Mohamed Gharib





Multiple impact problems arise in many practical applications, such as airplane undercarriages, impact crushers, and assembly robots. Solution of rigid body impact problems that involve simultaneous collisions is a challenging problem in mechanics and dynamics. This problem was studied by many researchers and several models were introduced; most notably the Impulse-Momentum and the Compliance contact theory based methods. These two methods have been around for many years but only few models produce energetically consistent solutions. The two approaches yield similar solutions for elastic and nearly elastic collisions, yet can diverge significantly for inelastic impacts. One can find few experimental studies that address multi-impact problems.

Impact-Contact models



Correlation between Impulse and Compliance Based Methods in Rigid Body Collisions:

In this work, we numerically solved the two ball inelastic impact problem using the impulse momentum and the compliance based models. We also established a correlation between the coefficient of restitution (in the impulse momentum model) and the contact stiffness (in compliance models). Two correlations were obtained. The former correlation for the compliance models with hysterias damping and the other one is for the compliance models with permanent indentation. In the first correlation, we used the force end condition to derive a new relationship for the damping factor. In the latter correlation, we used the definition of the energetic coefficient of restitution to derive a relationship among coefficient of restitution and the compression and restitution stiffness. The numerical results showed that the impulse momentum and the compliance model with permanent indentation produce identical results. While the common compliance model with hysteresis damping is valid only for nearly elastic impacts (e > 0.9). The new model proposed here produces good agreement for cases with a lower range of coefficient of restitution (e < 0.4).

Final velocities comparison for different impact/contact models



Solving the frictionless multiple impact problem using the Impulse Momentum and Compliance Based Methods:

In this work we numerically solved the multiple impact problem in a linear chain composed of elastic balls. Two methods were used, the impulse momentum method and the Hertz contact theory based method. We compared the post impact velocities from the two methods and the results showed that the methods produce similar solutions for elastic impacts. Hence one can use the Hertz contact theory based method to determine the impulse correlation ratio (ICR), which is an essential parameter in the impulse momentum method in elastic collision.

Velocity-Impulse diagrams for three steel balls



Solving frictionless rocking block problem with multiple impacts:

In this work, we developed a new approach to solve the multiple impact problem of a rocking block. The methodology is based on the use of impulse–momentum methods. The approach uses the ICR that was developed previously to solve the multiple impact problems in a linear chain of balls (
Ceanga & Hurmuzlu 2001). The method also uses the energetic coefficient of restitution and yields energetically consistent solutions. We have analyzed the separation of the non-contacting end. Then, we have derived a parameter map in terms of the surface inclinations and the height-to width ratio. It has been shown that one may have single and/or simultaneous multiple collisions at the surface contact points. Finally, a set of experiments were conducted to demonstrate the validity of the proposed methodology. We have shown that the experimental outcomes agree with the theoretical results. In addition, as far as the separation at the non-contacting end is concerned, the experiments exhibit the same trend that is predicted by the theory. The problem considered in this study was simplified by neglecting friction at the contact points. This was a necessary simplification at this initial stage of the development. Yet, including friction would be a good natural step for future research efforts.

Rocking block modeling and experiment



Solving the Baton Problem using Impulse Correlation Ratio:

This work presents the solution of the impact problem for a sliding/bouncing baton on flat and inclined planes considering the surface friction. The baton is assumed to have unilaterally constrained motion, which means one end slides on the ground while the other end collides with the ground. We use the impulse momentum approach and incorporate the impulse correlation ratio (ICR) hypothesis to solve the ground impact problem when the system has unilaterally constrained dynamics. Parametric investigations carried out to examine the effect of the baton's length and the inclined wall slope angle on the impulse correlation ratio. Numerical simulation and experiments carried out to validate the model.

Baton modeling and experiment



Multiple Impact Based Energy Absorption:

In this work, we proposed a scheme for energy absorption using a linear chain of steel balls. This scheme relies on dissipating the kinetic energy by using the appropriate arrangement of large and small balls. The numerical results showed that placing one or more small balls between the large balls will result in high energy absorption (84%). We propose to use one small ball placed between every pair of large balls, which roughly results in 80% energy absorption. Single small ball versus multiple small ball placements simplifies the design and resolves alignment problems. Also, we examined the effect of the coefficient of restitution and we found that using steel balls (e = 0.94) is efficient for most applications. Finally, we showed that increasing the number of large balls will lead to extra energy absorption. From the point of view of energy dissipation, the benefit of increasing the number of large balls diminishes as we increase the size of the chain. The adverse effect of increasing the number of balls is creating a bulkier shock absorber. A set of experiments was done for 3Large- 2Small and up to, 7Large-6Small balls. The results clearly demonstrate that the experimental outcomes are in agreement with the theoretical results.

Energy absorption linear chain experiment



Solving Particular 2D Multiple Impact Problems Using the Impulse Correlation Ratio:

In this work we developed a new approach to solve the non-frictional multiple impact problems that arise in a specific set of two dimensional cases. The approach uses Impulse Momentum based method, the Energetic Coefficient of Restitution, and the Impulse Correlation Ratio (ICR) hypothesis.
We presented a solution method for the collision of one ball that strikes one end of a stationary ball in contact with: (i)a third stationary ball placed at an arbitrary angle
θ, and (ii)two identical balls placed symmetrically at angles ±θ. First, we proved the existence of the ICR by considering springs on the contact points. Then, the solution method was generalized to solve the impact problem in three general cases: (a)a linear chain of r balls with one tilted chain of N-r balls placed at an angle θ, (b)a linear chain of $r$ balls with two symmetrically tilted chains each with N-r balls and placed at angles ±θ, and (c)a triangular arrangement of balls similar to those in the pool game but not with limited number of balls. Finally, a set of experiments were conducted to demonstrate the validity of the proposed methodology. We have shown that the experimental outcomes agree very well with the theoretical results. The problem considered in this study was simplified by neglecting friction at the contact points. Also, the two branches and the triangular configurations were assumed to have identical balls and placed in symmetrical configurations. This was necessary simplifications in this initial stage of the development.


Two dimensional multiple impact modeling and experiment



Elastic Impact of Spheres on Large Rectangular Plates:

In this work, we numerically solve the elastic impact problem of flat plate resting on a linear chain of spherical balls. The system’s governing equations are derived using the Hertz contact theory based approach. We use numerical analysis to demonstrate that one can obtain a very high number of intermittent collisions in the chain by placing small balls between the larger ones. Increasing the number of collisions significantly increases energy absorption due to impact. We exploit this feature to develop an impact based vibration suppression scheme of rectangular plate.

Plate vibration damping experiment



Ultrasound attenuation using spherical particles with different sizes:

In this work, we present an experimental study of ultrasound wave propagation in granular media made of poly sized spherical beads packed in a rigid container and randomly arranged. Two cases have been considered, the case with air is the surrounding medium, and the other case with oil surrounding medium. The results show that using different size spherical balls can achieve significant wave attenuation in ultrasound regime. Also, the medium surrounding the particles is highly affecting the wave transmission. The mass ratio effect has been studied for the spherical lead shots used in the experiments.

Ultrasound attenuation experiment




Last Update: May 2011