3.1. Peak deceleration of the flexible impactor

Figure 2(a) shows the impactor body deceleration as a function of time after the moment of first impact for experiments with $V=4\ {\rm m}\ {\rm s}^{-1}$ and $R=22.23$ mm. As a baseline, we conduct experiments with an equivalent rigid impactor which has the same total mass (within 2.4 %) as the flexible impactor. The rigid impactor is created by omitting the flexure assembly and rigidly fixing the nose to the impactor body. The rigid impactor experiences a sharp peak force during early times in the slamming phase which occurs up to a submergence depth of approximately one radius. However, despite the large forces during the slamming phase, the speed of the rigid impactor decreases only minimally – on average 1.6 % by $t = R/V$ (one radius depth) – and it proceeds at a high rate into the bath, forming a crown splash and eventually a trailing air cavity as it pierces deeper into the water (Supplementary movie 2). The impactor speed is plotted directly against time for these experimental cases in Appendix B. The splash and cavity formation for the flexible impactor, as seen in figure 2(b) or Supplementary movie 3, are similar to the rigid case during the slamming phase, though the cases with the 3-D printed noses sometimes feature larger splashes and enhanced cavity size, likely due to the higher surface roughness and lower wettability compared with the machined aluminium noses (Duez et al. Reference Duez, Ybert, Clanet and Bocquet2007; Aristoff & Bush Reference Aristoff and Bush2009; Speirs et al. Reference Speirs, Mansoor, Belden and Truscott2019b; Watson et al. Reference Watson, Bom, Weinberg, Souchik and Dickerson2021). Increased hydrophobicity has also been shown to increase the force of impact on a sphere throughout the cavity-forming phase (Truscott, Epps & Techet Reference Truscott, Epps and Techet2012), though the effect is predominantly isolated to larger depths than focused on in the present work. Although the interfacial physics of the cavity-forming phase are certainly rich, the deceleration of the rigid impactor during the time after the slamming phase in our experiments is relatively uninteresting, increasing only slowly as the steady state drag develops. The acceleration profiles of the flexible impactor body entering at $4\ {\rm m}\ {\rm s}^{-1}$ with $k=4.29\ {\rm N}\ {\rm mm}^{-1}$ and $7.80\ {\rm N}\ {\rm mm}^{-1}$ (figure 2a) follow the typical intuition for a ‘cushioned’ impact; the impulse from the water entry ‘shock’ is spread out over a longer time. Consequently, the maximum deceleration is decreased compared with the rigid case and the peak occurs at a later time, around one radius of submergence depth. The oscillations of the impactor persist for several cycles, dissipating slightly due to damping as the impactor continues its descent. Like in the rigid case, the mean of the flexible impactor deceleration curve increases gradually at later times as the steady state drag develops. However, surprisingly, the peak deceleration for the flexible impactor with $k=13.98\ {\rm N}\ {\rm mm}^{-1}$ is higher than the rigid case, indicating that, in general, the body acceleration during impact can either increase or decrease as a result of adding elasticity. Furthermore, we find that whether the impact acceleration increases or decreases for a given stiffness depends on the impact speed. The peak deceleration of the rigid and flexible impactors is plotted against impact speed in figure 2(c). For all experiments in this figure, the total impactor mass is held constant and, in the flexible impacts, the nose contains 12 % of the total mass. The peak deceleration of the rigid impactor increases like the square of the impact speed as shown in the inset in figure 2(c), confirming the anticipated inertially dominated regime. At the highest speed, all of the flexible impactors experience a peak deceleration reduction compared with the rigid case but the opposite is true at the lowest speed, where all of the flexible impactors experience a peak deceleration increase. This result portends an important subtlety in the design of elastic force reduction mechanisms for applications. Namely, the stiffness of the impactor must be carefully matched to the operating conditions or else the addition of elasticity can have significant deleterious effects. In order to rationalize, interpret and synthesize these observations, we develop a reduced mathematical model of the impact forces in what follows.

Figure 2.

(a) Plots of impact deceleration versus time for the rigid ($M=0.578$ kg) and flexible ($M=0.592$ kg and $k=4.29$, 7.80, $13.98\ {\rm N}\ {\rm mm}^{-1}$) cases with $V=4\ {\rm m}\ {\rm s}^{-1}$ and $R=$ 22.23 mm. The results are averaged over five trials and the shaded regions indicate the standard deviation between trials. The corresponding impactor velocities are presented in Appendix B. (b) High-speed images of the flexible impactor ($k=4.29\ {\rm N}\ {\rm mm}^{-1}, V= 4\ {\rm m}\ {\rm s}^{-1}, R=22.23\ {\rm mm}$) as it enters the water. The time of each photograph corresponds to the time axis label to the left of the image in (a). A video version is available as Supplementary movie 3. (c) Peak deceleration of the flexible impactor body depends on both stiffness and impact speed, so it can experience either a peak acceleration increase or decrease compared with the rigid case. The error bars show the standard deviation between five trials. The inset log–log plot of the rigid data shows that the peak acceleration increases like the square of the impact velocity.

3.2. Rigid added mass model

The hydrodynamic force during the slamming phase can be understood in the context of the added mass effect, originally applied by von Kármán (Reference von Kármán1929) to the problem of water entry. Since the impact is inertially dominated, the impact force may be thought of as the rate at which the impactor must transfer momentum to a virtual quantity of fluid mass – $m(x)$ in figure 3(a) – in order to accelerate that added fluid mass to the current impactor speed $\dot {x}(t)$. Conservation of momentum on the impactor and added mass system dictates that

(3.1)\begin{equation} \frac{\textrm{d}}{\textrm{d} t} \left[(M+m)\dot{x} \right] = 0, \end{equation}

where $M$ is the mass of the impactor. Integrating with the initial conditions that $m(0) = 0$ and $\dot {x}(0)=V$ yields the following expression for the acceleration during impact (Abrate Reference Abrate2011):

(3.2)\begin{equation} \ddot{x} ={-} \frac{(M V)^2}{(M+m)^3} \frac{\textrm{d} m}{\textrm{d}\kern0.7pt x}. \end{equation}

Figure 3.

(a) Schematic of the rigid added mass model. The impinging body must accelerate an effective mass $m(x)$ of fluid as it enters the water; hence the impact force can be obtained from an expression of momentum conservation for the outlined system. (b) The forces from rigid impact experiments at several speeds with different nose radii ($R=22.23$, $M=0.578$ kg or $R=29.64$, $M=0.537$ kg) collapse onto a single curve when using an inertial scaling. The Shiffman & Spencer ${C}_{F}$ curve (dashed line) agrees excellently with the experiments. The shaded regions around the experimental curves indicate standard deviation of at least three trials with the lower speed experiments exhibiting greater variation between trials due to the lower signal-to-noise ratio. The nose used for the $R=29.64$ mm rigid experiments is not a complete hemisphere so the data is truncated accordingly (importantly, the peak force is captured accurately). (c) We extend the classic added mass model to the case of a flexible impactor by introducing a trailing spring and mass. The outlined system for which we write conservation of momentum now includes the external contribution from the spring. (d) The deceleration of the flexible impactor body measured in experiments (solid coloured lines, $V=4\ {\rm m}\ {\rm s}^{-1}$, $R=22.23$ mm, $\alpha =0.12$, $M=0.592$ kg) is captured well by the flexible added mass model (dashed lines). Furthermore, the model predicts that the deceleration of the impactor centre of mass (dotted lines) deviates only slightly from the rigid theoretical ${C}_{F}$ curve (solid black line), suggesting that the elasticity does not have a strong influence on the hydrodynamics in this regime. The shaded region around the experimental curves indicates the standard deviation between five trials.

Thus the impact force profile can be calculated given only the added mass $m$ as a function of depth $x$. The added mass function for a sphere was derived by Shiffman & Spencer (Reference Shiffman and Spencer1945a, Reference Shiffman and Spencer1947) from potential flow around an axisymmetric lens. By neglecting deformation of the interface and higher-order velocity terms, the dynamic boundary condition reduces to a statement of zero potential at $x=0$, and hence the impact problem is equivalent to uniform flow around the submerged portion of the body, mirrored about the undisturbed interface. Shiffman & Spencer later refined the model with additional theoretical and experimental corrections which account for the deformation of the interface and wetting of the sphere (Shiffman & Spencer Reference Shiffman and Spencer1945b). This corrected added mass function $m(x)$ is used in the present work and is reproduced along with its derivative $\textrm {d} m / \textrm {d}\kern 0.06em x$ in the supporting datasets. Although the corrected curve is only available from Shiffman & Spencer (Reference Shiffman and Spencer1945b) up to a depth of $x=R$, we assume it smoothly tapers to zero at $x=1.15R$ and remains zero thereafter. Whether the value of $\textrm {d} m / \textrm {d}\kern 0.06em x$ for $x>R$ is left constant, set abruptly to zero or smoothly tapered to zero makes no significant quantitative difference in our predictions presented here. These functions are the only externally derived components of our model, which is otherwise self-contained and described completely herein. Shiffman & Spencer also define a dimensionless number

(3.3)\begin{equation} \sigma = M \left/ \left(\tfrac{4}{3} {\rm \pi}R^3 \rho \right)\right. \end{equation}

which compares the impactor mass with the mass of an equivalent volume of fluid assuming a spherical impactor. When $\sigma$ is sufficiently large, as in the current experiments, $M \gg m$ (that is, the impactor mass is much larger than the peak added mass) and (3.2) can be reduced to

(3.4)\begin{equation} F ={-}V^2 \frac{\textrm{d} m}{\textrm{d}\kern0.7pt x}, \end{equation}

where $F$ is the impact force on the body. As a consequence of large $\sigma$, the speed of the body does not change appreciably during the slamming phase and the impact force reaches a limiting curve (in practice, when $\sigma =3$, the maximum impact force is already 96 % of the infinite $\sigma$ case). Hence, we can define an impact drag coefficient ${C}_{F}$ based on $\textrm {d} m / \textrm {d}\kern 0.06em x$ as

(3.5)\begin{equation} F(t) = \tfrac{1}{2} {C}_{F}(t) \rho V^2 {\rm \pi}R^2. \end{equation}

Assuming a heavy impactor with a given nose shape, ${C}_{F}$ is a function of time (or, interchangeably, depth) alone. The impact drag coefficient for the high $\sigma$ limit reported by Shiffman & Spencer (Reference Shiffman and Spencer1945b) agrees excellently with rigid experiments performed with $V=2$ to $6\ {\rm m}\ {\rm s}^{-1}$ and $R=22.23$ mm or 29.64 mm, as shown in figure 3(b). The experimental curves collapse with the inertial force scale $\frac {1}{2} \rho V^2 {\rm \pi}R^2$ and impact time scale $R/V$, indicating that the added mass during the slamming phase is independent of both the impactor speed and the size of the splash and air cavity, which vary throughout the experimental range of impact speeds. Because the nose is not a complete hemisphere in the $R=29.64$ mm rigid experiments (it is a complete hemisphere in all other experiments), the force curves in figure 3(b) are truncated at the point where the outer edge of the nose would first contact the undisturbed free surface. However, the peak force, which is the primary quantity of interest and used for comparison with the flexible experiments, is still captured accurately. Furthermore, the rigid results for both radii agree well with the numerous classic experimental studies for the force of impact on a rigid sphere in terms of both scaling and peak impact drag coefficient (Watanabe Reference Watanabe1934; May & Woodhull Reference May and Woodhull1948, Reference May and Woodhull1950; Richardson Reference Richardson1948; Moghisi & Squire Reference Moghisi and Squire1981).

3.3. Flexible added mass model

We extend the added mass model to the case of an impacting simple harmonic oscillator by considering an external spring force on the nose-plus-added-mass system, as illustrated in figure 3(c). The parameter $\alpha$ is introduced which equals the ratio of the nose mass to the total impactor mass, $M$. Hence conservation of momentum for the impactor nose-plus-added-mass system is written as

(3.6)\begin{equation} \frac{\textrm{d}}{\textrm{d}t} \left[ (\alpha M + m) \dot{x}_n \right] = k (x_b - x_n), \end{equation}

where $x_b$ and $x_n$ are the positions of the impactor body and nose, respectively. Integrating as before, the equation of motion for the nose is

(3.7)\begin{equation} \ddot{x}_n = \frac{\textrm{d}}{\textrm{d}t} \left[ \frac{1}{\alpha M + m} \int_0^t k (x_b-x_n) \, \textrm{d} \tau \right] - \frac{\alpha^2 M^2 V^2}{(\alpha M + m)^3} \frac{\textrm{d}m}{\textrm{d}\kern0.7pt x_n}. \end{equation}

The second term on the right-hand side is equivalent to the hydrodynamic resistance from added mass as presented in (3.2), while the first term is new and accounts for an additional momentum exchange via the spring element. Since the only force felt by the body comes from the spring, the equation of motion for the body is simply

(3.8)\begin{equation} \ddot{x}_b = \frac{1}{(1-\alpha) M} k (x_n - x_b). \end{equation}

Using the Shiffman & Spencer added mass function $m$, (3.7) and (3.8) are numerically integrated with an RK4 scheme with trapezoidal rule for the integral terms and the deceleration of the body is compared with experimental results at $V = 4\ {\rm m}\ {\rm s}^{-1}$ and $\alpha =0.12$ in figure 3(d). The model agrees well with the experimental data and captures the transition in the peak deceleration for experiments with $k=13.98\ {\rm N}\ {\rm mm}^{-1}$ although it tends to underpredict the peak deceleration, most significantly for the $k=4.29\ {\rm N}\ {\rm mm}^{-1}$ experiments. In this case, the peak deceleration occurs after $tV/R=1$ in the region where Shiffman & Spencer's potential flow theory predicts that ${C}_{F}$ goes to zero. Experimentally, however, the contribution of form drag leads to non-zero impact force at late times as seen in figures 2(a) and 3(b). When the peak deceleration occurs at later non-dimensional times (such as with low stiffness or high speed), the accuracy of the model can be improved by including the contribution of form drag – directly from the experimental curves in figure 3(b), for instance, as shown in Appendix C – in the added mass function $m$. Figure 3(d) also shows the prediction for the deceleration of the flexible impactor centre of mass, $\ddot {x}_c = \alpha \ddot {x}_n + (1-\alpha ) \ddot {x}_b$, which is equal to the overall deceleration due to the hydrodynamic force. Despite its simplicity, this model captures the two-way coupling between the hydrodynamic force and impactor elasticity in the problem, with the added mass force term varying both explicitly as a function of nose depth and implicitly based on the nose and body velocities through the structural coupling. However, the hydrodynamic force deviates only slightly from the rigid case, indicating that a further simplified one-way coupled model may be adequate. By formally assuming $\alpha M \gg m$ (that is, the nose mass is much larger than the peak added mass), (3.7) and (3.8) simplify to

(3.9)\begin{gather} (1-\alpha) M \ddot{x}_b = k (x_n - x_b) , \end{gather}

(3.10)\begin{gather}\alpha M \ddot{x}_n = k (x_b - x_n) + F(t) , \end{gather}

where $F(t)$ is the hydrodynamic force in the rigid case given by (3.5). In our experiments, the ratio $\alpha M / \max [m]$ takes values from 1.7 to 9.3. A solution to (3.9) and (3.10) essentially comprises the structural response of the flexible impactor to the hydrodynamic forcing associated with the impact of a rigid sphere at constant velocity.

3.4. Peak force transition

In order to understand the mechanism by which a high stiffness impactor can experience increased force compared with the equivalent rigid impactor, (3.9) and (3.10) can be recast in modal coordinates, resulting in equations of motion for the rigid body mode (centre of mass) and elastic mode as

(3.11)$$\begin{gather} \ddot{x}_c = \frac{F(t)}{M}, \end{gather}$$

(3.12)$$\begin{gather}\ddot{\delta} + \frac{k}{\alpha (1-\alpha) M} \delta = \frac{-F(t)}{\alpha M}, \end{gather}$$

where $\delta = x_b-x_n$. When $\delta =0$, the spring is at its natural length. By combining (3.9) and (3.12), the acceleration of the impactor body can be written as

(3.13)\begin{equation} \ddot{x}_b = \frac{F(t)}{M} + \alpha \ddot{\delta}. \end{equation}

Consequently, the acceleration of the impactor body (the quantity of interest, and directly measured in experiment), can be understood as the sum of contributions from the hydrodynamic force and the elastic mode. Furthermore, from (3.12), it can be seen that $\ddot {\delta }>0$ in very early times (as $F(t)<0$), and thus the spring initially serves to isolate the body from the hydrodynamic forcing. For all of the presently studied impacts, the hydrodynamic force has the same characteristic shape with a sharp increase to the peak followed by a slower decay (figure 3d solid black and dotted lines). For impacts with low stiffness springs, the contribution from the elastic mode counteracts the peak in hydrodynamic force throughout the slamming phase and, as a result, the body experiences reduced peak deceleration. On the other hand, in the high stiffness case, the impactor body experiences the high frequency oscillations of the elastic mode (with $\ddot {\delta }<0$ earlier in the slamming phase) on top of the hydrodynamic forcing and thus the peak deceleration is increased. This interpretation suggests that the relationship between the hydrodynamic time scale and the impactor oscillation frequency plays a key role in determining whether the peak force will increase or decrease: if the elastic mode begins to oscillate before the hydrodynamic force decays, the impactor body will experience increased force. This ratio of time scales emerges directly when non-dimensionalizing the governing equations (3.12) and (3.13) using the length scale $R$ and the time scale $R/V$ in order to reach

(3.14)$$\begin{gather} \ddot{\tilde{x}}_b = \frac{3}{8} \frac{{C}_{F}}{\sigma} + \alpha \ddot{\tilde{\delta}}, \end{gather}$$

(3.15)$$\begin{gather}\ddot{\tilde{\delta}} + {R}_{F}^2 \tilde{\delta} ={-} \frac{3}{8} \frac{{C}_{F}}{\alpha \sigma}, \end{gather}$$

where the non-dimensional variables are marked with tildes. The new non-dimensional parameter ${R}_{F}$ is defined as

(3.16)\begin{equation} {R}_{F} = \sqrt{\frac{k}{M \alpha (1-\alpha)}} \frac{R}{V}. \end{equation}

This so-called hydroelastic factor is the ratio of the time scale of the hydrodynamic loading to the free fundamental oscillation period of the elastic impactor. Since (3.15) has the same form as an undamped simple harmonic oscillator subjected to external forcing, the non-dimensional force on the impactor body may be predicted by computing the convolution of the hydrodynamic forcing and the elastic unit impulse response as

(3.17)\begin{equation} \frac{M \ddot{x}_b}{\frac{1}{2} \rho V^2 {\rm \pi}R^2} = {R}_{F} \int_0^{\tilde{t}} {C}_{F} (\tau) \sin ({R}_{F} (\tilde{t} - \tau)) \, \textrm{d} \tau.\end{equation}

Figures 4(a) and 4(b) show the non-dimensional maximum impact force $M \max [\ddot {x}_b]=M [\ddot {x}_b]_m$ and time of peak force $t_{m}$ as a function of ${R}_{F}$ for flexible impact experiments with $V=2$ to $6\ {\rm m}\ {\rm s}^{-1}$, $k=4.29$ to $13.98\ {\rm N}\ {\rm mm}^{-1}$, $\alpha =0.12$ or 0.24 and $R=22.23$ mm or 29.64 mm. The experiments collapse to a single curve which agrees excellently with the model in (3.17). At some critical value near ${R}_{F} \approx 2$, the peak force on the flexible impactor exceeds the non-dimensional peak force on the equivalent spherical rigid impactor which is approximately 1.05. The results of the two-way coupled model in (3.7) and (3.8) are plotted as well; as previously discussed, the two-way coupled model tends to underpredict the maximum impact force but captures the time of peak force more accurately than the convolution integral. Thus, we have shown that the force on the body depends only on the impact drag coefficient ${C}_{F}$ for the equivalent rigid case, which is a consequence of the nose geometry, and the hydroelastic factor ${R}_{F}$, which depends on the design of the elastic structure and impact velocity.

Figure 4.

The scaled maximum impact force (a) and the time of the peak force (b) collapse along a single curve against the hydroelastic number ${R}_{F}$ for experiments in which the impact speed, stiffness, nose radius and mass ratio are varied. The error bars, which are sometimes smaller than the marker size, show the standard deviation between at least three trials. The simplified prediction from the convolution integral in (3.17) (solid black lines) agrees well with the experiments (markers) and captures the critical hydroelastic factor near ${R}_{F}\approx 2$ at which the peak force in the flexible case equals the peak force in the equivalent rigid case (horizontal line). The marker shape indicates the impactor mass ratio and nose radius in a given experiment while the colour and opacity indicate the stiffness and impact speed, respectively. The two-way coupled added mass model from (3.7) and (3.8) is also shown, which more accurately predicts the time of the peak force. The two-way model line style (dashed, dotted, or dash–dotted) indicates the mass ratio and nose radius corresponding to a particular predicted curve as shown in the legend.

Two notable comments remain. The first regards the data point which furthest deviates from the theory in figure 4(a), corresponding to experiments with $V=2\ {\rm m}\ {\rm s}^{-1}$, $k=4.29\ {\rm N}\ {\rm mm}^{-1}$ and $\alpha =0.24$. The elastic mode of the flexible impactor is not constrained during free fall but, in all other cases, the oscillations are so slight that they have no noticeable effect on the peak acceleration. However, this data point represents the ‘worst’ case scenario with the heaviest nose, weakest spring and least time during free fall for the oscillations to dissipate. As such, we observe that the phase of the free-fall oscillation at impact significantly influences the force. In particular, $\delta (0)<0$ for these experiments – the impactor is ‘prestretched’ – which increases the impact force compared with the $\delta (0)=0$ case. This effect can be captured by changing the initial conditions of the two-way coupled model. Conversely, according to the model, a ‘precompressed’ ($\delta (0)>0$) impactor would experience reduced force at these operating parameters. This point is explored in more detail in Appendix D. The second comment is a reminder that only the slamming phase of impact is considered in the present study; in some rare cases (particularly at low $k$ or low $V$), the maximum force during slamming is exceeded at much later times when the trailing air cavity pinches off. Consequently, additional considerations must be made in these cases if the goal is to predict the maximum force during the entire impact.

3.5. High stiffness limit

Intuitively, as the stiffness of the impactor increases to sufficiently large values (corresponding to high ${R}_{F}$), its behaviour should eventually return to the ‘rigid’ case. In order to observe this behaviour, we conducted experiments with two additional flexure spring designs with high stiffness values of $k=760\ {\rm N}\ {\rm mm}^{-1}$ and $21\,600\ {\rm N}\ {\rm mm}^{-1}$. Since these values exceed the capabilities of our tensile testing machine, the stiffness values were instead extracted from the impactor natural frequencies which were measured by suspending the impactor from a low stiffness bungee and exciting the axial mode with an impact hammer. As shown in figure 5, the high stiffness experiments performed with $V=2$ to $6\ {\rm m}\ {\rm s}^{-1}$, $R=22.23$ mm and $\alpha = 0.12$ or 0.24 also collapse nicely onto the theoretical curve predicted by (3.17) despite the fact that the axial mode no longer represents the fundamental mode, with bending modes predicted to occur at lower frequencies (Appendix A). By substantially increasing the impactor's axial stiffness, it is possible to extend our experimental ${R}_{F}$ values by more than an order of magnitude while keeping other experimental parameters the same. For impacts near ${R}_{F}\approx 10$, the non-dimensional peak force achieves values that are approximately double those in the equivalent rigid case before steadily decreasing as ${R}_{F}$ is further increased. However, despite spanning nearly four orders of magnitude in stiffness in our flexible experiments, the peak of the equivalent rigid case was not recovered: even at ${R}_{F}\approx 200$, the measured (and predicted) peak force exceeds the rigid case by more than 30 % as seen in figure 5(a). In fact, according to the model, it is not until ${R}_{F}$ achieves a value near 4500 that the peak force returns to within 5 % of the rigid case. Past ${R}_{F}\approx {20}$, the time of the peak force is essentially the same as in the equivalent rigid case as demonstrated in figure 5(b). Note that the odd bumpiness in the theoretical curves at high ${R}_{F}$ is physical in origin – once ${R}_{F}$ is high enough for several impactor oscillations to occur before the hydrodynamic force reaches its peak, the time and magnitude of the peak force are highly sensitive to the relative phase of the impactor oscillations and hydrodynamic force.

Figure 5.

Additional experiments with two high stiffness impactors demonstrate that the scaled maximum impact force (a) and the time of the peak force (b) approach the equivalent rigid case (horizontal line) as ${R}_{F}$ becomes large. While the time of the peak effectively returns to the rigid case past ${R}_{F}\approx {20}$, the non-dimensional peak force still exceeds the rigid case by more than 30 % at ${R}_{F}\approx 200$. The simplified prediction from the convolution integral (solid black line) accurately captures the impactor behaviour at large ${R}_{F}$. The error bars, which are sometimes smaller than the marker size, show the standard deviation between at least three trials.

3.6. Effect of damping

A final question of high practical relevance that can now be addressed with our validated model is the influence of damping on the behaviours elucidated herein. The model culminating in (3.17) can be readily extended to an impactor which is a linearly damped harmonic oscillator. We replace the undamped unit impulse response function in the convolution with the appropriate damped unit impulse response function $g(t)$ and let

(3.18)\begin{equation} I = \int_0^{\tilde{t}} {C}_{F}(\tau) g(\tilde{t} - \tau) \, \textrm{d}\tau. \end{equation}

The unit impulse response functions $g(t)$ for the damped impactor used in (3.18) are solutions to

(3.19a–c)\begin{equation} \ddot{\tilde{\delta}} + 2 \zeta {R}_{F} \dot{\tilde{\delta}} + {R}_{F}^2 \tilde{\delta} = 0 \quad \tilde{\delta}(0)=0 \quad \dot{\tilde{\delta}}(0)=1 \end{equation}

which are given by

(3.20)\begin{gather} g(\tilde{t}) = \frac{1}{{R}_{F}}\sin({R}_{F} \tilde{t} ) \quad \textrm{undamped ($\zeta=0$)}, \end{gather}

(3.21)\begin{gather}g(\tilde{t}) = \frac{1}{{R}_{F}\sqrt{1-\zeta^2}} \exp (-\zeta {R}_{F} \tilde{t}) \sin\left({R}_{F} \sqrt{1-\zeta^2} \tilde{t} \right) \quad \textrm{underdamped ($\zeta<1$)}, \end{gather}

(3.22) \begin{gather} g(\tilde{t}) = \tilde{t} \exp (-{R}_{F} \tilde{t} ) \quad \textrm{crit. damped ($\zeta=1$)}, \end{gather}

(3.23) \begin{gather} g(\tilde{t}) = \frac{1}{2 {R}_{F}\sqrt{\zeta^2-1}}\left[ \exp\left({{R}_{F} \tilde{t}\left(\sqrt{\zeta^2-1}-\zeta \right) }\right) \right.\nonumber\\ - \left.\, \exp\left({-{R}_{F} \tilde{t}\left(\sqrt{\zeta^2-1}+\zeta \right) }\right) \right] \quad \textrm{overdamped ($\zeta>1$)}. \end{gather}

Then, the dimensionless impact force on the body is given by

(3.24)\begin{equation} \frac{M \ddot{x}_b}{\frac{1}{2} \rho V^2 {\rm \pi}R^2} = {R}_{F}^2 I + 2 \zeta {R}_{F} \dot{I}. \end{equation}

Here, $\zeta$ is the impactor damping ratio defined as

(3.25)\begin{equation} \zeta = \frac{c}{2 \sqrt{M k \alpha (1-\alpha)}} \end{equation}

where $c$ is the damping coefficient. The theoretical maximum impact force versus ${R}_{F}$ is plotted for several values of $\zeta$ in figure 6. For the impactors with stiffness values $k=4.29$, 7.80 and $13.98\ {\rm N}\ {\rm mm}^{-1}$, the damping ratio $\zeta$ is $0.007 \pm 0.001$ as measured from ring down tests. Equation (3.24) predicts that damping tends to lower the peak force at high ${R}_{F}$, but increases the peak force at low ${R}_{F}$. This general trend is consistent with the physics of passive vibration isolation. As $\zeta$ increases, the critical ${R}_{F}$ where the peak force is equal to the equivalent rigid case shifts in a non-monotonic way, first increasing and then starting to decrease near critical damping. At high damping, the curve returns to the rigid case. For an impactor with a fixed stiffness that must perform over a wide range of operating conditions, an underdamped system could be designed to maintain significant peak force reduction at high speeds (low ${R}_{F}$) without incurring such a large penalty at low speeds (high ${R}_{F}$) as the undamped system.

Figure 6.

The theoretical prediction for the non-dimensional maximum impact force on a linearly damped impactor is plotted as a function of hydroelastic number ${R}_{F}$ and compared with the undamped case ($\zeta =0$) and the equivalent rigid impactor (dashed line). The $\zeta =0.007$ curve corresponds to the measured damping ratio of the flexible impactor design and differs minimally from the undamped case. An underdamped impactor could still experience substantial peak force reduction at low ${R}_{F}$ without incurring such a large peak force increase at large ${R}_{F}$ as compared with the undamped case.