Use of Connectors in ABAQUS/Explicit to Simulate - Simulia

15 Pages·2004·1.49 MB·English

Use of Connectors in ABAQUS/Explicit to Simulate
Dynamic Motion Along a Curved Path
Steven M. Cron
Michelin Americas Research and Development Corp.
Abstract: This paper presents a technique for imposing high speed motion along an arbitrary
curved path using Abaqus connectors to prescribe piece-wise continuous circular motion about
arbitrary centers of rotation plus translations. An example of such motion is observed in a rolling
tire or similar loaded rolling structure. In the case of the tire, points near the rim follow
essentially circular paths whereas points far from the rim follow much more complicated paths.
This technique provides the capability to simulate the dynamic response of an elastic body
connecting two such paths where the tangential velocities are indicative of highway speeds.
1. Introduction
One of the challenges of any dynamic simulation is to efficiently specify appropriate boundary
conditions on the problem of interest. This becomes particularly apparent when the boundary
condition to be enforced is described by velocity along a curved path. A general description of
such a situation is illustrated in Figure 1.
In Figure 1, the elastic body, denoted with cross-hatching, has its motion constrained to move such
that point A follows path 1 and point B follows path 2. The figure is taken as a snapshot of the
body at some arbitrary point in time. In this illustration point A is moving along a path comprised
of a circular arc plus a line while point B is moving along a path comprised of two circular arcs.
While this description is given for coplanar motion, in principal, the paths could be non-coplanar.
This paper will outline an example where this type of boundary condition is enforced with
constant tangential velocity along each of two coplanar paths.
2. Example problem
2.1 Description
A practical example where we find the situation described above is shown in Figure 2. Here an
array of deformable “spokes” is used to connect a deformable exterior ring to a non-deformable
interior hub to form a deformable wheel-like structure.
The deformations shown in Figure 2 are indicative of static loading or very slow rolling
conditions. As the speed of rotation increases, however, the deformations of the spokes become
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increasingly influenced by inertial forces. It is these inertially influenced deformations that we
desire to simulate. The discussion will focus first on analysis of an existing spoke design and then
briefly on an updated design.
It is important to note that while the outer ring in the example is deformable, it is significantly
more rigid than the spokes connecting it to the rigid hub. As such, it is possible to consider a
single spoke independently subjected to deformations applied by the motion of the hub and ring
without being concerned with the influence the single spoke has on the ring.
An additional characteristic of the ring is that its nominal circumferential length is essentially
unchanged during deformation. This allows us to assume that when loaded and rolling, the
tangential velocity of points on the ring remain unchanged as vertical deflection increases.
Finally, it should be noted that the spokes outside of the contact region are carrying the load in
tension while spokes in the contact region carry essentially zero load. The simulation will account
for this static tension condition.
2.2 Simplified boundary conditions
Given that a single spoke can be considered independently from the remainder of the structure, a
simplified set of dynamic boundary conditions can now be proposed. An approximate description
of the path seen by the spoke connections to the hub and the ring are shown in Figure 3.
The inner connection of the spoke to the non-deformable hub will be assumed to move along a
simple circular path of radius, R . The outer connection of the spoke to the deformable ring will
hub
be assumed to be moving along a path comprised of a circular arc of radius R , a transition circular
o
arc of radius R and then along a straight line path in the contact region.
t
The hub will be assumed to be rotating with constant angular velocity, ω . The ring will be
hub
assumed to be rotating with constant tangential velocity, V , where V is equal to the angular
g g
velocity of the hub multiplied by the free external radius of the undeformed ring. That is simply to
say that the ring and the hub complete one revolution synchronously.
2.3 Model setup in Abaqus/CAE
Our attention now turns to implementing the simplified model and boundary conditions in
Abaqus/CAE. It is assumed that the reader is familiar with the use of Abaqus/CAE and, as such,
the discussion will follow the module hierarchy. The reader is referred to Table 1 for a summary
of key numerical values used in the model definition.
2.3.1 Part and Property modules
The geometry for the spoke model is a simple 3d solid as shown in Figure 4. The spoke is 230
mm wide, 87 mm tall, and 2.4 mm thick. It has a slight curvature out of plane to predispose
bending to occur consistently in one direction. The material is linear elastic with a Young’s
modulus of 37 Mpa.
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2.3.2 Assembly module
At the assembly level the part is positioned at a convenient starting position. In the present
example, the starting position will be taken to be horizontally level with the axis of rotation (the
wheel axle) as was shown earlier in Figure 3.
2.3.3 Step module
The problem is subdivided into six precisely defined time periods. These time periods correspond
to the times required to traverse each unique boundary condition defining the spoke’s deformation
and motion:
Step 1: Slowly stretch the spoke to impose static tensile strain of ~3%.
Step 2: Accelerate the spoke linearly from rest to full speed. This step ends at
the entry to the transition zone characterized by radius, R.
t
Step 3: Pass through the R transition zone at constant hub angular velocity and
t
constant ring tangential velocity.
Step 4: Pass through the contact zone at constant hub angular velocity and
constant ring tangential velocity.
Step 5: Pass through the second R transition zone at constant hub angular
t
velocity and constant ring tangential velocity.
Step 6: Continue at constant hub angular velocity and constant ring tangential
velocity until stopping point 180° opposite starting position is reached.
Refer to Table 1 for numerical values for the times and velocities used for each step. The reader
will note in Table 1 that the time for Step 2 is twice the time given for Step 6 to allow for the
linear ramp up to full velocity.
2.3.4 Interaction module
The real substance of the method being described is realized in the interaction and load modules.
In the interaction module we use Abaqus connectors and coupling constraints to facilitate
application of the simplified boundary conditions already described.
The first action taken in the interaction module will be to define coupling constraints for the spoke
faces that correspond to the attachments to the hub and the ring, as shown in Figure 5. To
accomplish this, for each face a reference point is created at the face center point. RP-2 is located
at the center of the face corresponding to the connection to the non-deformable hub while RP-3 is
located at the center of the face corresponding to the connection to the deformable ring. The
coupling constraints tie all six degrees of freedom for each face to its corresponding reference
point. An additional reference point, RP-1, will also be required at the center of rotation.
The next step in the interaction module is to attach three connectors of type “beam” to the
deformable spoke. The first beam will connect RP-3 to the center of rotation, RP-1, as shown in
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Figure 6. This beam will be used to enforce the circular motion of RP-3 about RP-1 during Step 2
of the simulation.
Before the second and third beams can be attached, two additional reference points are required.
The first point, RP-4, will be located 2.74mm from RP-1 in the direction of points RP-2 and RP-3
(in this case the direction also happens to be along the positive x-axis). The position of RP-4 is
shown in Figure 7. The distance between RP-4 and the center of rotation (RP-1) will be used to
stretch the spoke during Step 1 to a state of approximately 3% tensile strain.
The final reference point to be located is RP-5. It will be located a distance equal to R (243 mm)
t
toward the center of rotation away from RP-3.
The second and third beam connectors can now be attached to the spoke. The second beam
connector will attach RP-2 to RP-4. It will be used during Step 1 to stretch the spoke 2.74mm and
during Steps 2 through 6 to enforce motion at RP-2 along the circular arc of radius, R . The
hub
third beam connector will attach RP-3 to RP-5 and it will be used to enforce circular motion at
RP-3 along the two circular arcs of radius R.
t
The spoke with all three beams attached is shown in Figure 8.
2.3.5 Load module
At the load level, we will specify displacement and velocity boundary conditions at the reference
points already defined. The initial boundary condition to be applied will be to stretch the spoke to
about 3% elongation. This could be done in an implicit analysis step to save computational time
but for simplicity we will complete it here with an explicit analysis step prior to beginning the
rotation steps. The “stretch” step is carried out by simply specifying a displacement of 2.74mm to
move RP-4 to the position of RP-1. Thus, at the end of Step 1, both RP-1 and RP-4 will be at the
center of rotation.
For Step 2 we will ramp up angular velocities at RP-1 and RP-4 from zero to full speed. In this
case our full speed condition is 80 kph (22,222 mm/sec). The corresponding angular velocities
applied to RP-1 and RP-4 are 76.104 rad/sec and 76.654 rad/sec, respectively. Note that the
angular velocity applied to RP-1 is slightly less than the angular velocity applied to RP-4 since
less than half of the deformable ring is present below the horizontal starting point of the spoke.
Also note that the angular velocity applied to RP-4 will remain unchanged for the balance of the
simulation.
Step 2 ends when the spoke has reached full speed and is about to enter the transition arc
(characterized by R) into the contact region. At this time (start of Step 3) the boundary condition
t
on RP-1 is deactivated and a new boundary condition is applied to RP-5 to take its place.
Recalling that the tangential velocity (80 kph) is to remain constant at RP-3, the angular velocity
applied to RP-5 will increase to a value of 91.449 rad/sec to account for the smaller arc radius.
Step 3 ends and Step 4 begins when the spoke is about to enter the contact region and RP-3 must
now translate horizontally with a constant velocity of 80kph. Note that during this step the
boundary conditions on both RP-1 and RP-5 have been deactivated.
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At the end of Step 4, the boundary condition on RP-5 is reactivated (for Step 5) with the same
angular velocity as it had earlier. To finish the simulation (Step 6) the RP-5 boundary condition is
again deactivated, the boundary condition on RP-1 is reactivated (at full speed), and the simulation
is allowed to run out to a convenient end point. In this case the simulation is stopped when the
spoke is 180° opposite its starting position.
2.3.6 Mesh module
The mesh used is shown in Figure 9. The elements used are C3D8R: 8-node linear brick, reduced
integration, hourglass control.
2.4 Simulation results
Snapshots of the spoke at various points of the analysis are shown in Figures 10 and 11. Of
course, the analysis results are best viewed using the animation feature in the Abaqus/CAE
visualization module.
A typical solution time on an HP c3750 workstation was 40 minutes.
2.5 Experimental comparison – Original design
A qualitative comparison of the simulated and experimental responses was made by video taping
the physical deformable wheels while being exposed to a strobe light flashing in synch with the
rotation speed of the wheel. Snapshots of the wheel at 40 kph and 120 kph are shown in Figures
12 and 13. While the snapshots do not paint a complete picture of the spoke oscillations the reader
can get a general feel for the comparison.
3. Use of technique to optimize a new spoke design
3.1 New design objective
An update to the deformable spoke design was undertaken using the techniques described above to
simulate various spoke designs. The objective of the redesign was to reduce the deformations
caused by the high-speed inertial effects such that the deformations at 120 kph were visually
indistinguishable from the deformations at 40 kph.
Analyses made of the optimized designs indicated that the inertially driven deformation had been
significantly reduced.
3.2 New design experimental results
Deformable wheels with the updated spoke design were built and tested under the same conditions
as the original configuration. Figures 14 and 15 show the new design at 40 kph and 120 kph. It is
apparent from the photos that the deformations at 120 kph are visually identical to the
deformations at low speed.
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4. Summary and conclusion
A simple technique has been outlined wherein prescribed motion along a pair of paths comprised
of circular and linear parts has been enforced using coupling constraints and connectors in
Abaqus/CAE and Abaqus/EXPLICIT. An example of a deformable wheel has been presented
along with qualitative comparisons to experimental results. The comparison was deemed
sufficiently accurate to warrant use of the new technique as a tool to reoptimize the deformable
wheel spoke design to eliminate excess deformations caused by the large inertial forces present
during high-speed rolling. Qualitative assessment of the updated design indicates that the
optimization was successful.

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5. Tables
Table 1. Global modeling parameters.
Applicable
Variable Steps Parameter Value Units
Geometry
Hub radius All R 204 mm
hub
Ring radius at spoke connection All R 292 mm
o
Transition Radius of curvature All R 243 mm
t
Velocities
Ground (ring tangential) velocity 4 V 22,222 mm/sec
g
Hub angular velocity 2,3,4,5,6 ω 76.654 rad/sec
hub
Angular velocity in outer ring R zone 2,6 n/a 76.104 rad/sec
o
Angular velocity in transition zone, R 3,5 n/a 91.449 rad/sec
t
Step Times
Slow spoke stretch 1 n/a 0.05 sec
Linearly accelerate from rest to start of R 2 n/a 0.0216864 sec
t
Constant velocity through entry R zone 3 n/a 0.00815299 sec
t
Constant velocity through contact zone 4 n/a 0.00299173 sec
Constant velocity through exit R zone 5 n/a 0.00815299 sec
t
Constant velocity in R zone to stopping point 6 n/a 0.0108432 sec
o
Miscellaneous
Spoke "stretch" 1 n/a 2.74 mm
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6. Figures
R2
R1
A
B
R3
line segment
path 1
path 2

Figure 1. General description of problem.

Figure 2. Deformable wheel example.
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V
g
R
o
R
hub
spoke
ω
hub
R
t
V
g
Figure 3. Approximate spoke connection paths.

Figure 4. Spoke part geometry.

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Figure 5. Spoke part with coupling constraints applied.

Figure 6. Spoke with first beam connector attached.
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