# Calibration of the low-velocity impact experiments

Ramzi Othman, U. Khashaba

► To cite this version:

Ramzi Othman, U. Khashaba. Calibration of the low-velocity impact experiments. Polymer Composites, 2025, 46 (14), pp.12765-12779. 10.1002/pc.29896. hal-05515259

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# Calibration of the low-velocity impact experiments

Ramzi Othman<fnref n="1" />,<fnref n="2" /> ID

U. A. Khashaba<fnref n="1" />,<fnref n="3" /> ID

## Correspondence

Ramzi Othman and U. A. Khashaba,
Mechanical Engineering Department,
Faculty of Engineering, King Abdulaziz
University, P.O. Box 80248, Jeddah 21589,
Saudi Arabia.

Email: rothman1@kau.edu.sa and

ukhashaba@kau.edu.sa

## Abstract

Low-velocity impact experiments are essential for evaluating the mechanical behavior and impact resistance of composite materials in engineering applications. These experiments are primarily based on the measurement of the force-time signal and a few parameters, including impact energy, impact velocity, and mass. All other parameters, such as displacement, velocity, and absorbed energy, are subsequently derived from these fundamental inputs. This paper first investigates the influence of miscalibration of the force signals, highlighting its significant impact on the reliability of the calculated displacement, velocity, and absorbed energy. Specifically, a miscalibration error of 10% can result in more than 20% error in maximum deflection and over 40% error in absorbed energy. Second, this study introduces a novel force calibration method based on the principle that composite materials absorb negligible energy during low-energy impacts below the damage threshold. The proposed method is validated using a theoretical mass-spring model, demonstrating its effectiveness in improving force signal accuracy. Calibration enhances the consistency of maximum force values, reducing deviation by an average of 6.45%, and aligning force-displacement curves with theoretical predictions. Additionally, the calibration procedure significantly reduces the residual absorbed energy from an initial range of 1.084–1.226 J to a negligible 0.013–0.030 J, yielding values that are more reliable for tests conducted at impact energies below the damage threshold. Hence, this study highlights the importance of proper calibration in improving the reliability of impact experiments and provides a robust framework for achieving accurate and consistent results.

## Highlights

* A new force calibration method for low-velocity impact tests is proposed.

* Incorrect calibration significantly affects the accuracy of impact energy and displacement.

* The method ensures accurate energy absorption estimates for polymer composites.

* It is validated using a mass-spring model, improving force signal reliability.

* The approach improves the repeatability and accuracy of impact experiments.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2025 The Author(s). *Polymer Composites* published by Wiley Periodicals LLC on behalf of Society of Plastics Engineers.

<footnote n="1">¹Mechanical Engineering Department,
Faculty of Engineering, King Abdulaziz
University, Jeddah, Saudi Arabia</footnote>

<footnote n="2">²Ecole Centrale de Nantes, Institut de
Recherche en Génie Civil et Mécanique,
UMR CNRS 6183, Nantes, France</footnote>

<footnote n="3">³Mechanical Design and Production
Engineering Department, Faculty of
Engineering, Zagazig University,
Zagazig, Egypt</footnote>
### KEYWORDS

experimental validation, force calibration, low-velocity impact, mass-spring model, mis-calibration effects, polymer composites

# 1 | INTRODUCTION

Composites are increasingly used across various engineering sectors, including automotive, aerospace, and marine applications. By combining two or more materials, composites exploit the strengths of each component to produce materials with tailored properties, enabling the design of solutions optimized for specific applications. Their widespread use is largely due to their versatility and ability to deliver a diverse range of properties, making them highly adaptable and effective for numerous engineering challenges.¹

With increasing concerns about environmental sustainability and global warming intensifying, there is a growing demand for lightweight materials that can reduce vehicle mass and energy consumption. Composites are particularly promising in this context due to their exceptional specific mechanical properties. These materials offer high stiffness and strength while maintaining low density, making them ideal for lightweight applications.² However, composites are susceptible to low-velocity impacts (LVIs), which can cause damage that may not be immediately visible but significantly impairs their mechanical performance.³ Such impacts can result in substantial reductions in stiffness and strength due to the induced damage.

Low-velocity impact (LVI) testing is commonly employed to evaluate impact-induced damage in composite materials.⁴ Gunoz et al.⁵ examined the impact of various environmental conditions on the LVI behavior of hybrid composites, demonstrating that exposure to seawater and acidic environments leads to stiffness reduction and heightened damage susceptibility. Furthermore, repeated LVI events have been shown to induce progressive damage in composite structures, especially when subjected to internal stresses like hydrostatic pressure, as observed in glass-reinforced plastic (GRP) composite pipes.⁶

Khashaba observed that, compared to the low-velocity impact (LVI) performance of woven GFRP at room temperature,⁷ the mismatch in the coefficients of thermal expansion between the fibers and the matrix at elevated temperatures⁹ induces significant interlaminar thermal stresses. This stress accelerates the initiation and propagation of matrix cracks, particularly under higher impact loads. The damage diameter, perimeter length, and area were measured using a new AutoCAD Image Processing (ACIP) technique, US Patent No. US11373296B1.¹⁰

Key mechanical characteristics, such as absorbed energy, damage threshold force, and damage threshold energy, are typically derived from the contact force at the mass-specimen interface.¹¹,¹² The force-time signal is measured using a strain-gage Wheatstone bridge mounted on the falling striker. However, to the best of the authors' knowledge, no systematic calibration technique for the force sensor has been proposed in the literature. Typically, the calibration of force measurement is performed by the machine manufacturer and may be subject to periodic review. However, there is an urgent need for a systematic calibration procedure.

The split Hopkinson pressure bar device, another impact testing machine, also employs Wheatstone strain gage bridges to measure impulse forces. For this technique, several calibration procedures have been developed to systematically adjust fluctuations in strain gage outputs.¹³ These procedures rely on parameters such as the striker bar's kinetic energy or maximum strain and are simple enough to be conducted daily or at the beginning of an experimental program, ensuring high accuracy in strain gauge readings. Khan and Iqbal¹⁴ validated their calibration method using an analytical approach based on one-dimensional stress wave propagation. Zhou et al.¹⁵ proposed a calibration method that utilized a specially shaped striker. However, these calibration methods cannot be directly applied to drop-weight machines.

This paper proposes a new calibration method for measuring contact forces in low-velocity impact experiments. This technique is designed to be simple and systematically implemented at the start of each experimental program, utilizing the existing instrumentation in falling mass machines without requiring any modifications.

# 2 | LOW-VELOCITY IMPACT EXPERIMENTS

## 2.1 | Fundamentals

In low-velocity impact (LVI) experiments, also known as falling mass impact tests, a mass is raised to a specific height (h) and then released to fall freely onto a flat specimen, as illustrated in Figure 1. The height from which the mass is dropped determines the impact velocity (vᵢ), which is the speed of the mass upon reaching the specimen.
<figure><img src="image_2.png" /><figcaption>FIGURE 1 Simplified schematic of a falling mass impact.</figcaption></figure>

As the mass is released, its potential energy (mgh), is progressively transformed into kinetic energy. By the time the mass reaches the specimen, the impact velocity can be expressed as:

$$v_i = \sqrt{2gh} \qquad (1)$$

The kinetic energy of the mass just before impact, known as the impact energy (Ei), is determined by the mass and the height from which it is released. This relationship is expressed as:

$$E_i = \frac{1}{2}mv_i^2 = mgh \qquad (2)$$

For a given falling mass m and a desired impact energy Ei, the required falling height can be calculated using Equation (2) as

$$h = \frac{E_i}{mg} \qquad (3)$$

As the mass comes into contact with the specimen, its kinetic energy starts to decrease and is gradually transferred to the specimen. The mass continues to decelerate until it comes to a complete stop, at which point the entire impact energy is absorbed as strain energy within the specimen. The elastic portion of this strain energy is then released, propelling the mass upward once again.

Separation between the mass and the specimen occurs when the elastic strain energy of the specimen has been fully converted back into the mass's kinetic energy. At this moment, the mass's velocity and kinetic energy are referred to as the residual velocity (vr) and residual energy (Er), respectively. The difference between the initial impact energy and the residual energy represents the energy absorbed by the specimen:

$$E_a = E_i - E_r \qquad (4)$$

In LVI experiments, the primary measurement recorded is typically the contact force, F(t). All other parameters, such as acceleration, velocity, displacement, and energy, are subsequently derived from this measured force. Consequently, any miscalibration of the force sensor directly impacts all experimental results, potentially compromising the accuracy and reliability of the data obtained from these experiments.

## 2.2 | Processing

In low-velocity impact (LVI) experiments, the contact force F(t) is directly measured using a force sensor attached to the falling mass. This is the sole direct measurement recorded during the tests. Based on the contact force, critical parameters such as acceleration, velocity, displacement, and energy can be derived over time.

Once the falling mass contacts the specimen, it experiences two forces: the downward gravitational force and the upward rebound force. The impactor's acceleration can be expressed as:

$$a(t) = g - \frac{F(t)}{m} \qquad (5)$$

The impactor velocity is obtained by integrating the acceleration over time:

$$v(t) = v_i + \int_{0}^{t} a(\tau) d\tau = v_i + \int_{0}^{t} \left( g - \frac{F(\tau)}{m} \right) d\tau, \qquad (6)$$

where vi is the initial impact velocity that can be calculated from the falling height using Equation (1). The displacement is then determined by integrating the velocity:

$$u(t) = \int_{0}^{t} v(\tau) d\tau \qquad (7)$$

The displacement of the impactor corresponds to the deflection of the specimen during the time the impactor is in contact with it and before separation. Thus, a single measurement of the contact force provides valuable information about both the impactor and the specimen, which can also be confirmed in terms of energy analysis.

Knowing the contact force F(t) and using the derived velocity v(t) from Equation (6), the energy of the falling mass can be calculated as:
$$E(t) = \int F(\tau) du = \int_{0}^{t} F(\tau) v(\tau) d\tau \quad (8)$$

The residual energy ($E_r$) is defined as the energy of the impactor when it rebounds and separates from the specimen, at which point the contact force returns to zero. The absorbed energy ($E_a$) is then calculated from Equation (4) as the difference between the initial impact energy ($E_i = \frac{1}{2}mv_i^2$) and the residual energy ($E_r$).

The absorbed energy ratio $\rho$ is expressed as:

$$\rho = 100 \frac{E_i - E_r}{E_i} = 100 \frac{E_a}{E_i} \quad (9)$$

Additionally, the contact force measurement allows for the determination of other important parameters, such as the peak contact force ($F_{max} = \max F(t)$) and the maximum deformation of the specimen ($\delta_{max} = \max u(t)$).

## 2.3 | Mis-calibration effects

This section examines the effects of mis-calibration in force measurement on the parameters derived from it, as detailed in Section 2.2. To achieve this, low-velocity impact tests were conducted, followed by the introduction of artificial mis-calibration, and the resulting errors were quantified.

### 2.3.1 | Method

The CFRE laminates (500 × 500 × 5.1 mm³) were fabricated from 25 layers of T300-3 k plain-woven carbon fiber fabrics (200 g/m²) combined with a YPH-120-23A/B epoxy matrix using the prepreg technique. The mechanical properties of the CFRE composite laminates were previously documented by Khashaba et al.¹⁶⁻¹⁸ Impact tests were conducted at room temperature (RT) using a CEAST 9340 falling weight impact machine to perform low-velocity impact tests on carbon fiber-reinforced epoxy (CFRE) composite laminates in accordance with ASTM D5628 standards. Figure 2A,B shows the experimental setup. The specimens were prepared in accordance with ASTM D7136 standards and cut using an abrasive waterjet technique to achieve precise dimensions of 101.6 × 152.4 × 5.1 ± 0.1 mm, as shown in Figure 2C. Appropriate cutting parameters were selected through the machine software to minimize kerf damage, delamination, and eliminate matrix burning at the specimen edges, providing advantages over other traditional cutting methods.¹⁹ The specimen was clamped using four fixtures on a steel plate with a 125 × 75 mm² rectangular window, placed inside the thermal conditioning chamber, as shown in Figure 2B. The specimens were subjected to energy levels of 20, 40, and 60 J. Specific details about the impact mass, velocity, and corresponding falling height are provided in Table 1.

For each impact test, the contact force $F(t)$ was recorded, and an artificial mis-calibration factor $\epsilon_c$, ranging between 0.8 and 1.2 was applied. This corresponds to

<figure><img src="image_3.png" /><figcaption>FIGURE 2 (A) Falling weight impact tester model CEAST 9340, (B) fixing the specimen on the machine table with open windows of 125 × 75 mm using four clamps, and (C) dimensions of impact specimen, ASTM D 7136.</figcaption></figure>
<table><caption>TABLE 1 Inputs of low-velocity impact tests on CFRE composite plates.</caption><thead><tr><th>Impact energy<br>E<sub>i</sub>(J)</th><th>Falling mass<br>m(kg)</th><th>Impact velocity<br>v<sub>i</sub>(m/s)</th><th>Falling height<br>h(mm)</th></tr></thead><tbody><tr><td>20</td><td>7.732</td><td>2.37</td><td>286.4</td></tr><tr><td>40</td><td>7.732</td><td>3.35</td><td>572.2</td></tr><tr><td>60</td><td>7.732</td><td>4.10</td><td>857.1</td></tr></tbody></table>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>FIGURE 3 Force versus time: (A) $E_i = 20J$, (B) $E_i = 40J$, and (C) $E_i = 60J$.</figcaption></figure>

a calibration error between -20% and +20%. Examples of correctly calibrated and mis-calibrated force signals are shown in Figure 3. To evaluate the effects of calibration errors, Equations (5) through (8) were utilized to calculate the mass acceleration, velocity, displacement, and energy. However, $\epsilon_c F(t)$ was substituted for $F(t)$ to introduce the calibration error. The factor $\epsilon_c$ simulates either an amplification or damping error in the actual force $F(t)$. Such errors can arise from electrical or environmental artifacts.

* ($\epsilon_c = 1$): Represents a correctly calibrated force signal.

* ($\epsilon_c > 1$): Simulates an amplification (over-estimation) error in the force measurement.

* ($\epsilon_c < 1$): Simulates a damping (under-estimation) error in the force measurement.

## 3 | RESULTS AND DISCUSSION

### 3.1 | Effects of mis-calibration on the displacement

This section examines the impact of mis-calibration on the measurement of displacement. Figure 4A presents displacement signals calculated using Equations (5) through (7) for five different mis-calibration factors $\epsilon_c$: 0.8, 0.95,
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>FIGURE 4 Displacement versus time: (A) $E_i = 20J$, (B) $E_i = 40J$, and (C) $E_i = 60J$.</figcaption></figure>

1, 1.05, and 1.2 (Figure 3). These values correspond to calibration errors of ±5% and ±20% on the force signal. The analysis is conducted for three impact energy levels: 20 J (Figure 4A), 40 J (Figure 4B), and 60 J (Figure 4C).

The results demonstrate that the impact of mis-calibration is negligible at the beginning of the test but becomes increasingly significant over time. In the early stages, the displacement curves are nearly identical, indicating that calibration errors initially have little effect on the calculated displacement signals. However, as the test progresses, the discrepancy between the displacement curves steadily increases, becoming substantial by the end of the test. This phenomenon can be attributed to the method of displacement calculation, which involves double-time integration of the force signal, as presented in Equations (6) and (7). While these integrations act as a filter during short durations, they also lead to the accumulation of errors over time. Consequently, the displacement error at a given time $t$ incorporates not only the errors in the force signal at $t$ but also the cumulative errors from preceding moments, resulting in significant deviations for longer durations.

Figure 5 illustrates the relationship between maximum displacement error and force mis-calibration error.

<figure><img src="image_9.png" /><figcaption>FIGURE 5 Effect of the calibration error on the error in the maximum deformation.</figcaption></figure>

The results in this figure demonstrate that under-calibration of the force leads to an overestimation of the maximum deformation of the composite plate, whereas over-calibration results in an underestimation. This error is asymmetric, with under-calibration causing more significant errors than over-calibration. Furthermore, it is notable that maximum displacement errors become more
pronounced at higher impact energies, where the dis-
placement deviations surpass the artificially introduced
mis-calibration errors.

The findings of this study underscore the importance
of precise force calibration, as even minor inaccuracies
can lead to significant deviations in displacement mea-
surements, particularly over extended time durations.
Calibration errors can result in data that are nearly
impossible to interpret within a physical framework. Spe-
cifically, scientific literature extensively documents that,
in force-displacement curves obtained from low-velocity
impacts, the unloading curve lies beneath the loading
curve.⁴,⁷⁻⁹,¹¹,¹²,²⁰⁻²⁹ The unloading curve should never
exceed the loading curve, as such behavior implies nega-
tive energy absorption. This implies that the impacted
specimen returns more energy to the dropped impactor
than it initially received. Such a scenario suggests energy
generation, which contradicts the fundamental principle
of energy conservation. Similarly, the unloading curve
should not intersect the loading curve, as this would con-
tradict the principle of energy conservation.

However, Santulli and Caruso³⁰ reported instances in
which the unloading path crosses the loading path
in force-displacement curves (see Figure 2 in Santulli and
Caruso³⁰). In these cases, the residual displacement was
also found to be negative. This suggests that, following the
separation of the specimen and the dropped impactor, the
position of the impacted specimen was higher than its
position prior to impact. Such a phenomenon is challeng-
ing to explain within the context of hysteresis loops.

In this study, this peculiar behavior is simulated by
introducing force miscalibration. Figure 6 shows a compar-
ison between force-displacement curves obtained from cor-
rectly calibrated and miscalibrated (+20% amplification)
force signals. It is evident that the correctly calibrated
force-displacement curve exhibits the expected behavior

<figure><img src="image_10.png" /><figcaption>FIGURE 6 Force-displacement curves showing unloading path crossing loading one.</figcaption></figure>

commonly observed in the literature. The unloading path
remains below the loading path, and the residual displace-
ment (i.e., the displacement when the force drops to zero)
is positive. On the other hand, the miscalibrated force-
displacement curve exhibits anomalous behavior, with the
unloading path crossing the loading path and the residual
displacement becoming negative. This can be explained by
the amplified force signal causing an overestimation of the
upward force decelerating the dropped impactor. As a
result, the impactor experiences initial over-deceleration
during its downward motion and subsequent over-
acceleration during its upward motion. Consequently, the
upward motion of both the impactor and the specimen,
just before separation, is overestimated, leading to a nega-
tive residual displacement.

**3.2** | Effects of mis-calibration on energy

This section explores the impact of mis-calibration on
energy measurements, extending the analysis conducted
earlier for displacement. Five mis-calibration factors,
ε<sub>c</sub> = 0.8, 0.95, 1, 1.05, and 1.2, are considered, and
Equation (8) is applied to the mis-calibrated force data to
compute impact energy. The results are presented in
Figure 7A–C for impact energies of 20, 40, and 60 J,
respectively.

Figure 7 reveals that force mis-calibration introduces sig-
nificant errors in energy calculations derived from force
measurements. Unlike displacement errors, which emerge
only after a certain duration, energy errors are evident from
the beginning of the test. The mis-calibrated energy curves
alternate between overestimation and underestimation,
lying above the correct energy curve in some regions and
below it in others. Specifically, when the force is amplified
(ε<sub>c</sub> > 1), the energy curves initially exceed the correct energy
curve, indicating an overestimation, but later fall below it,
indicating underestimation. Conversely, when the force is
damp or under-calibrated (ε<sub>c</sub> < 1), the energy curves ini-
tially fall below the correct curve and subsequently rise
above it, leading to an underestimation at the beginning
and an overestimation by the end of the test.

This behavior can be attributed to the fact that energy
calculations rely on both force and displacement. In the
early stages of the test, displacement errors are negligible,
and only the force mis-calibration affects the energy.
Amplified forces result in overestimated energy, while
damped forces lead to underestimated energy. Over time,
however, displacement errors become significant and
counteract the force mis-calibration errors. Consequently,
displacement errors dominate, causing underestimated
energy when the force is amplified and overestimated
energy when the force is under-calibrated.
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>FIGURE 7 Energy versus time: (A) $E_i = 20J$, (B) $E_i = 40J$, and (C) $E_i = 60J$.</figcaption></figure>

Notably, for low-impact energies, Figure 7A, significant miscalibration can yield non-physical results. Specifically, the energy may become negative, suggesting that the composite plate restitutes more energy to the impactor than it initially absorbed. This outcome implies an unrealistic creation of energy by the composite plate, highlighting the critical importance of accurate force calibration to maintain physical validity in energy measurements.

Figure 8 illustrates the correlation between absorbed energy error and force mis-calibration error. Specifically, an under-calibration of the force results in an overestimation of the absorbed energy, while over-calibration leads to an underestimation. This error distribution is asymmetric, as over-calibration induces more pronounced errors compared to under-calibration. Moreover, the impact of these errors is magnified at low impact energies, where absorbed energy errors reach exceptionally high levels.

Errors in energy calculations result from inaccuracies in both force and displacement, which have opposing effects. While force errors remain constant throughout the test, displacement errors are negligible at the beginning but increase significantly over time. Additionally, displacement tends to be underestimated when force is overestimated, and vice versa. Figure 8 illustrates the

<figure><img src="image_15.png" /><figcaption>FIGURE 8 Effect of the calibration error on the error in the absorbed energy.</figcaption></figure>

errors in absorbed energy, defined as the energy value recorded after mass-specimen separation. At this stage, energy errors are primarily influenced by displacement errors. Since displacement errors accumulate due to time integration, they are not significantly affected by the impact energy. Therefore, the absolute errors recorded on the absorbed energy are slightly affected by the initial
impact energy. In terms of relative values, the errors increase as the impact energy decreases.

It is worth noting that miscalibration effects, such as negative absorbed energy, have also been reported in the literature. For example, Singh et al.³¹ observed negative energy absorption during low-velocity impact (LVI) tests on asymmetric and symmetric GFRP laminates (see Figure 6 in Singh et al.³¹). This behavior aligns with the observations in Figure 7A, where a negative energy absorption was recorded when the force signal was mis-calibrated and amplified by 20%.

Over-amplification of the force signal leads to an underestimation of energy absorption, which can result in negative energy absorption values. This outcome, however, contradicts the principle of energy conservation. Absorbed energy is calculated by subtracting the restituted energy from the incident energy. The incident energy is transferred from the impactor to the specimen, while the restituted energy is the amount returned from the specimen to the impactor.

The energy restituted to the impactor can never exceed the energy transferred to the specimen. Therefore, the absorbed energy should always be positive. A negative absorbed energy value indicates that the specimen is returning more energy to the impactor than it received, which is physically impossible. Such a result cannot be accepted within the framework of energy conservation. Negative absorbed energy is a clear indicator of force mis-calibration and should serve as a warning sign that the measurement process requires correction.

# 4 | CALIBRATION PROCEDURE

## 4.1 | Method

In the previous section, we demonstrated the significant impact of force miscalibration on the accuracy of displacement, velocity, and energy predictions in LVI experiments. This highlights the necessity of developing a reliable calibration technique for such experiments. Calibration requires reference values for comparison. The approach used in this work relies solely on the measurements available within the machine, eliminating the need for additional sensors. However, it requires a separate test with predictable behavior.

Our calibration method is based on the well-defined behavior of elastic impact observed at low impact energy. More precisely, the calibration technique involves using low-impact energy LVI tests, as it is widely accepted that damage occurs only above a certain energy threshold.⁴,⁷⁻⁹,¹¹,³²⁻³⁴ At low-impact energies, no absorbed energy is typically recorded. The suggested approach involves the following steps:

1. Conduct low-impact energy LVI experiments and record force-time signals.

2. Process the force-time signals using Equations (5) to (8) to determine the impactor's energy and Equation (9) to calculate the absorbed energy.

3. Adjust the calibration factor and repeat step 2 iteratively until the absorbed energy becomes negligible.

The calibration procedure is summarized in Figure 9. Initially, correct calibration is assumed, with the calibration factor $c_F$ set to 1. Equations (5) to (9) are then used to calculate the absorbed energy. If the absorbed energy is not negligible, the calibration factor is adjusted to a value different from 1. The absorbed energy is recalculated using Equations (5) to (9).

This iterative process—adjusting the calibration factor and recalculating the absorbed energy—continues until the absorbed energy becomes insignificant.

The main advantage of the proposed calibration procedure is that it does not require additional sensors to be installed on drop-weight machines, as it utilizes the existing instrumentation. However, its primary limitation is the assumption that no damage occurs during the calibration test. This necessitates prior knowledge or an estimation of the damage threshold energy. Therefore, it is

<figure><img src="image_16.png" /><figcaption>FIGURE 9 Calibration procedure.</figcaption></figure>
advisable to use a specimen with well-characterized
behavior for the calibration test. Notably, the calibration
test material does not need to be the same as the one
used in the experimental campaign. Additionally, the
method assumes minimal variation in the calibration
coefficient between tests. To ensure reliability, it is
strongly recommended to perform a calibration test at
the beginning of each experimental program.

## 4.2 | Low-impact energy LVI tests

The experimental procedure described in Section 2.3.1 was also employed in this study. Low-velocity impact (LVI) tests were conducted on carbon fiber-reinforced epoxy (CFRE) composite laminates using the CEAST 9340 falling-weight impact machine, following ASTM D5628 standards. The specimen material and geometry remained consistent with those detailed in Section 2.3.1.

The key difference in this study is the use of a lower impact energy level of 5 J. Khashaba and Othman⁴ previously investigated the same composite material over a wide range of impact energies. Their findings indicated that the threshold damage impact energy was approximately 9.6 J, as determined through an extrapolation of the absorbed energy ratio. Therefore, the impact energy of 5 J employed here is expected to result in an insignificant amount of absorbed energy. Table 2 presents the

mass, impact velocity, and drop height for the four
experiments considered in this calibration procedure.

## 4.3 | Calibration results

The calibration procedure described in Section 4.1 was applied to the low-energy, low-velocity impact experiments conducted as outlined in Section 4.2. For each test, the calibration factor $c_F$ was calibrated to minimize the absorbed energy post-calibration, with the objective of achieving a value as close to zero as possible. This approach assumes that the composite plates used in the tests fully restitute energy for an impact energy of 5 J, as the damage threshold impact energy for these specimens is approximately 9.6 J.⁴

Figure 10 illustrates the force-time and energy-time curves before calibration. As shown in Figure 10B, the absorbed energy is positive. While the absorbed energy is not high, it is also not zero, as it should be for this range of impact energy (below the damage threshold energy). Using the calibration method presented in Figure 9, new force-time and energy-time curves were obtained and shown in Figure 11. Notably, Figure 11B demonstrates that the absorbed energy after calibration is negligible, reflecting a more accurate representation of the composite plate behavior before damage. The corrected force-time curves in Figure 11A display a more **symmetric profile** than the

<table><caption>TABLE 2 LVI experiments at 5 J of impact energy.</caption><thead><tr><th>Test number</th><th>Falling mass<br>m(kg)</th><th>Impact velocity<br>v<sub>i</sub>(m/s)</th><th>Falling height<br>h(mm)</th></tr></thead><tbody><tr><td>Test #1</td><td>3.132</td><td>1.79</td><td>163.4</td></tr><tr><td>Test #2</td><td>3.132</td><td>1.79</td><td>163.4</td></tr><tr><td>Test #3</td><td>4.132</td><td>1.56</td><td>124.1</td></tr><tr><td>Test #4</td><td>4.132</td><td>1.56</td><td>124.1</td></tr></tbody></table>

<figure><img src="image_17.png" /><figcaption>FIGURE 10 Force and energy signals before calibration: (A) Force versus time, and (B) Energy versus time.</figcaption></figure>
<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>FIGURE 11 Force and energy signals after calibration: (A) Force versus time, and (B) Energy versus time.</figcaption></figure>

<table><caption>TABLE 3 Calibration results.</caption><thead><tr><th>Test</th><th>Calibration factor c<sub>F</sub></th><th>Absorbed energy before calibration (J)</th><th>Absorbed energy after calibration (J)</th></tr></thead><tbody><tr><td>Test #1</td><td>1.069</td><td>1.226</td><td>0.021</td></tr><tr><td>Test #2</td><td>1.065</td><td>1.161</td><td>0.013</td></tr><tr><td>Test #3</td><td>1.060</td><td>1.084</td><td>0.022</td></tr><tr><td>Test #4</td><td>1.063</td><td>1.138</td><td>0.030</td></tr><tr><td>Average</td><td>1.064</td><td>1.152</td><td>0.022</td></tr><tr><td>Range</td><td>0.009</td><td>0.142</td><td>0.017</td></tr></tbody></table>

uncalibrated data, indicating enhanced accuracy in force measurement. This symmetry strengthens the reliability of the correction method and results in more precise energy calculations, as shown in Figure 11B.

The calibration results, presented in Table 3, include the calibration factor, the absorbed energy before calibration, and the absorbed energy after calibration. The calibration factor ranges from 1.06 to 1.069, with an average value of 1.064. The absorbed energy prior to calibration ranges from 1.084 to 1.226 J, while the absorbed energy post-calibration ranges from 0.013 to 0.03 J. These narrow ranges across the three parameters—calibration factor, absorbed energy prior to calibration, and absorbed energy post-calibration—demonstrate the good repeatability of the results. Additionally, the calibration factor exceeding 1 indicates that the original force signals were under-calibrated, resulting in an overestimation of the absorbed energy.

## 4.4 | Validation of the calibration procedure

In this section, we aim to validate the calibration results. In Section 4.4.1, a theoretical elastic mass-spring model is developed for the low-velocity impact (LVI) experiments. Since the impact energy is low, it is expected that little to no damage occurs as a result of the impact. Using the

analytical model, key characteristics are identified and analyzed in Section 4.4.2 for both calibrated and uncalibrated force signals. The results demonstrate that the calibrated signals align more closely with the predictions of the analytical model compared to the uncalibrated signals.

### 4.4.1 | Mass-spring model

The LVI test, illustrated in Figure 1, can be simplified as a mass-and-spring system, as shown in Figure 12. In this representation, the mass corresponds to the falling impactor, while the spring represents the equivalent stiffness of the impacted specimen, including contact stiffness, bending stiffness, and membrane stiffness.35–37 The target's mass is considered negligible compared to the impactor's mass, reducing the LVI test to a single degree-of-freedom mass-and-spring system.

For this calibration procedure, the impact energy is assumed to be low enough to treat the spring as linearly elastic. Under this assumption, the displacement, velocity, acceleration, and force are described by the following equations:

$$u(t) = -\frac{v_i}{\omega_n} \sin \omega_n t, \quad (10)$$
<figure><img src="image_20.png" /><figcaption>FIGURE 12 Mass-and-spring representation of the LVI test.</figcaption></figure>

$$v(t) = -v_i \cos \omega_n t, \quad (11)$$

$$a(t) = v_i \omega_n \sin \omega_n t, \quad (12)$$

and

$$F(t) = m v_i \omega_n \sin \omega_n t, \quad (13)$$

where $v_i$ is the initial impact velocity, and $\omega_n$ is the natural frequency. These equations represent the solution to the free vibration of a mass-and-spring system, where the impact velocity is treated as the initial velocity of the mass. They are valid for the first half-period of oscillation, after which the impactor rebounds off the specimen. As a result, the force-time curve resembles a half-sine wave. The time $T_{max}$, at which the force reaches its peak, corresponds to a quarter of the natural period. The natural frequency is then given by:

$$\omega_n = \frac{\pi}{2T_{max}}, \quad (14)$$

From Equation (13), the maximum force $F_{max}$ is related to the natural frequency and can be expressed as:

$$F_{max} = m v_i \omega_n = \frac{\pi m v_i}{2 T_{max}}, \quad (15)$$

Equation (15) reveals an essential relationship: the maximum force $F_{max}$ is directly linked to $T_{max}$, the time at which the force peaks. In other words, the measurement of $F_{max}$ must be consistent with that of $T_{max}$. Any inconsistency could indicate a calibration issue in the force measurement.

### 4.4.2 | Comparison of uncalibrated and calibrated results

Equation (15) is particularly significant. For both the uncalibrated and calibrated force signals, it is possible to

identify the maximum force, $F_{max}$, and the time, $T_{max}$, at which this maximum force occurs. Using these parameters, a theoretical maximum force, $F_{max,th}$, can be calculated from Equation (16). Ideally, $F_{max}$ and $F_{max,th}$ should closely match. To evaluate this alignment, a ratio is defined as follows:

$$\xi_F = \frac{F_{max}}{F_{max,th}} = \frac{F_{max}}{\frac{\pi m v_i}{2 T_{max}}} = \frac{2 T_{max} F_{max}}{\pi m v_i} \quad (16)$$

The closer this ratio is to 1, the more accurate the force signal.

Table 4 presents the results for the maximum force obtained from both uncalibrated and calibrated signals. It includes the time $T_{max}$, which remains unaffected by any calibration, as well as the maximum force, $F_{max}$, before and after calibration. Finally, it provides the ratio $\xi_F$, calculated using both uncalibrated and calibrated signals.

For the uncalibrated force signals, the ratio $\xi_F$ ranges between 0.825 and 0.857, with an average value of 0.836. For the calibrated force signals, the ratio $\xi_F$ ranges between 0.875 and 0.911, with an average value of 0.89. The calibration factors derived in Section 4.3 consistently yield values closer to 1 compared to the raw, uncalibrated signals. This demonstrates that the calibrated force signals are more consistent with the elastic mass-spring model. On average, the calibration improves the estimation of the maximum force by 6.45%. These findings confirm the effectiveness of the calibration procedure in enhancing the accuracy of force signals. The closer agreement between the calibrated signals and the theoretical model highlights the importance of calibration using low-energy, low-velocity impact experiments, ensuring reliable and repeatable results.

To further validate the calibration technique, a comparison is made of the times associated with three key events: the time of maximum force ($T_{max}$), the time of maximum displacement ($T_{max,d}$), and the time of maximum energy ($T_{max,e}$). According to the mass-spring model, these three times should ideally coincide.

Table 5 presents the values of these times for both uncalibrated and calibrated force signals. From the table, it is evident that the times for maximum displacement and maximum energy are consistent, both before and after calibration. However, these times differ from the time of maximum force. It is noteworthy that the values for $T_{max,d}$ and $T_{max,e}$ move closer to $T_{max}$ after the calibration of the force signals.

The observed results provide additional evidence for the effectiveness of the calibration procedure. While a perfect match between the three times is not achieved, the closer alignment of $T_{max,d}$ and $T_{max,e}$ with $T_{max}$ after calibration indicates an improvement in the accuracy
<table><caption>**TABLE 4** Maximum force: Comparison of the uncalibrated and calibrated signals.</caption><thead><tr><th rowspan="2">Test</th><th rowspan="2">T<sub>max</sub> (ms)</th><th colspan="2">F<sub>max</sub> (kN)</th><th colspan="2">Ratio ξ<sub>F</sub></th></tr><tr><th>Before Calib.</th><th>After Calib.</th><th>Before Calib.</th><th>After Calib.</th></tr></thead><tbody><tr><td>Test #1</td><td>1.274</td><td>5.734</td><td>6.129</td><td>0.829</td><td>0.887</td></tr><tr><td>Test #2</td><td>1.273</td><td>5.766</td><td>6.140</td><td>0.834</td><td>0.888</td></tr><tr><td>Test #3</td><td>1.510</td><td>5.532</td><td>5.864</td><td>0.825</td><td>0.875</td></tr><tr><td>Test #4</td><td>1.552</td><td>5.592</td><td>5.945</td><td>0.857</td><td>0.911</td></tr><tr><td>Average</td><td></td><td></td><td></td><td>0.836</td><td>0.890</td></tr></tbody></table>

<table><caption>**TABLE 5** Time at maximum force, displacement, and energy: Comparison of the uncalibrated and calibrated signals.</caption><thead><tr><th rowspan="2">Test</th><th rowspan="2">T<sub>max</sub> (ms)</th><th colspan="2">T<sub>max,d</sub> (ms)</th><th colspan="2">T<sub>max,e</sub> (ms)</th></tr><tr><th>Before Calib.</th><th>After Calib.</th><th>Before Calib.</th><th>After Calib.</th></tr></thead><tbody><tr><td>Test #1</td><td>1.274</td><td>1.451</td><td>1.383</td><td>1.451</td><td>1.383</td></tr><tr><td>Test #2</td><td>1.273</td><td>1.466</td><td>1.402</td><td>1.467</td><td>1.402</td></tr><tr><td>Test #3</td><td>1.510</td><td>1.710</td><td>1.643</td><td>1.710</td><td>1.643</td></tr><tr><td>Test #4</td><td>1.552</td><td>1.714</td><td>1.645</td><td>1.715</td><td>1.645</td></tr></tbody></table>

and consistency of the experimental data. This reinforces the validity of the calibration technique in refining low-energy, low-velocity impact measurements.

## 5 | CONCLUSIONS

In low-velocity impact (LVI) experiments, the force signal serves as the foundational metric from which all critical parameters, including displacement and absorbed energy, are derived. Given its central role, any miscalibration in the force signal can propagate significant errors, undermining the accuracy and reliability of experimental data. This paper addressed two key objectives: investigating the effects of force signal miscalibration on displacement and energy measurements and proposing a novel calibration method to mitigate these inaccuracies.

The study provided a comprehensive analysis of the impact of miscalibration, emphasizing the cumulative nature of displacement errors over time and the complex interplay between force and displacement errors in energy measurements. Notably, the results highlighted how even minor calibration errors could lead to significant deviations, particularly at higher impact energies, underscoring the critical need for accurate force calibration. It was also demonstrated that displacement errors are negligible at the early stages but accumulate over time due to double-time integration, leading to substantial deviations at longer durations. The study shows that under-calibration amplifies displacement errors, resulting in overestimated maximum displacements, while over-

calibration suppresses displacement signals. These errors are particularly pronounced at higher impact energies. For energy measurements, miscalibration introduces errors from the onset of the tests. Amplified forces initially lead to overestimated energy values, while damped forces cause underestimation. However, over time, displacement errors become predominant, reversing these trends.

Building on these findings, a robust calibration procedure was introduced and validated through its application in low-energy, low-velocity impact experiments. By adjusting the calibration factor to minimize the absorbed energy for impacts below the damage threshold, the proposed method effectively corrected under-calibrated force signals. The analysis of key parameters, such as maximum force and the times associated with maximum force, displacement, and energy, further validated the procedure, demonstrating its ability to enhance the accuracy and consistency of experimental data. The main advantage of the calibration method is that it does not require additional sensors. However, it assumes that no damage occurs during the calibration test. Therefore, a specimen with well-known behavior should be used for calibration.

In conclusion, this study not only highlights the pivotal role of force signal calibration in LVI experiments but also introduces an effective calibration technique to ensure reliable and repeatable results. By addressing the challenges of miscalibration, the findings contribute to advancing the accuracy of impact measurements in composite material testing, offering valuable insights for future experimental methodologies.
ACKNOWLEDGMENT

Open Access publishing facilitated by the Deanship of Scientific Research (DSR) at King Abdulaziz University, as part of the Wiley – King Abdulaziz University agreement.

CONFLICT OF INTEREST STATEMENT

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

DATA AVAILABILITY STATEMENT

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

ORCID

Ramzi Othman https://orcid.org/0000-0001-9037-5671

U. A. Khashaba https://orcid.org/0000-0002-7148-2807

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**How to cite this article:** Othman R, Khashaba UA. Calibration of the low-velocity impact experiments. *Polym Compos.* 2025;46(14): 12765-12779. doi:10.1002/pc.29896