The Determination of the Velocities after Impact for the Constrained Bar Problem

1 Center for Engineering, Modeling and Applied Social Sciences (CECS), Federal University of ABC (UFABC), Aerospace Engineering, Av. dos Estados, 5001. Bloco B/Sala 936, 09210-580 Santo André, SP, Brazil 2 National Institute for Space Research—INPE, Av. dos Astronautas, 1758, 12201-940 São José dos Campos, SP, Brazil 3 German Aerospace Center—DLR, Institute of Robotics and Mechatronics, 82234 Wessling, Germany


Introduction
There are several ways to deal with the problem of interaction between bodies.Impact dynamics and continuous contact between bodies can both be included in the mathematical model of the constrained problem, or just one of these effects can be considered.It depends, obviously, on the characteristics of the studied problem.
The investigations about the contact between bodies include at least two different kind of analysis 1 : one associated with the beginning of contact and one associated with its termination.In the first analysis, the distance between the bodies must be checked in order to know when contact occurs; in the second analysis, once the contact is established, the reaction normal; compression force between the bodies must be checked.In the second analysis, contact finishes when the contact force is equal to zero.
One of the hardest parts in the study of contact problems involves the different models that must be developed for contact and noncontact situations and the switching between these models when integrating the equations of motion 2, 3 .The unconstrained problem and the constrained problem do not have the same number of degrees of freedom.Dynamic systems when constrained have less degrees of freedom than when unconstrained.
The transition between constrained and unconstrained motions is sometimes called contact including impact and sometimes called just impact mostly when the bodies separate after the collision .When contact occurs, the new velocities of the bodies involved must be known in order to generate the initial conditions to the second part constrained problem of the numerical integration.In the constrained problem, the concept of coefficient of restitution is very important 4 .

Geometric Model of the System and Governing Equations of Motion
The problem discussed here is depicted in Figure 1.According to this figure, in a part of its trajectory, the free end of the bar moves along the constraint represented by the mass named m w .All the movements occur in the horizontal plane.When contact occurs, impact and bouncing are also allowed to occur.
The mass in which the rigid bar is pivoted m s oscillates when excited by the movement of the bar free and constrained .In the axis Z, passing through the connection between the bar and m s perpendicular to the paper sheet , there is a prescribed moment, M θ , acting to turn the bar.
The dashed lines represent the position of the masses in which the springs and dampers are free of forces.The dotted line represents the position from which one starts to count the angular displacement, θ.
In physical terms, this system may represent a robot with a translational joint and a rotational joint; m w can be thought as an obstructing wall on the robot's trajectory or some object this robot must handle or interact with , and M θ can be thought as an external torque provided by a dc motor.
According to 5 , the constrained governing equations of motion for this system are given by and the constraint condition is given by d − y s y w − sin θ 0, 2.2  where I b,cm represents the bar moment of inertia around its center of mass, m b represents the mass of the bar, d Acmb represents the distance from A to the cm of the bar, c w represents the damping coefficient of m w , c s represents the damping coefficient associated with mass m s , k w represents the stiffness coefficient of mass m w , k s represents the stiffness coefficient associated with m s , and F N represents the amplitude of the normal force.It is assumed the there are no friction forces involved and represents the total length of the bar.Equations 2.1 are the equations of motion for y s , y w , and θ.Equation 2.2 is an additional relationship between the generalized coordinates y s , θ and y w when contact occurs.Equations from 2.1 to 2.2 provide four equations and four unknowns y s , θ, y w , and F N considering the constrained problem and three equations and three unknowns y s , θ, and y w considering the unconstrained problem.In the unconstrained case, 2.2 does not apply and F N 0.

The Contact Case
In contact, for this problem, there is the loss of one degree of freedom.In other words, one of the variables is dependent on all the others.The best choice is the elimination of the generalized coordinate y w , which is not always present into the system represented by the oscillating bar 6 .The new set of equations 5 is given by The fully plastic impact case is considered here for the calculation of the velocities immediately after contact.Separation will take place when the normal force is zero.As soon as these two variables are known, the remaining variable, y w , is also known through 2.2 .Equations 3.1 represent, respectively, the time behavior of the generalized coordinates y s and θ during the contact condition.In 5 , an analytical expression to the reaction force, F N , is also presented.

The Determination of the Velocities after Contact (Impact)
The equations for the impact are formulated for point P see Figure 2 for the representation of the velocities of the three bodies where, for sake of clarity, it is distinguished between Point P 1 belonging to the wall and point P 2 belonging to the bar. Figure 3 shows the free body diagram for the three rigid bodies indicating not the forces at the points of connection or contact but rather indicating the equivalent linear impulses due to impact.All these quantities are marked with an overhead symbol "hat", for example, P x , which is the linear impulse of the equivalent force P x .The physical dimension is the same as the linear momentum, that is, N • s, except for the angular impulse M θ whose unit is Nm • s.For each of the three rigid bodies, we can formulate now the linear impulse/linear momentum equations in the two directions x and y.Additionally, for the rotating bodie s , we have the equivalent angular impulse/angular momentum equation in z-direction, formulated w.r.t. to the respective centre of mass.
To better distinguish between velocities right before and right after impact, they are denoted with superscripts " " after and "-" before .Their two components in x-and ydirections are indicated by corresponding subscripts "x" and "y".
And, to be more general, it is also allowed initially for the rigid bodies with masses m s and m w to rotate as well.The respective angular velocities therefore will be denoted by ω with appropriate indices.Later, this additional degree of freedom will be kinematically constrained.
For the wall, it is obtained that  For the bar, it is obtained

4.2
And, finally, for the lower rigid body with mass m s , it is obtained that assuming that the directions of C x and F s are going through the center of mass.The geometric quantities b sx and b sy , not shown in Figure 3, denote the distances of the respective linear impulses measured from the center of mass.These equations simplified if the following assumptions are made.

4.12
In order to calculate the velocities at the point of impact, P , only 4.4 and 4.7 to 4.10 are of interest.Additionally, it is needed to establish some kinematic relationships.For the bar center of mass, one has

4.13
where the length of the vector r Acmb is just |r Acmb | d Acmb .Equation 4.13 is valid for the velocity right before and after impact.For the free end of the bar, it is obtained equivalently . In the same way as 4.13 , equation 4.14 is valid for the velocity right before impact and right after.During impact, one has the additional equation, which relates the velocities before and after impact at point P , in the direction normal to the contact surface, that is, in y-direction:

4.16
In the following, it is assumed that there is a fully plastic impact, that is, the impacting bodies maintain steady contact as far as the contact force is repulsive otherwise, they will separate .This leaves ε y 0, and hence v P 1y v P 2y 4.17

Mathematical Problems in Engineering
The denominator of these two equations then is written as

4.34
In order to check 4.33 , one case is investigated; that is, for θ 90 • , we should maintain the simple translational impact between the combined rigid body consisting of the two masses m s and m b and the wall with mass m w .For the fully plastic impact, one then obtains from 4.33 with α 1 0 and α 2 0:

4.35
where the first equation for the translational motion coincides with the result governed from simple impact of two rigid bodies.

Numerical Results
The values for the parameters used in the numerical simulations that follow are presented in Tables 1 and 2. The time step considered in the integration of the governing equations of motion is kept constant and equal to 0.0001 s.The fourth-order Runge-Kutta is the numerical integrator used.Two different classes of simulation are investigated.The constant torque with different amplitudes was chosen because it is the simplest one, and in order to make the bar rotate always in the same direction and fulfill 360 • .Any other kind of excitation e.g., like a sinusoidal one with maximum amplitude of 180 o , for instance can be chosen without problem.In the simulation runs, the motion of the bar starts always in its horizontal position to the right, that is, with θ 0 • .
The very beginning of contact is considered here as a fully plastic impact with impact time Δt ≈ 0 and with e 0, where e represents the coefficient of restitution.Contact finishes when F N 0. No friction or contact is considered, up to this point of the investigation, between m s and the guide it slides through or between m b and m w .

Considering Different Values of k w
When first contact takes place, m w is at rest.The second contact only shown here for the simulations varying M θ will happen with m w presenting some velocity.The bar is able to develop many turns and, in fact, there are possibilities for it to reach many contact conditions as the time evolves.
According to Figures 4,5,6,7,8, and 9, the amplitude of F N jumps at the beginning of contact, from zero no contact to a value associated with the impact force between the bodies.The contact force evolves with time according to the system states and properties.The value of F N at the instant of impact does not necessarily represent the biggest value for  the contact force, as can be seen in these figures.A sudden change in velocity, when collision takes place, can be verified clearly in these figures.

Considering Different Values of M θ
Table 2 shows numerical values considered in the numerical simulations for different values of M θ .

Conclusions
To conclude, it is important to say that the time step used in the numerical integration and the choice of the integrator are very important aspects to be considered.New numerical Mathematical Problems in Engineering integrators can be tested in the course of this investigation and results compared to the ones presented here.An important consideration not to be forgotten when dealing with problems presenting some sort of constraint is that more than one set of governing equations of motion must be integrated to cover all the system dynamics.The set of equations that governs the system dynamics when the constraint condition is active is different from the one that governs the unconstrained movement of the system.One of these sets is always generating the states for the other.
In this context, the determination of the velocities after contact impact is very important.The velocity expressions presented in 4.33 are the necessary corrections one must do when considering the fully plastic impact case.If this correction is not taken into consideration in the numerical integration of the governing equations, the system will gain energy after impact, which is not true.
It is important to realize also that the number of degrees of freedom involved changes from one set of equations to the other.The necessity for changing from one set of governing equations to another according to the system's requirements of contact or noncontact conditions represents a source of integration errors, since the integrator is faced with singularities.
The problem presented in this paper and the procedures developed for its analysis can be extended to many other systems and situations including more complex ones .The theory presented here can be applied to problems in which robots have to follow some prescribed patterns or trajectories when in contact with the environment like in painting activities, for instance, or the ROKVISS experiment at DLR .
The next steps are the development of the analytical expressions for the velocities after impact considering any value for the coefficient of restitution and the inclusion of friction forces between m s and the left and right vertical guiding surfaces; and between the free end of the bar and m w .
m w v wy − m w v − wy P y − F w , m w v wx − m w v − wx − P x B x , I w ω w − I w ω − w − B y w 2 P y sin θ.
v wy y w .
or v wy v sy ω b cos θ v by ω b − d Acmb cos θ. 4.18 Replacing v bx , v by and v wy , one arrives at the two equations for the unknown velocities v sy and ω b : m s m b m w v sy m s v − sy m b v − by m w v − wy − ω b m b d Acmb m w cos θ, I b,cm m b d 2 Acmb sin 2 θ m w − d Acmb cos 2 θ ω b v sy m s d Acmb − m w − d Acmb cos θ I b,cm ω − b − m s v − sy d Acmb cos θ − m b v − bx d Acmb sin θ m w v − wy − d Acmb cos θ.Acmb cos θ, and v − bx −ω − b d Acmb sin θ, these equations can finally be expressed by means of the independent velocities, v − sy , v − wy , and ω − b , right before impact: m s m b m w v sy m s m b v − sy m w v − wy m b ω − b d Acmb cos θ − ω b m b d Acmb m w cos θ.I b,cm m b d 2 Acmb sin 2 θ m w − d Acmb cos 2 θ ω b v sy m s d Acmb − m w − d Acmb cos θ I b,cm m b d 2 Acmb sin 2 θ ω − b − m s v − sy d Acmb cos θ m w v − wy − d Acmb cos θ 4.31 With the abbreviations m tot m s m b m w ,

Table 1 :
Numerical values considered in the numerical simulations for different values of k w .

Table 2 :
Numerical values considered in the numerical simulations for different values of M θ .