Shock Energy Conversion from Translation to Rotation

Abstract

To protect structures from short duration shock load in various engineering applications, a novel energy conversion mechanism with concept design is proposed. Different from conventional methods with cellular solid/structure dissipating input translational kinetic energy to plastic strain energy with large compressive deformation, the proposed approach converts part of incident translational kinetic energy to rotational kinetic energy, which is not detrimental to the protected structure. The mechanism of energy conversion is analyzed and formulated, with key factors governing the conversion efficiency identified and discussed, which sheds light on alternative approach for short duration load mitigation.

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

1. Introduction

Structures subjected to potential hazards such as accidental explosions and bomb attacks may result in significant loss of life and property and thus should be properly protected. As an alternative to the conventional method of strengthening the structure itself or adding shock damper [1], the approach of attaching a protective cladding to structure exterior emerged [e.g., [2]]. The idea is that the cladding serves as a sacrificial layer to protect structural components during an extreme event, such as blast and impact [3, 4]. After that, replacing the damaged cladding with a new one will resume the protection capacity immediately, promoting structural resilience. The idea of sacrificial cladding not only is employed in structure protection against blast load [5], but also applies to protection of bridges against ship impact with low velocity [6]. In fact, in addition to the applications in civil engineering [e.g., [7]], this idea finds its applications in various other fields such as vehicle impact protection [8].

Almost all the claddings in this approach are sandwich structures with a solid face plate and cellular solid core in the broadest sense, including versatile cellular materials such as foams, as well as various hollow structures such as honeycomb/lattice/truss [9, 10]. Typically, the energy incident to the cladding or other lightweight structure from a shock load such as blast and impact is translational (or linear) kinetic energy [11, 12]. The cellular cores or spring [13] undergo large compressive deformation while loaded, absorbing a considerable amount of energy. As a result, the energy transmitted to the protected structure is remarkably reduced, provided that the cladding is properly designed for a specific threat level so that full densification does not occur. From the perspective of energy conversion, in this process, a large part of the blast/impact induced translational kinetic energy is converted to plastic strain energy of the cellular core.

In the present study, the concept design of a cladding system with a different energy conversion mechanism is proposed. Instead of absorbing incident translational kinetic energy as plastic strain energy, it converts part of the incident translational kinetic energy into rotational kinetic energy, which is not detrimental to the protected structure. In addition, if the cladding is properly designed, the direction of some translational kinetic energy can be reversed by spring, to further cancel some input translational kinetic energy. These two energy conversion and cancellation mechanisms combined together reduce translational kinetic energy transferred to the protected structure. In particular, this proposed methodology of kinetic energy conversion from translation to rotation may provide a different view and complement the existing shock energy mitigation strategies.

2. Mechanism of Shock Energy Conversion from Translation to Rotation

Consider a representative volume element (RVE) of the proposed cladding, separated by two dash lines, as shown in Figure 1. A series of tailored springs connecting the protected structural member and front plate maintain the designed configuration of the cladding, and rollers are used to support the plate. In the present study, a blast induced by an airburst is taken as an example of the general short duration load (the analysis with other shock loads is similar) to illustrate how the proposed mechanism works. It is assumed that the explosion is relatively distant from the cladding, so that the curvature of the blast/shock load near the cladding can be neglected. Subsequently the incident load to the cladding is reasonably simplified uniform. If a close-in strong blast/shock is applied to the cladding, localized pressure may cause significant bulging of the plate and invalidate the current model.

[figure omitted; refer to PDF]

The input energy is translational kinetic energy gained during the fluid-structure interaction between the blast and face plate (with nose, as shown in Figure 1). Subsequently, face plate and flywheel interact through gears, in which process a portion of the translational kinetic energy of the face plate is converted to the rotational kinetic energy of the flywheels. It is worth noting that the rotational energy is not detrimental to the protected structure.

There are interlocking mechanisms of various types between the flywheels and the face plate nose, one of which is typical gears. In fact, during the gear-flywheel interaction, the flywheels may undergo slight lateral swing. In detail, there are 4 sequential phases for the cladding to respond to a short duration shock load. Phase 1: fluid-structure interaction (the interaction between the blast and the face plate); Phase 2: face plate-flywheel interaction through gears; Phase 3: flywheel-spring interaction and flywheels bounce back; Phase 4: rebounding flywheel-face plate interaction (collision).

The mechanism of the first, third, and fourth phases are simple and straightforward; therefore in the present study, emphasis is placed on the second phase, the interaction between the face plate and the flywheels through the gears, in which part of the translational kinetic energy is converted to rotational kinetic energy.

3. Formulation

3.1. Phase 1: Fluid-Structure Interaction

Based on the fact that the duration of blast applied on the face plate is negligibly short compared to the structural response time, it is always assumed that the velocity of the face plate is gained impulsively in no time without any displacement [14]. Assuming the blast pressure is uniformly distributed, the reflected impulse on the RVE is$$\begin{array}{}\text{(1)}& I=m_b{v}_b\end{array}$$where $m_{\mathrm{b}}$ and $v_{\mathrm{b}}$ are the mass (unit thickness in out-of-plane direction) and impulsively gained velocity of the RVE face plate with gear, respectively.

3.2. Phase 2: Face Plate-Flywheel Interaction

This phase is most instrumental in the entire process as part of the translational kinetic energy of the face plate is converted to rotational kinetic energy of the flywheels. Similar to the treatment of linear impact problem in which the velocity is gained impulsively without any displacement, the angular velocity is assumed to be gained impulsively, without any angular displacement (rotation) [14]. To determine the angular velocity of the flywheels and linear velocity of the face plate after the interaction, several physical principles are applied.

(i) Conservation of the Linear Momentum in the Load Incident Direction:$$\begin{array}{}\text{(2)}& m_b{v}_b=m_b{v}_b^{\ast }+\mathrm{2}{m}_f{v}_f\end{array}$$where $m_{\mathrm{f}}$ and $v_{\mathrm{f}}$ are the mass of each flywheel and linear velocity at each flywheel center, respectively. ${v_{\mathrm{b}}}^{\ast }$ is the velocity of face plate after the interaction.

In this direction, linear momentum conserves since there is no external force acting on the RVE consisting of the face plate and two flywheels, and during the interaction, the forces between the face plate and the flywheels are internal forces. In fact, due to the inertia and the short interaction duration, the flywheels undergo almost no displacement, while it gains velocity almost impulsively [14], leading to negligible spring force. In nature, this process is essentially a collision process. It is assumed that face plate velocity before and after the interaction $v_{\mathrm{b}}$ and ${v_{\mathrm{b}}}^{\ast }$ is in the same direction, as shown in Figure 2.

[figure omitted; refer to PDF]

(ii) Conservation of Angular Momentum. The angular momentum of the system consisting of the face plate (the gear) and the flywheel conserves, as there is no external moment. Taking point A as reference, shown in Figure 2, the angular momentum of the system is zero. During the interaction, the angular momentum conserves; then$$\begin{array}{}\text{(3)}& m_f{v}_{f}R=I_f{\omega }_f\end{array}$$where $m_{\mathrm{f}}$ and $v_{\mathrm{f}}$ are the mass and linear velocity of flywheel center, respectively. $I_{\mathrm{f}}$ is the moment inertia of each flywheel and ${\omega }_{\mathrm{f}}$ is the angular velocity gained during the interaction.

(iii) Coefficient of Restitution. In addition, due to the short duration impact nature of the interaction between the flywheels and face plate, the coefficient of restitution is introduced for the contact point A, which is defined as the ratio of restitution impulse to compacting impulse, always simplified as the ratio of separating speed to approaching speed of two interacting objects [14]. In the present study, the interacting bodies are the face plate nose and flywheels:$$\begin{array}{}\text{(4)}& e=\frac{v_{f/A}-v_b^{\ast }}{v_b-0}\end{array}$$where $v_{\mathrm{f}/\mathrm{A}}$ is the linear velocity of the flywheel at point A after interaction, while ${v_{\mathrm{b}}}^{\ast }$ and $v_{\mathrm{b}}$ denote the speed of face plate (with nose) after and before the interaction, respectively. The zero implies that the linear velocity of flywheel at point A before the interaction is zero.

(iv) Geometrical Constraint. Further, the force applied on point A of the flywheel can be substituted with the sum of the same force applied on the flywheel center and a moment equal to the product of the force and flywheel radius, with the same resultant effect. Correspondingly, the force applied on point A results in both the translational velocity of the flywheel center and rotational velocity with respect to its center. Therefore, the linear velocity at point A is equal to the sum of linear velocity at flywheel center and relative linear velocity of point A to the flywheel center, the latter being the product of the radius and angular velocity of the flywheel after interactions, due to the geometrical constraint:$$\begin{array}{}\text{(5)}& v_{f/A}=v_f+{\omega }_{f}R\end{array}$$where $R$ is the radius of the flywheel. The combination of (2) through (5) leads to$$\begin{array}{}\text{(6a)}& {\omega }_f=\frac{\left(1+e\right)m_b{m}_{f}R}{\left(m_b+2m_f\right)I_f+m_b{m}_f{R}^2}{v}_b\text{(6b)}& v_f=\frac{\left(1+e\right)m_b{I}_f}{\left(m_b+2m_f\right)I_f+m_b{m}_f{R}^2}{v}_b\text{(6c)}& v_b^{\ast }=v_b-\frac{2\left(1+e\right)m_f{I}_f}{\left(m_b+2m_f\right)I_f+m_b{m}_f{R}^2}{v}_b\end{array}$$From (6a) and (6b), the angular and linear velocities of the flywheel after the interaction are always positive. This implies that the direction of flywheel linear velocity after the interaction is the same as that of incident face plate velocity, and the flywheel starts to rotate accordingly. From (6c), the direction of face plate linear velocity after the interaction relative to that before interaction depends on the relative relationship between the face plate mass, the radius, and rotational inertia of the flywheel, as well as the restitution coefficient between the face plate and flywheel.

To further examine the predictions, two limit situations are considered: on the one hand, if the mass and rotational inertia of the flywheel is negligibly small (both $m_{\mathrm{f}}$ and $I_{\mathrm{f}}$ approach zero), according to (6c), the direction and magnitude of the face plate after the interaction remain the same as those before the interaction; on the other hand, if the mass and moment inertia of the flywheel are extremely large (both $m_{\mathrm{f}}$ and $I_{\mathrm{f}}$ approach infinity), according to (6c), the face plate will be bounced back in the opposite direction, while the speed will be reduced by applying the coefficient of restitution. The predictions reasonably represent these two limit scenarios.

It is worth noting that as the duration of interaction between the plate and flywheel is very short, the nature of the interaction is collision. According to impact dynamics [14], in such a process, the velocity is gained with negligible displacement. The “velocity” here includes both linear and angular velocity, while the “displacement” includes both translational and rotational movement. In the present study, the plate/flywheel interaction results in both translational and rotational movement simultaneously, in an impulsive manner.

3.3. Phase 3: Flywheel-Spring Interaction and Phase 4: Possible Rebounding Flywheel-Face Plate Interaction

After the interaction with the face plate, each flywheel, together with the supporting spring, forms a single-degree-of-freedom (SDOF) system connected to the protected structural member with a spring (if the minor lateral movement of the flywheels due to the face plate, flywheel interaction is neglected). The motion in this phase is the vibration of the SDOF system with an initial velocity. It is worth noting that the protected structural member also undergoes vibration. However, due to the substantial difference between the deformation magnitude of the protected structure and this protective cladding, it is reasonable to consider the protected structure rigid and unmovable. In fact, if the cladding is properly designed for a specific blast, the deformation of the protected structure should remain in the elastic regime and negligibly small [e.g., [15]]. Subsequently, a collision between the rebounding flywheels and face plate will occur if the velocity of face plate after Phase 2 is positive.

4. Discussion with Numerical Simulation

During short duration interactions between the face plate and flywheels, mechanical energy does not conserve. Some energy is dissipated as plastic energy, as the interaction is not perfectly elastic. However, the momentum conserves, both translational and rotational, as there is no external force or moment applied to the system of concern. Therefore, the interaction process is described with combined approach of impulse-momentum method and the coefficient of restitution, rather than the work-energy relationship. The energy loss is calculated as the difference before and after the interaction:$$\begin{array}{}\text{(7)}& E_{loss}=\frac{1}{2}{m}_b{v}_b^2-\frac{1}{2}{m}_b{v}_b^{\ast 2}-2\times \frac{1}{2}{I}_f{\omega }_f^2-2\times \frac{1}{2}{m}_f{v}_f^2\end{array}$$With ${v_{\mathrm{b}}}^{\ast }$, ${\omega }_{\mathrm{f}}$, and mf expressed in terms of $v_{\mathrm{b}}$ in (6a)–(6c), in a limit case, if the interaction is assumed perfectly elastic with coefficient of restitution e=1, the energy loss is evaluated as zero. This implies that if the interaction is perfectly elastic, the process can be also described with the work-energy relationship as mechanical energy conserves, in addition to the combination of momentum method and coefficient of restitution. This ideal situation can be considered as a special case of the general scenario formulated in the present study. In the real world, the coefficient of restitution is always smaller than unity, implying inevitable energy loss thus damping effect.

For convenience of analysis, the flywheels are assumed as solid cylinders made of uniform material. In this case, $I_{\mathrm{f}}$=1/2$m_{\mathrm{f}}$R². Then the ratio of converted rotational energy to the total energy input can be simplified as$$\begin{array}{}\text{(8)}& \frac{E_{rotation}}{E_{total}}=\frac{2\times \left(1/2\right)I_f{\omega }_f^2}{\left(1/2\right)m_b{v}_b^2}=\frac{{\left(1+e\right)}^2\left(m_f/m_b\right)}{{\left(3/2+m_f/m_b\right)}^2}\end{array}$$As expected, the ratio is independent of the incident velocity of the face plate. With typical range of mass ratio of flywheel to face plate (with nose, as shown in Figure 1), and the range of coefficient of restitution between the flywheel and gear, the energy conversion ratio is plotted in Figure 3. It is seen that, with a relatively large coefficient of restitution, and a proper mass ratio of flywheel to face plate, a considerable portion of the energy input is converted into rotational kinetic energy.

[figure omitted; refer to PDF]

From (8), the ratio of rotation energy to total energy input is a function of only the mass ratio of flywheel to face plate when the coefficient of restitution is known, illustrated in Figure 4. It is obvious from (8) with some simple calculation that the maximum value without considering the coefficient of restitution is 1/6 at a mass ratio of 3/2, which is the upper bound of the conversion efficiency from translation to rotation (without consideration of coefficient of restitution).

[figure omitted; refer to PDF]

The coefficient of restitution between flywheel and face plate is associated with properties of two interacting material and can be determined with known flywheel and face plate materials. If the interaction between the flywheel and face plate is perfectly elastic, the coefficient of restitution is 1, such as the coefficient of glass to glass, not feasible in engineering practice. In this ideal case, theoretically, 2/3 of the total energy input is converted into nondetrimental rotational kinetic energy, which establishes the upper bound of shock energy conversion from translation to rotation.

A numerical study is conducted to validate the effectiveness of the proposed mechanism with a commercial hydrocode ANSYS AUTODYN. To simplify the problem and capture its nature, a 2-dimensional model is established. Each flywheel has a diameter of 40 mm with 4 rectangular gears of 6 mm by 10 mm (wide). The area of each flywheel is identical to that of the plate, implying each flywheel has the same mass as plate, provided that the materials of these three parts are the same. When subjected to a typical impulsive load such as a blast, the face plate gains velocity in a very short period and then interacts with the flywheels. In the model, the plate impacts the wheels with an initial velocity of 10 m/s, while both flywheels, separate with 2 mm gap in between, are at rest. Some gauges are set to monitor the linear velocity at some points so that the angular velocity can be calculated: one in the plate while the others are in the upper flywheel, due to symmetry, as shown in Figure 5(a). Gauge 1 is at the flywheel center, while Gauges 2 and 3 are at half radius and whole radius of the flywheel, respectively.

[figures omitted; refer to PDF]

The configuration of the plate-flywheel system changes after the interaction. Immediately following the collision, slightly rotated flywheels separate from the plate, as shown in Figure 5(b). As angular velocity cannot be directly read from the hydrocode, it is calculated based on the linear velocity of the gauges assuming that the configuration change of each part is negligible during the interaction, which is valid in the current numerical model.

In the formulation of the proposed mechanism, the linear and angular velocity are assumed to be gained impulsively, implying that in the interaction process the flywheels are accelerated to certain linear and angular velocity in a very short time in which the displacement and rotation can be neglected. This assumption is validated with the numerical model with all parts set as modified steel, as indicated in Figure 6 (directly captured from the program). It can be seen from Figure 6(b) that, after the interaction, the flywheel gained a low velocity in vertical direction, which also affects the vertical velocity of the associated gauges points. This observation more or less agrees with the assumption of neglecting the lateral velocity of the flywheel. The angular velocity of the flywheel can be readily calculated from the components of linear velocity (when $\theta$ is very small, which is true in the present study):$$\begin{array}{}\text{(9)}& \omega =\frac{d\theta }{dt}=\frac{d}{dt}\left(\frac{v_y}{v_x}\right)=\frac{d}{dt}\left[\frac{1}{v_x}\left(v_{y0}+kt\right)\right]=\frac{k}{v_x}\end{array}$$where ω and θ are the angular velocity and rotation angle of the flywheel; $v_{\mathrm{y}}$ and $v_{\mathrm{x}}$ are the linear velocity components in vertical and horizontal direction of a certain point; $v_{\mathrm{y}\mathrm{0}}$ is the initial vertical velocity gained during the interaction; $k$ is the gradient of the vertical velocity time history. It can be seen from the figure that the gradient of the vertical velocity of Gauge 3 is twice that of Gauge 2, proportional to their radius ratio to center, thus reasonable.

[figures omitted; refer to PDF]

With the gradient in Figure 6(b) and the horizontal velocity in Figure 6(a), the angular velocity of flywheel based on Gauge 2 after interaction is calculated as 189 rad/s. With the same approach, the angular velocity is calculated as 192 rad/s based on Gauge 3. Therefore, the angular velocity can be taken as 190 rad/s. A separate numerical study on two colliding objects with the same materials used in the current model is conducted to determine the coefficient of restitution as 0.25. With all of the data above, the angular velocity is calculated as 195 rad/s, which agrees well with the numerical simulation. However, the calculated flywheel linear speed is 18% greater, and the predicted plate velocity after interaction is 15% smaller than those of the corresponding parameters of the numerical simulation. The most important factor is that the ratio of rotational energy to total input energy is predicted as 0.25, comparing well with 0.22 of the numerical study. The main reason accounting for the discrepancies of flywheel linear velocity and plate velocity is that the flywheels gain vertical velocity, which is not modeled in the theoretical formulation. Although the vertical velocity is not significant compared to its horizontal counterpart, it does take some energy, thus reducing the flywheel linear velocity in horizontal direction.

Moreover, the proposed cladding can be stacked layer by layer to provide extra shock energy conversion/mitigation, if space is not restricted in some application scenarios. Although currently the claddings with the proposed energy conversion mechanism may not be cost-effective for typical structure protection, it provides an alternative protection strategy and sheds some light on different mechanisms for energy dissipation of short duration shock load.

5. Conclusions

A novel mechanism is proposed to dissipate energy for short duration shock load. Different from energy dissipation with large deformation crushing of cellular solid claddings, it converts part of the input translational kinetic energy into rotational kinetic energy, which is not detrimental to the protected structure. The governing factors of conversion efficiency are identified, and theoretical upper bound of the efficiency is determined. To maximize the efficiency, the coefficient of restitution should be as large as possible provided that it is practical for engineering application, while the mass ratio of the flywheel to face plate should be adopted around 3/2 with balanced energy conversion rate and mass efficiency. The present study provides an alternative strategy to protect structures from short duration shock load for a wide spectrum of applications.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The funding support from the National Natural Science Foundation of China [Project no. 51778028] is gratefully acknowledged.

References

[1] K. S. Arsava, Y. Kim, "Modeling of magnetorheological dampers under various impact loads," Shock and Vibration, vol. 2015, 2015.

[2] C. Q. Wu, L. Huang, D. J. Oehlers, "Blast testing of aluminum foam-protected reinforced concrete slabs," Journal of Performance of Constructed Facilities, vol. 25 no. 5, pp. 464-474, DOI: 10.1061/(asce)cf.1943-5509.0000163, 2011.

[3] H. Zhou, X. Wang, Z. Zhao, "High velocity impact mitigation with gradient cellular solids," Composites Part B: Engineering, vol. 85, pp. 93-101, DOI: 10.1016/j.compositesb.2015.09.042, 2016.

[4] M. F. Ashby, A. Evans, N. A. Fleck, L. J. Gibson, J. W. Hutchinson, H. N. G. Wadley, Metal Foams: A Design Guide, 2000.

[5] F. Zhu, Z. Wang, G. Lu, G. Nurick, "Some theoretical considerations on the dynamic response of sandwich structures under impulsive loading," International Journal of Impact Engineering, vol. 37 no. 6, pp. 635-637, DOI: 10.1016/j.ijimpeng.2009.11.003, 2010.

[6] D. Jiang, D. Shu, "Local displacement of core in two-layer sandwich composite structures subjected to low velocity impact," Composite Structures, vol. 71 no. 1, pp. 53-60, DOI: 10.1016/j.compstruct.2004.09.019, 2005.

[7] H. Zhou, Z. Zhao, G. Ma, "Mitigating ground shocks with cellular solids," Journal of Engineering Mechanics-ASCE, vol. 139 no. 10, pp. 1362-1371, DOI: 10.1061/(ASCE)EM.1943-7889.0000585, 2013.

[8] G. X. Lu, T. X. Yu, Energy Absorption of Structures and Materials, 2003.

[9] Q. Qin, X. Zheng, J. Zhang, C. Yuan, T. J. Wang, "Dynamic response of square sandwich plates with a metal foam core subjected to low-velocity impact," International Journal of Impact Engineering, vol. 111, pp. 222-235, DOI: 10.1016/j.ijimpeng.2017.09.011, 2018.

[10] J. Zhang, Q. Qin, C. Xiang, Z. Wang, T. J. Wang, "A theoretical study of low-velocity impact of geometrically asymmetric sandwich beams," International Journal of Impact Engineering, vol. 96, pp. 35-49, DOI: 10.1016/j.ijimpeng.2016.05.011, 2016.

[11] V. Birman, "Mechanics and energy absorption of a functionally graded cylinder subjected to axial loading," International Journal of Engineering Science, vol. 78, pp. 18-26, DOI: 10.1016/j.ijengsci.2014.01.002, 2014.

[12] C. J. Shen, G. Lu, T. X. Yu, D. Ruan, "Dynamic response of a cellular block with varying cross-section," International Journal of Impact Engineering, vol. 79, pp. 53-64, DOI: 10.1016/j.ijimpeng.2014.08.017, 2015.

[13] M. Leblouba, S. Altoubat, M. Ekhlasur Rahman, B. Palani Selvaraj, "Elliptical leaf spring shock and vibration mounts with enhanced damping and energy dissipation capabilities using lead spring," Shock and Vibration, vol. 2015, 2015.

[14] W. J. Stronge, Impact Dynamics, 2000.

[15] H. Y. Zhou, G. W. Ma, J. D. Li, Z. Y. Zhao, "Design of metal foam cladding subjected to close-range blast," Journal of Performance of Constructed Facilities,DOI: 10.1061/(asce)cf.1943-5509.0000606, 2015.