior to commencing the fluid experiments. The rationale for conducting this initial test with a solid mass is fundamental. In contrast to the complex and a priori unknown sloshing forces within an IBC, the inertial forces of a rigid body are theoretically predictable with Newton’s Second Law (F [N] = m [kg] · a [m/s2]). This allows for a definitive verification of whether the measurement system is recording forces accurately and with the correct order of magnitude.
For this validation a concrete block on a EUR-pallet, constituting a total calibrated mass of 374 kg, was mounted onto the PTM. This rigid payload was then subjected to a sinusoidal displacement profile with an amplitude of 30 mm and a period of one second.
Figure 4 presents the results of this test, showing a superposition of the directly measured force and the theoretical force (calculated as the acceleration profile multiplied by the known mass). The excellent correlation between the two signals validates the measurement setup. It is important to note that the scope of this preliminary test is strictly confined to validating the measurement apparatus itself.
FIGURE 4
2.3 Test sequence
The experimental setup was designed to characterize the sloshing dynamics across the full operational range of the IBC. Tests were conducted for fill levels ranging from 100 L to 1000 L, in increments of 100 L. Additionally, a baseline measurement was performed with an empty IBC (0 L, representing the rigid body reference) and a completely full IBC (approx. 1050 L). Water was selected as the working fluid due to its well-defined properties, its low viscosity which promotes pronounced sloshing effects, and its inherent safety and availability.
2.3.1 Test 1: identification of natural frequencies
Objective: To experimentally determine the natural resonant frequencies (sloshing modes) for each fill level and compare them to theoretical values.
Theoretical Framework: The natural frequencies of a fluid in a rectangular tank can be theoretically approximated using linear potential flow theory (). The angular frequency for each mode n is given by Formula 1:
Hereby is:
: angular frequency [rad/s]
: mode number (n = 2k-1 for horizontal excitation with k = 1,2,3, …) [-]
: Pi (=3.14159) [-]
: gravitational constant (=9.81) [m/s2]
: length of the IBC in the direction of the movement [m]
: fluid height [m]
This model assumes a rigid rectangular tank, small wave amplitudes, and an ideal (inviscid, incompressible) fluid. A primary goal of this test is to quantify the deviation of this ideal model from the real-world measurements on an IBC with water. Because the experimentally measured frequencies correspond to the damped natural frequencies, they should, in principle, be compared to the computed damped natural frequencies. These can be determined using Formula 2, however, the damping ratio of water is typically very low (ζ<<0.1), for which . For example, a damping ratio of 0.0713 was reported for a small container (200 × 150 mm) with a water depth of 30 mm (). Via a small experiment the damping ratio of water in an IBC is determined and it is around 0.001–0.005. For these reasons, the calculated is used directly for comparison.
Method: To excite multiple sloshing modes simultaneously, the IBC was subjected to a single-cycle sinusoidal excitation. The strategy was to center the excitation frequency around the highest-order mode considered to be of practical significance. For this investigation, the fifth mode (f5) was defined as the upper limit, as higher-order modes are significantly more difficult to excite and their contribution to the total sloshing force is generally negligible.
Consequently, the motion profile was designed as a single sinusoidal cycle with a period of 0.5 s (2 Hz), which approximates the theoretically expected period of the fifth mode (T5), and an amplitude of 15 mm. The underlying principle is that while this excitation primarily targets f5, its short-duration, broadband nature ensures that the lower, more energetically dominant modes (such as f1 and f3) - which are inherently easier to excite - are also effectively energized.
Analysis: A Fast Fourier Transform (FFT) was applied to the resulting force-time signal of the free vibration to obtain the frequency spectrum (Figure 5). The peaks in the FFT reveal the actual resonant frequencies measured in the experiment. These values were compared to the calculated values from formula 1 to evaluate the model’s accuracy (Table 1). The experimentally determined fundamental resonant frequency (f1) for each fill level was then used as the primary input for subsequent tests.
FIGURE 5
TABLE 1
| Mode number n [-] | Calculated frequency [Hz] | Measured frequency [Hz] |
|---|---|---|
| 1 | 0.792 | 0.763 |
| 3 | 1.574 | 1.558 |
| 5 | 2.038 | 2.012 & 2.031 |
Calculated versus measured frequencies with 300 L liquid.
A primary qualitative observation from the spectral analysis is that exclusively odd-numbered sloshing modes (n = 1, 3, 5, …) were excited. This is an expected and direct consequence of the input motion. A purely horizontal, translational motion constitutes an anti-symmetric excitation, where the fluid rises on one side of the tank while falling on the other. Such a motion can, by definition, only excite anti-symmetric wave modes (the odd modes), as illustrated in Figure 6.
FIGURE 6
2.3.2 Test 2: characterization of the fundamental mode (f1)
Objective: To quantify the maximum forces, RMS force, and damping ratio of the force, associated with the first and most dominant resonant mode.
Method: For each fill level, the IBC was subjected to a single-cycle sinusoidal excitation at its experimentally determined fundamental frequency, f1 (from Test 1). A fixed amplitude of 30 mm was selected for this excitation.
This value was established during preliminary trials conducted at a 300 L fill level. The objective was to determine an optimal balance: the motion had to be sufficiently energetic to generate a distinct and slowly decaying wave with a measurable force response, yet moderate enough to avoid inducing highly non-linear behavior such as wave breaking. This approach ensures a “clean” decay signal (Figure 7a) ideal for subsequent analysis. The total measurement duration was 160 s to capture the decay of this oscillation.
FIGURE 7
Analysis: The analysis of the resulting force-time signal follows a sequential, multi-step process. First, a Fast Fourier Transform (FFT) is applied to the signal to confirm that the response is, as expected, almost entirely dominated by the excited fundamental resonant frequency, f1 (Figure 7b). This high degree of modal purity is critical, as it allows for the confident determination of the damping ratio (ζ) that is specifically associated with this fundamental mode.
The damping is calculated using the logarithmic decrement method (Formula 3 and 4), which is applied to the free decaying horizontal wave forces in the time domain. Using horizontal forces to estimate slosh damping is an established practice, utilizing the linear relationship between lateral force and wave amplitude described in ().
Hereby is:
logarithmic decrement [-]
number of oscillations between wave peak X and Y [-]
amplitude of one wave [mm]
Y: amplitude of n waves later [mm]
Hereby is:
damping ratio [-]
Pi (=3.14159)
Finally, in addition to the damping characteristics, two key force metrics are extracted from the free decay signal: the maximum peak force and the Root Mean Square (RMS) force, calculated over a 120-s window. These metrics are essential for quantifying the severity of the sloshing and serve as the primary indicators for identifying the worst-case fill levels in the subsequent tests.
2.3.3 Test 3: response to sustained resonant excitation (f1)
Objective: To investigate the energy accumulation and the resulting force amplification when the IBC is subjected to sustained excitation at its fundamental resonant frequency.
Method: The procedure from ‘Test 2’ was repeated, but instead of a single cycle, the system was excited with two and five consecutive sinusoidal cycles at frequency f1.
Analysis: The primary focus of the analysis was on the evolution of the maximum force and the RMS force. These measurements are critical for understanding how forces can escalate when an AGV performs a maneuver that excites the fluid’s natural frequency for a prolonged duration, and help identifying the worst-case fill levels in the subsequent tests.
Based on the analysis of these three tests, a comprehensive understanding of the sloshing dynamics can be established. If necessary, further targeted tests can be defined.
3 Results and discussion
3.1 Results resonance frequencies
Following the application of a Fast Fourier Transform (FFT) to the ‘test 1’ data for each fluid level, the resonant frequencies were identified and are summarized in ‘Table 2’. In the table, a “-” indicates that a mode does not theoretically exist, while a '/' indicates that the mode could not be unambiguously identified in the measurement signal. The frequency resolution (Δf) of the FFT analysis, given by Δf = f_sample/N_samples, is at most 0.00637 Hz (7850 samples due to one second delay until movement starts and a maximum shaking period of 2 seconds at 100 L), ensuring high precision. Furthermore, the 50 Hz sampling rate provides a Nyquist frequency of 25 Hz, which is way higher than the highest observed sloshing frequency (∼2 Hz), thereby eliminating the risk of aliasing in the spectral analysis.
TABLE 2
| Volume [L] | Fluid height [m] | f1 [Hz] | f3 [Hz] | f5 [Hz] | f7 [Hz] |
|---|---|---|---|---|---|
| 0 | 0 | - | - | - | - |
| 100 | 0,108 | 0,492 | 1,363 | 1,975 | 2,379 |
| 200 | 0,205 | 0,662 | 1,552 | 2,032 & 2,013 | 2,397 |
| 300 | 0,295 | 0,763 | 1,558 | 2,012 & 2,031 | / |
| 400 | 0,385 | 0,826 | 1,564 | 2,012 | / |
| 500 | 0,476 | 0,858 | 1,564 | 2,012 & 2,031 | 2,397 |
| 600 | 0,562 | 0,877 | 1,564 | 2,012 & 2,038 | 2,378 & 2,397 |
| 700 | 0,652 | 0,889 | 1,564 | 2,012 & 2,037 | 2,378 & 2,379 |
| 800 | 0,745 | 0,896 | 1,564 | 2,012 & 2,044 | / |
| 900 | 0,832 | 0,902 | 1,571 | 2,018 & 2,044 | / |
| 1000 | 0,923 | 0,908 | 1,583 | 2,012 | 2,403 |
| 1050 | 0,96 | - | - | - | - |
FFT results from “test 1” at different liquid levels.
For a quantitative comparison, the theoretical frequencies were calculated using formula 1. A critical parameter in this model is the effective length L of the IBC in the direction of motion. Due to hydrostatic pressure, the plastic walls of the IBC bulge slightly, causing L to increase with fluid height and vary with measurement position (at the top or bottom of the IBC, in the middle or at the edges). Also the impact forces of the sloshing liquid cause the wall to deform. To account for this uncertainties, a realistic interval for L, from 0.94 m to 0.97 m, was used for the calculations. This approach yields a theoretical frequency band rather than a single theoretical line.
Figure 8 plots the measured fundamental resonant frequencies (f1) together with the calculated band. For fluid levels of 400 L and above, there is excellent agreement. At lower levels (<400 L), a systematic deviation is observed, with the measured frequency falling below the theoretical band. This can be attributed to the IBC’s geometry: the rounded corners and the presence of the outlet valve at the bottom significantly disturb the wave pattern at low fluid depths. Nevertheless, the deviation remains limited; at 100 L, the difference is only 5.4% relative to the calculated lower bound.
FIGURE 8
The asymptotic trend of the curve at higher fill levels is a direct consequence of the tanh(nπh/L) term in Equation 1. Once its argument nπh/L exceeds approximately 2.6, the function approaches unity, and the frequency becomes essentially independent of the fluid height h. For the fundamental mode (n = 1), this occurs at a fluid height of roughly 78 cm. For higher modes, this asymptote is reached even more rapidly due to the n multiplier in the numerator, which is evident in the analysis of f3 (Figure 8). Here, the agreement between measurement and theory is better even at lower levels, presumably because the smaller wave amplitude of the third mode is less affected by the geometric disturbances at the tank bottom. At 100 L, the difference is only 0.8% relative to the calculated lower bound.
A notable phenomenon was observed at the fifth resonant mode (f5), where the FFT analysis consistently revealed two distinct, closely spaced frequency peaks. This behavior suggests a splitting of the resonant response, which typically occurs when two eigenmodes with nearly coincident natural frequencies interact dynamically. Such modal interaction is characteristic of systems exhibiting near-degenerate sloshing modes, as discussed by .
Ideally, pure transverse excitation of a rectangular tank does not induce longitudinal wave motion. However, the specific geometry of the IBC introduces asymmetries, most notably the rounded vertical corners and the discharge valve, which act as coupling mechanisms. These geometric imperfections deflect the primary transverse flow (shake direction), introducing a secondary velocity component in the longitudinal direction (perpendicular on the shake direction).
While such geometric perturbations are present at all frequencies, they only result in significant energy transfer when the induced longitudinal excitation aligns with a natural frequency of the fluid. Theoretical analysis identifies the situation at f5 corresponds to a case of near modal degeneracy governed by the specific aspect ratio of the IBC (L = 1.15 m, W = 0.94 m). The eigenvalue calculation shows that the natural frequency of the targeted transverse mode (m = 0, n = 5) nearly coincides with that of the mixed mode (m = 6, n = 1). The spectral coefficients [(m/L)2 + (n/W)2] for these modes differ by less than 0.21% 28.2933 vs. 28.3529 .
Due to this spectral proximity, small geometric perturbations can trigger a dynamic interaction between the two modes. Instead of oscillating independently, the modes become coupled, causing the degenerate natural frequency to split into two hybrid frequencies, which are observed as the double peak in the spectrum. The validity of the geometric degeneracy hypothesis is further supported by the experimental data at the seventh resonant mode (f7). A similar peak-splitting phenomenon was observed, which corresponds to the coupling between the target transverse mode (0,7) and the mixed parasitic mode (6,5).
Conversely, for the lower resonant modes f1 and f3, calculations reveal a significantly larger spectral gap between the target transverse modes and parasitic modes. This explains why the FFT spectra for f1 and f3 display clean, single peaks, in stark contrast to the splitting observed at f5 and f7.
As a consistency check, the resonant frequencies were determined again from the single-cycle excitation data of Test 2, with the results presented in “Table 3”. A preliminary analysis using a standard FFT revealed a high degree of raw agreement: the identified peak frequencies for both test series were consistently found within the same or immediately adjacent frequency bins. The largest discrepancies were consistently noted at lower fill levels, which is ascribed to the pronounced influence of the outlet valve’s geometry on the fluid dynamics in shallow water conditions.
TABLE 3
| Volume [L] | f1 test 2 [Hz] | f1 test 1 [Hz] | f1 test 2 v2 [Hz] | f1 test 1 v2 [Hz] |
|---|---|---|---|---|
| 100 | 0,491 | 0,492 | 0,488 | 0,49 |
| 200 | 0,667 | 0,662 | 0,666 | 0,665 |
| 300 | 0,767 | 0,763 | 0,766 | 0,765 |
| 400 | 0,824 | 0,826 | 0,825 | 0,826 |
| 500 | 0,855 | 0,858 | 0,856 | 0,858 |
| 600 | 0,874 | 0,877 | 0,874 | 0,876 |
| 700 | 0,887 | 0,889 | 0,886 | 0,887 |
| 800 | 0,893 | 0,896 | 0,893 | 0,895 |
| 900 | 0,899 | 0,902 | 0,899 | 0,901 |
| 1000 | 0,906 | 0,908 | 0,907 | 0,907 |
FFT results from single-cycle excitation test 2 at different liquid levels.
However, this standard FFT analysis is limited by its theoretical resolution (Δf = 0.00637 Hz), which introduces a uncertainty to the exact peak location. To achieve a more precise and scientifically robust comparison, a refined signal processing protocol (FFT v2) was therefore employed. This protocol involved applying a Hanning window to reduce spectral leakage, followed by parabolic peak interpolation - which fits a parabola to a spectral peak and its two neighbors - to estimate the true frequency with sub-bin accuracy. This high-precision analysis confirmed the initial observation, revealing a maximum deviation between the two test methods of only 0.002 Hz. Given that this deviation is an order of magnitude smaller than the target accuracy of 0.01 Hz, the identified resonant frequencies are considered highly reliable and robust.
Further analysis of the single-cycle f1 excitation tests revealed another key insight. The FFT of the force response not only showed the expected dominant peak at the fundamental frequency but also exhibited measurable energy at higher-order odd modes, primarily f3 and f5. While the amplitudes of these higher harmonics were minor (at least 40 times smaller) compared to the fundamental, their presence is a signature of non-linearity in the system’s response. In a perfectly linear system, a sinusoidal input at f1 would induce a response exclusively at f1. The generation of these higher-frequency components, even at low amplitudes, indicates a mild deviation from pure linear behavior. This suggests the system operates in a weakly non-linear regime during sloshing of this magnitude.
Although these harmonic peaks were not significant at all fill levels, their frequencies, when identifiable, were in agreement with those determined from the initial “test 1”. The same observations regarding the correlation between the measured and theoretical frequencies for these modes, as discussed previously, remain valid here.
3.2 Results damping ratio
The damping ratio (ζ), which quantifies the rate of decay of the sloshing wave, was determined from the free-decay portion of the ‘Test 2' experiments. A methodological challenge arose during this analysis. Although the excitation was a pure sinusoid at f1, the resulting force signal exhibited higher-order harmonics (f3, f5), consistent with the weakly non-linear behavior previously discussed. As shown in the FFT spectrum for the 500 L test (Figure 9a), a small f3 component is present. This harmonic superposition creates a visible ‘ripple' on the decaying force signal (Figure 9b), where the f3 component alternately constructively and destructively interferes with the fundamental f1 wave. Directly applying the logarithmic decrement method to the raw peaks of this composite signal would therefore lead to an inaccurate estimation of the damping ratio for the fundamental mode.
FIGURE 9
To isolate the fundamental mode and enable an accurate calculation, a fourth-order Butterworth bandpass filter was applied to the signal. The filter was implemented using a zero-phase forward-backward filtering technique (filtfilt) to prevent any phase distortion of the wave. A filter bandwidth of 0.15 Hz, centered on the experimentally determined f1 for each fill level, was selected to effectively remove the higher-order harmonics and noise. The result is a clean, decaying sinusoid representing the pure fundamental mode, as shown in Figure 9c. The damping ratio was then reliably calculated using the logarithmic decrement method (Formula 3, 4) over the stable, free-decay portion of this filtered signal.
The calculated damping ratios for each fill level and number of excitation cycles (1, 2, and 5 shakes) are presented in “Table 4” and plotted in Figure 10. Several key physical phenomena can be identified from these results:
TABLE 4
| Volume [L] | ζ#1shake [-] | ζ#2shakes [-] | ζ#5shakes [-] |
|---|---|---|---|
| 100 | 0,00386 | 0,00381 | 0,00483 |
| 200 | 0,00262 | 0,0035 | 0,00438 |
| 300 | 0,00142 | 0,0016 | 0,00197 |
| 400 | 0,00149 | 0,00161 | 0,00183 |
| 500 | 0,00122 | 0,00135 | 0,00167 |
| 600 | 0,00125 | 0,00137 | 0,00167 |
| 700 | 0,00129 | 0,00139 | 0,00162 |
| 800 | 0,00149 | 0,00154 | 0,0016 |
| 900 | 0,0015 | 0,00153 | 0,00157 |
| 1000 | 0,00193 | 0,00195 | 0,002 |
Calculated damping ratios for 1, 2 and 5 shakes at different liquid levels.
FIGURE 10
Effect of wave amplitude: a clear trend is that a greater number of initial shakes, which induces a larger initial wave amplitude, consistently results in a higher damping ratio. This is attributed to the non-linear nature of fluid damping, larger wave amplitudes lead to increased turbulence. This is a effective mechanism for energy dissipation, which increases the rate of decay.
Influence of the outlet valve at low levels: at low fill levels (100–200 L), this amplitude-dependent effect is particularly pronounced. The outlet valve at the bottom of the IBC acts as a significant obstruction or baffle, causing premature wave breaking and a sharp increase in energy loss for larger waves, leading to significantly higher damping ratios. This effect becomes negligible for volumes above 200 L.
Influence of roof impact at high levels: at high fill levels, another non-linear phenomenon occurs: the wave impacts the roof of the IBC. This impact causes a rapid, initial energy loss that is not representative of the free-sloshing decay. Therefore, for these cases, the damping analysis was intentionally initiated only after this initial impact event, focusing on the subsequent, more gradual decay of the wave.
Convergence of damping at high levels: the roof impact also explains why the damping ratios at 800, 900, and 1000 L are very similar regardless of the number of shakes. At these high levels, even a single shake is sufficient to cause some interaction with the roof, creating a highly turbulent initial state. Consequently, the subsequent decay behavior is dominated by this turbulent initial condition rather than by the number of input cycles.
3.3 Results force measurements
The maximum and Root Mean Square forces (F
RMS) were determined from the data of the second (single-cycle) and third (multi-cycle) test series. A critical distinction was made when analyzing the peak forces:
Overall maximum force (Fmax,tot): the highest force peak recorded during the entire measurement, including both the forced excitation and the subsequent free decay.
Maximum free-slosh force (Fmax,free): the highest force peak recorded only during the free-decay portion of the signal, after the forced motion has ended.
The results for the FRMS, Fmax,free and Fmax,tot are presented in “Tables 5–7”, and plotted as a function of fill level in Figures 11a–c respectively.
TABLE 5
| Volume [L] | FRMS,#1shake [N] | FRMS,#2shakes [N] | FRMS,#5shakes [N] |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 100 | 9 | 12 | 15 |
| 200 | 36 | 52 | 78 |
| 300 | 102 | 186 | 246 |
| 400 | 139 | 242 | 340 |
| 500 | 171 | 309 | 482 |
| 600 | 185 | 333 | 536 |
| 700 | 193 | 352 | 521 |
| 800 | 193 | 356 | 435 |
| 900 | 194 | 292 | 292 |
| 1000 | 52 | 54 | 46 |
| 1050 | 25 | 21 | 16 |
RMS forces for respectively 1, 2 and 5 shakes at different filling levels.
TABLE 6
| Volume [L] | Fmax,free,#1shake [N] | Fmax,free,#2shakes [N] | Fmax,free,#5shakes [N] |
|---|---|---|---|
| 0 | 7 | 6 | 8 |
| 100 | 28 | 52 | 68 |
| 200 | 113 | 195 | 416 |
| 300 | 200 | 380 | 868 |
| 400 | 282 | 517 | 1103 |
| 500 | 348 | 609 | 1158 |
| 600 | 396 | 691 | 1255 |
| 700 | 434 | 737 | 1286 |
| 800 | 505 | 792 | 1027 |
| 900 | 548 | 837 | 760 |
| 1000 | 662 | 749 | 602 |
| 1050 | 553 | 404 | 241 |
Maximum free-slosh forces for respectively 1, 2 and 5 shakes at different filling levels.
TABLE 7
| Volume [L] | Fmax,tot,#1shake [N] | Fmax,tot,#2shakes [N] | Fmax,tot,#5shakes [N] |
|---|---|---|---|
| 0 | 25 | 25 | 26 |
| 100 | 55 | 71 | 107 |
| 200 | 113 | 213 | 483 |
| 300 | 220 | 388 | 932 |
| 400 | 302 | 532 | 1107 |
| 500 | 367 | 647 | 1211 |
| 600 | 443 | 742 | 1255 |
| 700 | 520 | 822 | 1450 |
| 800 | 566 | 846 | 1464 |
| 900 | 679 | 985 | 1579 |
| 1000 | 768 | 1200 | 1203 |
| 1050 | 899 | 897 | 1033 |
Overall maximum forces for respectively 1, 2 and 5 shakes at different filling levels.
FIGURE 11
The RMS force, which represents the average energy of the sloshing during the free-decay phase, first rises with increasing fill level before falling again (
Figure 11a). This behavior is governed by a fundamental trade-off:
Mass-limited regime (0–200 L): at low fill levels, the sloshing forces are limited by the small fluid mass available to generate force, even if the wave motion itself is vigorous.
Energy dissipation and suppression regime (≥300 L): as the fill level increases, the fluid possesses more and more mass to generate higher forces. However, two concurrent phenomena begin to limit and then reduce the sloshing energy:
Suppression by roof impact: the decreasing headspace acts as a hard physical limit. The sloshing wave is suppressed as it impacts the roof of the IBC, which prevents the wave from fully developing its amplitude. This effect occurs at progressively lower fill levels for more energetic inputs: at approximately 900 L for 1 shake, 800 L for 2 shakes, and 600 L for 5 shakes.
Non-linear energy dissipation: as the wave becomes larger and larger and/or interacts with the roof, the fluid motion becomes strongly non-linear and may undergo wave breaking. This leads to a dramatic increase in turbulence. This is an effective energy dissipation mechanism that convert the organized energy of the coherent wave into heat, causing the system’s damping to rise sharply.
The combined effect of this physical suppression and increased internal damping leads to a significant decrease in the sustained sloshing energy, resulting in the corresponding drop in the measured RMS force at higher fill levels.
The maximum force during free decay (Fmax,free) shows a similar trend (Figure 11b). However, the peak of this curve is consistently shifted approximately 100 L to the right (occurs at a higher fill level) compared to the RMS peak. This subtle but important difference highlights the distinction between the overall energy of the sloshing and its peak instantaneous force. Even as the roof interaction begins to suppress the wave and reduce its average energy (the RMS value), the initial free decay-motion can still permit the generation of a high-magnitude impact wave due to the increased fluid inertia. In this transitional regime, the peak force may continue to rise with added mass even though the subsequent oscillations are more rapidly damped. Only once geometric confinement becomes dominant does further mass addition cease to increase the maximum force during free decay.
The behavior of the overall maximum force (Fmax,tot), shown in Figure 11c, is markedly different from the free-slosh metrics. Instead of peaking early, this force continues to rise with the fill level across a much wider range. This behavior must be understood as the superposition of two distinct force components: the baseline inertial force (Finertial = mass(V) acceleration) and the dynamic sloshing force (Fslosh). Unlike high-acceleration scenarios where the inertial force would be overwhelmingly dominant, the resonant excitation used here generates a dynamic slosh force (Fslosh) that is of a comparable or even greater order of magnitude than the baseline inertial force. Therefore, the Fmax,tot curve is initially driven upwards by both the increasing mass for the inertial part and the developing wave for the sloshing part. However, the Fslosh component reaches a maximum and is then severely suppressed by roof impacts at high fill levels. The Fmax,tot curve only begins to decline at the point where this sharp reduction in the dynamic slosh force becomes greater than the steady, incremental increase of the inertial force.
The results from this initial, broad-range test series successfully map the general trends of the sloshing forces (FRMS & Fmax, free). Crucially, this initial mapping reveals that both force profiles are sensitive to the energy of the input motion. As the number of excitation cycles increases, the peaks of the FRMS and Fmax, free curves not only grow in magnitude but also consistently shift to lower fill levels. This observation creates a critical uncertainty: the absolute worst-case fill level for vehicle stability after movement has not yet been definitively identified, as it is unclear if this trend would continue with even more energetic inputs. The data strongly indicates that a high-risk zone lies within the 400 L–700 L range.
Simultaneously, while the Fmax,tot peak occurs at higher fill levels, it is essential to confirm that the forces in this region also reach a saturation point and do not continue to rise indefinitely with more cycles.
Therefore, a two-pronged approach for focused testing was devised:
A detailed test series within the 400–700 L range with an extended number of cycles (6-7-8-9-10-13) to pinpoint the saturation point of the free sloshing forces.
A complementary test series at high fill levels (800–1000 L), also with an extended number of cycles (6-7-8-9-10-13), to verify the saturation behavior during the movement also.
The results of these tests are summarized in Figures 12a–c for FRMS, Fmax, free and Fmax,tot respectively.
FIGURE 12
The detailed tests conducted within the 400–700 L range were designed to precisely identify the saturation point of the sustained disturbance forces (FRMS and Fmax, free). The results, shown in Figures 12a,b, confirm that both metrics reach a maximum at a fill level of 600 L when subjected to 6 excitation cycles.
Crucially, for a higher number of cycles (e.g., 10 and 13), both force metrics do not increase further, instead, they decrease significantly. This indicates that beyond a certain energy input threshold, the fluid motion transitions from a large, coherent resonant wave, into a highly turbulent chaotic state. In this regime energy is rapidly dissipated through internal vortices and the self-breaking of the wave, which loses its phase coherence with the driving frequency. This chaotic motion is less effective at generating a large, organized sloshing force, leading to the observed lower FRMS and Fmax, free values. This confirms that the absolute worst-case condition for post-maneuver vehicle stability has been successfully identified.
The focused tests also resolved the behavior of the overall maximum force (Figure 12c). The analysis confirms that the highest Fmax,tot values are also generated around the 6-cycle mark. This is explained by the metric’s definition: Fmax,tot records the peak at any point during the test. If the true maximum force is achieved after 6 cycles, the 13-cycle test simply captures the same value before the forces begin to diminish due to the chaotic state described above.
To conclude, the repeatability of the force measurements was evaluated. For this purpose, the test consisting of 5 shakes was repeated 10 times at both 300 L and 700 L. The mean and standard deviation of , , and were determined. Subsequently, the coefficient of variation (CV) was calculated. For 300 L, CV values of 0.92%, 1.76%, and 2.81% were obtained. For 700 L, the corresponding values were 1.02%, 2.32%, and 0.36%. Since all coefficients of variation are below 3%, the repeatability of the force measurements can be considered good, with several values below 1%, indicating very good repeatability.
4 Recommendations for future research
In this study, a 1000 L industrial IBC was tested for sloshing using excitation along the width direction (shortest side). This represents a limitation of the current work, as excitation along the longitudinal direction of the container was not investigated. In addition, only horizontal sloshing forces were measured. It would be valuable to also measure the free-surface elevation and correlate it with the measured forces. This would require the implementation of an additional measurement technique, such as a vision-based system, within the experimental setup.
Future experiments can extend the presented methodology by applying the same test approach to excitation along the length direction of the IBC. Moreover, different IBC volumes (e.g., 300 L, 600 L, 800 L) can be investigated to assess scaling effects. If only the worst-case fill level is of interest, the number of required experiments can potentially be reduced, as the present study already provides an indication of the critical range. The methodology can also be extended to other commonly used container types with different geometries, such as drums.
Furthermore, only water was used as the working fluid in this study due to its low viscosity and high susceptibility to sloshing. Future work could investigate the influence of more viscous fluids on the observed worst-case conditions. While some industrial liquids exhibit similar low-viscosity behavior, many others have higher viscosities, resulting in increased damping and reduced sloshing amplitudes. As testing with large volumes of such fluids may be impractical, a scaled version of the IBC and test setup could be considered for these investigations.
Finally, this work focuses on the characterization of sloshing in IBCs and does not provide direct solutions to mitigate it. Therefore, further research is required into active control strategies for reducing sloshing in practical intralogistics applications. In this context, techniques such as command shaping, and more specifically input shaping, appear to be promising approaches for future investigation.
5 Conclusion
First and foremost, a comparison was made between the theoretical and practical determination of the resonance frequencies. Since these are horizontal shaking movements in a direction perpendicular to the IBC wall, only the odd resonance modes will occur in the fluid wave. Test 1 also showed that mainly the first and third modes were significantly present in the FFT, while higher modes such as the fifth mode have amplitudes that are on average 30 times smaller.
The theoretically calculated and practically measured frequencies correspond very well with each other from a fill level of 400L. This is despite the fact that the IBC has rounded corners, the walls are not infinitely rigid, and water has internal friction. At lower levels, there is a small deviation due to the fact that the valve at the bottom of the IBC disturbs the wave. However, the deviation remains very limited, with a maximum of 5.4% for the first mode and 0.8% for the third mode.
The damping ratio for the first mode of water in the IBC is of course very low due to the properties of water. Nevertheless it is not the same for every fill level, for the low fill levels (<300 L) the damping is higher due to the outlet valve at the bottom of the IBC that disturbs the wave. Fill levels between 300 and 700 L are approximately constant and the high fill levels (>700 L) have an increased damping ratio due to the higher turbulence caused by the interaction with the lid of the IBC. If the number of shakes is increased the damping ratio will also increase because the turbulence in the liquid is higher.
The worst case fill level of the IBC, in terms of residual horizontal sloshing forces after the movement, is around 600L. Lower fill levels do have enough space to create high waves, but have less mass. Higher fill levels have extra mass but less space to fully develop into a large wave. The curves also show that from a fill level above 100 L, the forces start to increase significantly, with forces above 1200 N around the worst case fill level. These forces will play a role in whether or not the IBC can be accurately positioned with, for example, AGVs or AMRs.
Based on the findings of this study, sloshing mitigation strategies should primarily focus on limiting excitation near the first and third resonant modes, as these were found to contribute most significantly to the observed force amplitudes. Higher-order modes exhibited considerably lower amplitudes and have therefor negligible impact on the AGV and AMR systems.
In this context, control-based approaches such as command shaping, and more specifically input shaping, appear to be promising techniques to reduce sloshing by avoiding excitation of these dominant modes. The corresponding resonance frequencies can be estimated using analytical formulations, while experimental identification provides increased accuracy, particularly at lower fill levels.
For more viscous fluids, further investigation is required to detect potential deviations between theoretical predictions and experimentally observed behavior.
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, upon request and without undue reservation.
Author contributions
TH: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Project administration, Software, Validation, Visualization, Writing – original draft, Writing – review and editing. WN: Conceptualization, Writing – review and editing. CV: Writing – review and editing. RN: Writing – review and editing. MJ: Supervision, Writing – review and editing. DA: Supervision, Writing – review and editing.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
forces, intermediate bulk container, intralogistics, liquids, sloshing
Citation
Hellin T, Nica W, Vandecauter C, Noens R, Juwet M and Accoto D (2026) Experimental characterization of horizontal sloshing forces in an IBC under low-acceleration intralogistics motions. Front. Mech. Eng. 12:1829839. doi: 10.3389/fmech.2026.1829839
Received
13 March 2026
Revised
17 April 2026
Accepted
20 April 2026
Published
29 May 2026
Volume
12 - 2026
Edited by
Pavlo Krot, Wrocław University of Science and Technology, Poland
Updates
Copyright
© 2026 Hellin, Nica, Vandecauter, Noens, Juwet and Accoto.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Thomas Hellin, thomas.hellin@kuleuven.be
ORCID: Thomas Hellin, orcid.org/0009-0004-9816-5432; Ward Nica, orcid.org/0000-0001-9783-2126; Camille Vandecauter, orcid.org/0009-0008-0726-3336; Robbe Noens, orcid.org/0009-0007-5088-1474; Marc Juwet, orcid.org/0000-0002-0188-7411; Dino Accoto, orcid.org/0000-0001-9039-5488
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.