A Finite Element Method for Problems Involving Motion

Introduction

For years, linear Finite Element Analysis (FEA) has proven to be a valid tool in determining the stresses in bodies subjected to force or pressure loads. For purely static problems, there is little guesswork in determining the magnitude of such loads. For problems involving some sort of motion, engineers have typically relied on linear FEA, but have had to utilize rule-of-thumb methods to estimate the maximum value for these loads. Maximum values are required to insure that the maximum stresses are obtained.

In this paper, we show that the magnitude of loads caused by motion can greatly exceed conservative estimates. We reach our conclusions by considering a simple problem which is amenable to analytical classical physics methods. Specifically, we estimate the maximum load that a dropping rigid weight inflicts on a linearly flexible body upon impact. The primary drawback of the resulting analytical solution is that it is only applicable under highly restrictive conditions. Nevertheless, it is applicable to our problem of interest: to calculate the maximum force that a rigid weight generates when dropped above the center of a simply supported beam.

We then model this same problem using an FEA method that accounts for motion -- a Mechanical Event Simulation using Accupak/VE. This method is not subject to the highly restrictive conditions of our analytical solution. Furthermore, an event simulation of this problem does not require the maximum force as an input. Instead, the entire event is simulated: from the weight dropping, through impact, until the maximum stress in the beam is reached. Finally, we compare the results of the event simulation with those given by the analytical solution.

Maximum Force Upon Impact: Analytical Solution

In this section we derive an expression for the maximum force that a geometrically simple, flexible object experiences when impacted by a dropped rigid weight. This expression is obtained using conservation of energy principles from classical physics in conjunction with the following assumptions:

• The flexible object behaves like a linear spring. That is, when a force, F, is applied to the object, it deforms according to Hooke's law. Specifically, the deformation, , is in the direction of the force and has magnitude F/k, where k is the stiffness of the spring.
• The weight is rigid and is dropped in a uniform gravity field.
• No dissipative effects, such as friction, are present.

These assumptions limit the applicability of our analysis to flexible objects with both (1) simple geometries and (2) linear material behavior. Nevertheless, it is worth pursuing such a solution because of the valuable insight it provides into the problem of estimating maximum impact forces.

We begin our analysis by modeling the flexible body using two springs each of stiffness 2k connected by a point mass, m. Consider the two states depicted in Figure 1. State 1 is characterized by a stationary rigid weight, W, at a height, H, above the relaxed position of the mass-springs system. State 2 is characterized by the rigid weight at zero velocity; such a configuration occurs at the instant of maximum deflection, _max.

The total energy of the system at State 1 is comprised of the potential energy of both the rigid weight and of the point mass. At State 2, the total energy not only includes the new potential energies of the rigid weight and of the point mass, but it also includes the potential energy stored in the springs as well as the kinetic energy of the point mass. It is not acceptable to assume that the point mass is also at zero velocity; it is most likely experiencing (high frequency) oscillations at the instant of maximum deflection. The details of such oscillations are not obtainable from an energy analysis; thus, we are forced to make yet another assumption. That is, we must assume that the flexible body is massless, or that m is zero. To circumvent this assumption, we would have to solve a partial differential equation in space and time for the motion of the point mass. This limitation further highlights the difficulties encountered when analyzing problems involving motion. It is now possible to consider the two spring combinations as having an effective stiffness k.

We are finally in a position to write the energy of the system at both states. The energy at State 1 is given by

At State 2, the energy is given by

E2 = -Wmax + 1/2kmax2.

[2]

Because we have assumed that no dissipative effects are present, energy is conserved; thus E₁ = E₂, which after application of the quadratic formula yields an expression for _max:

[3]

Because physics dictates that the spring compresses upon impact, we can eliminate the negative root in Equation [3]; note from Figure 1 that we have defined compression of the spring to result in a positive value for _max. It should be noted that the negative root does represent a physical situation. Consider the situation where after contact the weight sticks to the spring and rebounds. The negative root gives the maximum height subsequently reached by the weight. For our interest, _max is unambiguously given by

[4]

Equation [4] can be combined with Hooke's law (F=k) to yield an expression for the maximum force,

[5]

Example: Dropping a Weight on a Flexible Beam

In this section we consider the problem of dropping a 4.0 lb. weight from a height of 1.0 in. onto the center of a simply supported steel beam with a length of 23 in. and a circular cross-section with a diameter of 1/2 in. The steel is taken to have a Young's modulus, E, of 30 ^. 10⁶ lb./in.² We need to calculate the stiffness of such a beam in order to apply Equation [5]. As long as we assume linear behavior (which is consistent with our approach), the stiffness of the beam is given by

[6]

where I is the area moment of inertia. For a circular cross-section of radius r, I is given by

[7]

Inserting the numerical values given above into Equations [6] and [7] yields: I =0.003068 in.⁴ and k = 363.1 lb./in. Inserting these values into Equation [5] yields a value of 58.1 lb. for the maximum force. This is a striking result; how can a 4.0-lb. weight generate more than 58 lb. of force upon such a short drop? It should be noted that the maximum deflection _max (given by Equation [4]) was only 0.160 in., thus we were well within the linear range.

To verify this result, we conducted an event simulation of the same problem. The FEA model utilized a cross-section of elements configured as shown in Figure 2.

Using 16 sets of such elements to model the length of the beam, we obtained a maximum deflection of 0.165 in. A more appropriate comparison of this numerical result and that given by Equation [4] can be obtained if we redo the analytical solution using a value of I for the cross-section utilized in the event simulation. This cross-section is not perfectly circular and has I=0.00280 in.⁴ Using this value of I, the analytical solution gives _max=0.168 in., which in turn corresponds to a maximum force of 55.7 lb.

Note how the analytical solution yields a slightly higher deflection and thus a higher maximum force. The reason for this is that the energy analysis assumes a perfectly rigid dropping weight and a weightless beam. In the event simulation, the dropped weight is not perfectly rigid (it has a finite stiffness) and the beam has a very small mass (to avoid stability problems). Furthermore, the numerical analysis includes contact elements, which because of their inherent stiffness consume some of the energy of the dropping mass. Nevertheless, because both the event simulation and the analytical analysis predict similar values for the maximum deflection, we are confident that our dropped weight generates more than 50 lb. of force after dropping just 1.0 inch. It should be noted that the maximum stress in the beam is about 24,700 lb./in.² Finally, if we account for the mass of the beam (using the density of steel) in the event simulation we obtain a maximum deflection of 0.156 in., which corresponds to 51.7 lb. of force.

Conclusion

In this paper, we showed how the motion of a part can lead to unexpectedly high stress values. These stresses can be large enough that not even conservative estimates may be appropriate for certain designs. These estimates can be avoided by directly modeling the motion of the part. MES provides the engineering designer with just such a modeling tool.