## Abstract

In this paper, we present a solution to the frictionless multiple impact problems that may arise in the rocking blocks. We use an approach based on the impulse–momentum methods, the energetic coefficient of restitution and the impulse correlation ratio. Subsequently, we present the results of an experimental study that is used to compare the results predicted by the proposed method with the experimental outcomes.

## 1. Introduction

Solution of multiple impact problems still involves many difficulties and unanswered questions (Marghitu & Hurmuzlu 1995; Brogliato 1996). These problems were studied by many researchers, and several models were introduced. Only a few models have produced unique and energetically consistent solutions (Ceanga & Hurmuzlu 2001; Glocker 2004, Liu *et al*. 2008, 2009).

The rocking block problem, in which multiple impacts are likely to occur, is one of the simplest rigid body impact problems that can involve multiple collisions. Understanding the physics of a rocking block is also important in applications such as robotics, buildings and tall structures subject to earthquakes and motion of water tanks. Housner (1963) introduced the first study to derive the mathematical equations of a free-standing rigid block under base excitation. This model was called the simple rocking model (SRM), in which plastic impact at the collision point is assumed (one degree of freedom). The block equations of motion were described by piecewise nonlinear equations depending on the sign of the rotation angle. Many researchers have analysed the block response due to different earthquake inputs using Housner’s approach (Yim *et al*. 1980; Spanos & Koh 1984; Hogan 1989). At the instant of impact, an angular velocity ratio is defined to relate the angular velocities before and after impact. The value of this ratio depends only on the slenderness ratio of the block and it has been shown that its theoretical values are not in agreement with experimental results (Muto *et al*. 1960; Priestly *et al*. 1978; Aslam *et al*. 1980). Lipscombe & Pellegrino (1993) considered rebounds at the colliding end by incorporating the coefficient of restitution in Housner’s formula. They showed that if bouncing is taken into account, the numerical and experimental results exhibit better agreement when compared with the SRM. This approach does not take into account the interaction between the two collision points. Thus, multiple impacts are ignored. Brach (1991) discussed the equivalent wrench value of the distributed forces acting on the block surface, used an approximated compliant model that had virtual springs and dampers and investigated the existence of a moment restitution coefficient. Moreau (1994) introduced a multiple impact law for rocking blocks with normal and tangential restitution coefficients. Recently, Ceanga & Hurmuzlu (2001) introduced a method that produced unique and energetically consistent solutions in frictionless multiple impact problems. They developed a new approach that used the energetic coefficient of restitution (Stronge 1998) and proposed a new constant that was called the impulse correlation ratio (ICR). This parameter is physically meaningful and more effective in dealing with multiple impact problems. They applied the method to the multiple impact problems in a linear *N*-ball chain.

The objective of this study is to extend the method given by Ceanga & Hurmuzlu (2001) to solve the rocking block problem. Here, we consider a rigid block with two rocking ends on smooth surfaces that are set at arbitrary angles. First, using a compliant model we show that the ICR concept is valid for the limiting cases. Then, we develop a solution method based on the rigid body approach and impulse–momentum methods. We proceed by investigating the effect of geometry on the post-impact bouncing patterns of the problem. Finally, we verify the approach by conducting a set of experiments and comparing the theoretical outcomes with the experimental ones.

## 2. Problem description

In this article, we consider the system shown in figure 1. The impact problem takes place as a result of the block striking the left surface (at *O*_{1}) while resting on the right surface (at *O*_{2}). The block is symmetrical with a width of 2*b*, a height of 2*h*, mass *m* and a centroidal moment of inertia, *I*_{cm}. To simplify the calculations, we choose an inertial coordinate system whose origin coincides with the centre of the block and its *x* and *y* axes are parallel to the respective edges of the block at the instant of impact (i.e. the block is always horizontal at the moment of impact). At the instant immediately before impact, the block is undergoing a non-centroidal rotation about *O*_{2} (where the initial velocity at *O*_{2} is equal to zero) with an angular velocity of ω^{−}. In addition, we consider only frictionless contacts and choose surface inclinations of *θ*_{1} and *θ*_{2} (see figure 1) at the respective contact points *O*_{1} and *O*_{2}.

The objective of solving the impact problem is to compute the angular velocity of the block and the linear velocity of its mass centre immediately after impact in terms of the pre-impact velocities. The problem can be cast in terms of five scalar unknowns for the frictionless case: three post-impact velocities and the magnitude of the normal impulses ( and ) at the two contact points. Here, as we consider the frictionless case, we will ignore the superscripts n in the remainder of the paper and denote the impulses at the contact points as *τ*_{1} and *τ*_{2}. The solution is not straightforward. One can obtain three equations from the conservation of linear and angular momenta. An additional equation can be obtained from the application of the concept of coefficient of restitution at the point of collision (*O*_{1}). Yet, one encounters difficulty in obtaining a fifth equation. A restitution equation cannot be written for the second contact point, because the pre-impact velocity of the block is zero at this point (it fails for all traditional definitions: kinematic, kinetic and energetic). Thus, the number of equations fall one short of the number of unknowns.

In this paper, we apply the ICR concept that was developed in Ceanga & Hurmuzlu (2001) to resolve the difficulty that is encountered in the present problem. In the following section, we use compliance contacts to derive the limits of the ICR for the rocking block problem.

## 3. Limits of the impulse correlation ratio

We use the compliant model that is shown in figure 2 to derive the limiting values of the ICR for the block problem with contact points modelled as linear springs. In general, we will assume that ICR is linear and can be experimentally measured, then we justify this assumption by conducting an experimental study. This is the same approach that was followed in Ceanga & Hurmuzlu (2001), for multiple collision problems of linear chains.

We may write the equations of motion from the block shown in figure 2. Solving these equations (using Mathematica) with the initial conditions , m s^{−1} and and ignoring non-impulsive terms (velocity-dependent terms in the acceleration equations) yields the following expressions for the *x* and *y* components of the displacements at contact points:
3.1
3.2
3.3
where and are the *x* components of the respective displacements and and are the *y* components of the respective displacements, and also
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
The changes in the impulses acting at the two contact points can be computed as
3.14
and
3.15
where Δ*τ*_{1} and Δ*τ*_{2} are the changes in the magnitude of the normal impulse vectors at *O*_{1} and *O*_{2}, respectively. To obtain the relationship between Δ*τ*_{1} and Δ*τ*_{2}, we form the following expression:
3.16
where *α* is the ICR. Substituting equations (3.14) and (3.15) into equation (3.16) and simplifying, we obtain
3.17

In general, *δ* and *α* are not constant for locally compliant models. Yet we can identify two limiting cases when they are constant. The first case where *γ*≪1 means that the first spring is much stiffer than the second one. The second case where *γ*≫1 means the second spring is much stiffer than the first one. Using *γ*≫1 in equation (3.17), we obtain the following expression for *α* such that *δ*=0:
3.18
On the other hand, when *γ*≪1, we obtain *α*=0 such that *δ*=0. Accordingly, we can specify the bounds for the ICR as follows:
3.19

In the following section, we assume that the ICR remains constant for the collision process. We assume that it is a physical constant that can be experimentally measured. We develop our solution method based on this assumption.

## 4. Velocity–impulse relationships

Using the law of impulse and momentum, we obtain the following equations:
4.1
and
4.2
where Δ**V**_{cm} and Δ**ω** are the changes in the linear and angular velocity vectors of the mass centre and the block, respectively, and **r**_{1} and **r**_{2} are the vectors from the mass centre to the contact points. The kinematic relationship among the velocities at the contact points and the mass centre can be written as
4.3
and
4.4
where **v**_{1} and **v**_{2} are the velocities of the block at *O*_{1} and *O*_{2}, respectively (see figure 1). For the remainder of the paper, we remove the vector notation. As we have no friction, we will use Δ*τ*_{i} and Δ*v*_{i} (*i*=1,2) for the changes in magnitudes of the normal components of the respective impulses and velocities. In addition, we use and (*i*=1,2) for the changes in the magnitudes of the velocity vectors in the *x* and *y* directions, respectively.

Now, we use equations (4.1)–(4.4) to derive the following velocity expressions in terms of the impulses at the contact points: 4.5 4.6 4.7 where 4.8

In this rigid block example, there are four possible cases that may arise during various phases of the collision process (see figure 3).

### (a) Impact at *O*_{1}–contact at *O*_{2}

Substituting Δ*τ*_{2}=*α*Δ*τ*_{1} into equations (4.5)–(4.7), we obtain the following equations:
4.9
4.10
4.11

### (b) Impact at *O*_{1}–no contact at *O*_{2}

Substituting Δ*τ*_{2}=0 into equations (4.5)–(4.7), we obtain the following equations:
4.12
4.13
4.14

### (c) Contact at *O*_{1}–impact at *O*_{2}

Substituting Δ*τ*_{1}=*α*Δ*τ*_{2} into equations (4.5)–(4.7), we obtain the following equations:
4.15
4.16
4.17

### (d) No contact at *O*_{1}–impact at *O*_{2}

Substituting Δ*τ*_{1}=0 into equations (4.5)–(4.7), we obtain the following equations:
4.18
4.19
4.20

For all the above four cases, we compute the changes in normal velocities of both ends as follows: 4.21 and 4.22

## 5. Bouncing patterns

In this section, we present the solution to the block multiple impact problems that are considered in this paper. We use the equations presented in the previous section to obtain a piecewise solution of the impact problem. At the onset of the collision, the block strikes the external surface at *O*_{1} while resting at *O*_{2} (this is the case given in §4*a*; where and , or more specifically ω^{−}=ω_{0}≠0). There are two possible bouncing patterns that result from the collision at *O*_{1}.

### (a) Single impact

This case arises when the non-impacting end bounces at the onset of the collision (i.e. its normal velocity becomes positive immediately). This means that the slope of the normal velocity *v*_{2} is positive at the onset of impact (case given in §4*b*). Thus, using equations (4.12) and (4.14), we may write the condition for the occurrence of this case as follows:
5.1

The maximum compression impulse at *O*_{1}, , can now be found by setting *v*_{1}=0 in equation (4.21). Then, we obtain
5.2

The impulse at the end of the collision at *O*_{1}, , can be found by using the energetic definition of the coefficient of restitution (Stronge 1990):
5.3
where *e*_{1} is the coefficient of restitution at *O*_{1}. By solving this equation, we obtain
5.4

The post-impact velocities can be found by substituting the final impulse into the respective velocity expressions. So, the final velocities in this case are 5.5 5.6 5.7 where 5.8

### (b) Simultaneous collision at both ends

If the condition in equation (5.1) is violated, simultaneous impacts at *O*_{1} and *O*_{2} take place. Initially, we have a case where there is an impact at *O*_{1} and contact at *O*_{2} (case given in §4*a*). Using equations (4.9), (4.10) and (4.21) and setting *v*_{1}=0 yields the maximum compression impulse for the first impact at *O*_{1} as follows:
5.9
where
5.10
and
5.11

We use the energetic definition of the coefficient of restitution to write
5.12
The final impulse for the first collision at *O*_{1} can be found by substituting the proper velocity expressions in equation (5.12) and solving for , which yields
5.13
where,
5.14
and
5.15
Now, the velocity expressions at the end of the *O*_{1} collision can be obtained by substituting in the respective velocity equations to yield
5.16
5.17
5.18
where
5.19
Although the impact at *O*_{1} ends, the collision at *O*_{2} continues. The impulse at *O*_{2} when the collision at *O*_{1} ends can be written as
5.20
Now that we have impact only at *O*_{2}, we consider the case where there is an impact at *O*_{2} and no contact at *O*_{1} (case given in §4*d*). The velocities during this interval can be written as
5.21
5.22
5.23
We use equations (4.22), (5.21) and (5.23), and along with the condition *v*_{2}=0 to obtain the maximum compression impulse for the first collision at *O*_{2} as follows:
5.24
where
5.25
Now, we use the energetic definition of the coefficient of restitution to write
5.26
We solve this equation to find the final impulse for the first *O*_{2} collision and we obtain
5.27
where
5.28
5.29
5.30
5.31
Substituting this final impulse in equations (5.21)–(5.23), we obtain the velocities for the end of this stage. Once again, we check the normal velocity at *O*_{1}. If it is positive, there would be no more impacts. Otherwise, additional impacts may emerge. In this case, the computations will continue in a similar manner. We just switch the notation for the two ends of the block and follow the procedure that we presented above. The process continues until all normal velocities at the contact points become positive. We have developed software routines in Mathematica that carry out this procedure and automatically stop when collisions at both ends cease. Finally, we may obtain the upper limit of the ICR by considering the inside of the square root in equation (5.27) as follows:
5.32
Simplifying the expression given in equation (5.32) yields the identical upper bound for the compliant case, which was presented in equation (3.18).

## 6. Numerical results

One of the important questions that we may ask in the present study is why the two conditions that are given in equation (3.18) and (5.1) are identical. This means that the upper limit of the ICR is positive, if and only if the condition in equation (5.1) is violated. On the other hand, when we have equation (5.1) satisfied, we have a negative upper limit for the ICR. This means that there is no valid ICR for these types of collisions. This is expected, because when the block undergoes single impact, the ICR is meaningless (there is no impulse at *O*_{2}) and hence no valid range for it can be found.

Now, we analyse the bouncing pattern for a rectangular block with a width-to-height ratio of *r*=*b*/*h*. Substituting *r* into equation (5.1) and simplifying yields
6.1
Using this inequality, we obtain the regions of single and multiple impacts depicted in figure 4). Setting the numerator of equation (6.1) equal to zero, we obtain the following relationship:
6.2
Using this relationship, we can draw the boundary curves and obtain the limiting angle values shown in figure 4. A choice of an angle pair in the unshaded regions will result in a single impact, whereas that in the shaded regions will result in multiple impacts. The partitioning of the *θ*_{1}–*θ*_{2} plane according to the bounce patterns at the contact points depends on the specific *r* values. Yet, we observe two different partitioning patterns depending on the value of *r*. When , we have disjoint multiple impact regions and connected single impact regions (see figure 4*a*). On the other hand, when , the situation is reversed as shown in figure 4*b*. In addition, as expected, we can observe that for wide blocks (large *r* values) multiple impacts are more likely to occur, whereas for a narrow block (*r* is small) we are more likely to have a single impact.

Figure 5 represents the velocity–impulse diagrams of four sample cases that are labelled as A, B, C and D in figure 4*a*. The velocities and in figure 5 are in the normal directions and normalized with respect to the initial vertical velocity at *O*_{1} (i.e. 2ω_{0}*b*). In case D, we observe a single impact at *O*_{1}, while *O*_{2} separates from the contact surface at the onset of the collision without any impact. Case C is almost in the transition from one to double impact region. One can observe that the normal velocity at *O*_{2} is almost equal to zero throughout the collision process. Thus, this end almost stays on the contact surface without collision. As we move on to case B, we observe that there are simultaneous impacts at *O*_{1} and *O*_{2}. Finally, for case A, multiple simultaneous impacts occur at both ends. For such cases, there will be ringing at the contact points of the block such that it settles into an almost complete rest after the collision process, despite the fact that the coefficient of restitution is non-zero at both ends.

## 7. Experimental study

We conducted a set of experiments to verify the methodology that we have presented in this paper. The experimental setup was designed such that the angular orientation of one contact surface could be freely adjusted and the second contact surface was fixed in a horizontal orientation. Figure 6 depicts a photograph of the experimental setup. Two heavy (much heavier than the mass of the block) rigid steel cylinders were used as contact surfaces. The angular orientation of the left cylinder and the vertical position of the right cylinder (the contact surface with fixed horizontal orientation) in the figure are adjustable. We ensured that the two contact points of the block would have the same altitude at the instance of impact. The drop mechanism consisted of a pneumatic cylinder with a Teflon-coated plastic attachment (to reduce horizontal motion of the released end of the block) mounted at the end of its piston. Each experiment was initiated by placing one end of the block on the Teflon coating and triggering the cylinder away from the block (see figure 6).

The experiments were conducted by using a 2.5 kg steel block that is shown in figure 7. The two edges of the block were rounded as shown in the figure to ensure the point contacts. Several markers were placed on the block and the contact surfaces. A high-speed video camera (1000 frames per second) was used to capture and digitize the motion of the markers during each experiment. Then, using the digitized data, the pre- and post-impact velocities of the block were computed.

The objective of the experimental study was to verify the theoretical outcomes. For this purpose, we first estimated the kinematic coefficients of restitution at both ends by dropping sample spheres made from the same material as the block on the two cylinders. Then, the ICR was estimated from a single experiment conducted at horizontal contact surfaces *θ*_{1}=*θ*_{2}=0^{°}. As a result, the coefficients were estimated as follows: *e*_{1}=0.43, *e*_{2}=0.64 and *α*=0.01. Subsequently, we have conducted a set of experiments by varying the surface orientations and using the three previously estimated coefficients.

The experiments were performed for *θ*_{1}={0^{°},5^{°},10^{°},15^{°},17.5^{°}, 20^{°},22.5^{°}} and *θ*_{2}=0^{°}. We conducted two sets of experiments. During the first set, the block was resting on the inclined surface at *O*_{1} and it impacted the horizontal surface at *O*_{2} (the configuration that is shown in figure 6). During the second set, the situation was reversed, the block was resting on the horizontal surface at *O*_{2} and it struck the inclined surface at *O*_{1}.

For each case, the experiments were repeated three times, releasing the block from various heights that varied between 30 and 60 mm. The theoretical and experimental results are depicted in figures 8 and 9. Each figure depicts the normalized (with respect to the pre-impact vertical velocity) post-impact velocities in the normal directions of the contact surfaces.

As one can observe from the figures, there is an excellent agreement among the experimental outcomes and the theoretical results. In addition, we observe a clear separation between the velocity slopes at the non-impacting ends. The separation takes place at an angular value of *θ*_{1}≈17^{°}. This value was computed analytically as the transition point from single to multiple collisions for the case at hand. In other words, the contacting end of the block separates without impact up to *θ*_{1}≈17^{°}. When this angular value is exceeded, the contacting end rebounds with interaction with the contact surface. The experimental data clearly exhibit this trend, attesting to the validity of the proposed methodology.

## 8. Conclusion

In this paper, 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 (although this issue is not valid in this paper because friction is neglected). To the best of our knowledge, no one before has either introduced such formulation or acknowledged the existence of multiple impacts. In addition, our formulation works or inclined foundations at both contact points, which has not been studied before. All the previous studies referenced in this article use Housner’s model where the colliding end is fixed in the horizontal direction (i.e. there is sufficient friction to hold this point off, or there is some sort of a notch). Our method is for non-frictional impacts (the block can slip freely in the horizontal direction), thus we cannot compare our results directly with their results.

We have analysed 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.

## Footnotes

- Received May 21, 2009.
- Accepted July 9, 2009.

- © 2009 The Royal Society