FINAL
5.8 LOGÍSTICA INTERNACIONAL
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Laplace transforms are particularly useful for control systems since differentiation of a signal is equivalent to multiplication of its Laplace
transform by s, while its integration is equivalent to multiplying its transform by s.
Block Diagrams
The Laplace transform of the transfer functions of the various elements of a control-loop are usually represented by a block diagram, such as that shown below:
Block diagrams are commonly used for modeling process-plants and electrical-systems. They are less common in modeling of mechanical systems.
With MotionView and MotionSolve, you can include control systems in your model, though not as a block diagram. Look up the online documentation for MotionSolve for details on how to build MBD models that include Single-Input-Single-Output (SISO) and Multiple Input Multiple Output (MIMO) systems using the Laplace transform and state-space representations.
Cams, Gears and other Higher Pairs
There are only 6 lower pairs, but any number of higher pairs can be constructed. Several higher pairs are fairly esoteric, which means their applications are restricted to specific domains. Modeling elements for tires, for instance, are called for almost exclusively by vehicle-dynamics designers.
Some higher pairs can be constructed using simpler modeling elements, if the modeling tool supports programmatic control. For instance, a one-way
CAE and Multi Body Dynamics Advanced Topics
clutch can be modeled using a bush together with an “if” statement to change properties based on the direction of rotation35.
Two higher-pairs that are extremely common are cams and gears.
Cams
A cam rotates about an axis and pushes a follower. The cam usually rotates at a uniform speed, and the profile of the cam is chosen so as to deliver the required motion to the follower. There are various classifications of both cams and followers, most of which reflect the topology or shape of the respective elements36. The follower is usually spring loaded to ensure that it stays in contact with the cam all through the rotation cycle.
Design interest centers principally around two things:
1. the profile the cam should have to achieve a required motion – the rise, dwell and return
2. the velocities and accelerations of the follower, and the resulting forces on the various components in the assembly
The first is usually the more interesting problem, but the second is no less challenging. Sometimes the cam profile is determined to match a specified follower-motion, but such cams can be expensive to manufacture. Often a predetermined cam profile is chosen and the follower of the motion is to be determined so that the design of the rest of the assembly can be tailored accordingly. In 4-stroke IC engines, for instance, designers need to determine the forces on the tappet.
The joint between the cam and its follower is maintained by contact. General contact can be used, but this approach is subject to the difficulties discussed above, in the section on Contact. It is usually more computationally efficient to use point-to-curve (PTCV) or point-to-surface (PTSF) constraints. This approach does sacrifice some of the generality offered by a full-fledged contact model. For instance, the PTCV constraint does not allow for contact to be broken. But at the concept design stage, the analysis is usually a kinematic analysis, since the goal is to derive the required profile of the cam.
Once this is done constraints like the PTCV can be used to verify that there
35 MotionView provides support both for bushes and for programmatic control. See the companion volume Managing the CAE Process – Basics.
36 Details can be found in any undergraduate-text on Machine Design.
Positive Return Cam, from the KMODDL
Advanced Topics CAE and Multi Body Dynamics
has been no loss of contact. If there is indeed loss of contact, full fledged contact modeling is essential.
Contact between the cam and follower can break if the spring-load is not enough to compensate for the inertial forces (that is, forces due to the accelerations the bodies experience). In engine-design this commonly called valve float, because cams are mainly used in the engine to control the valve-timing of four-stroke engines. The term lift-off is also used in several
applications.
Gears
There are two distinct problems posed by gears, which serve to transmit torque between different axes of rotation.
The transmission of torque is by positive engagement of the teeth.
Accordingly, the tooth itself needs to be designed for strength. The design of gear teeth is a subject that is normally not covered by MBD simulation. MBD analysis can help calculate the tooth-loads, and these loads can then be used as input for a stress analysis program – usually using Finite Element Analysis.
The other main class of problems deals with the design of the gear train itself. Gear trains range from the aptly named simple gear trains to the amazingly complex epicyclic gear trains. In these cases, analyzing the motion of the output shaft and calculating the ratio of input and output torques are the main areas of interest. An excellent range of models and animations at the KMODDL shows how complex the motion of gear trains can be. The images of a 4-bar mechanism with two gears, taken from an animation at the KMODDL, illustrate how complex the motion can be.
Designers of planetary gear trains need to calculate the loads on each gear.
Several gearboxes allow for multiple inversions of the gear train – that is, different gears are held “fixed” to generate different motion. MBD models go a long way towards eliminating the tedium and error in this demanding task.
MBD models also make it easier to estimate the efficiency of the gear train.
A detailed discussion of this aspect is beyond the scope of this book37.
37 See, for instance, Gear Handbook: The Design, Manufacture, and Application of Gears by Dudley, D
CAE and Multi Body Dynamics Advanced Topics
Epicyclic gears are over 300 years old, and are widely used today in a variety of applications, ranging from almost all propeller and turbine driven aircraft to lawn-mowers. While they are more challenging to design, the present a host of advantages, principally a lower weight and volume.
Calculating the efficiency of the gear train is an important but tedious task even for gears whose axes of rotation are fixed, like the worm-driven helical-rack-and-pinion shown alongside38.
Gear models in MBD are relatively easy to build.
Revolute joints define the axes of rotation of the shafts, while the gear joint represents the constraint between the two revolute joints.
38 This image too, is from a model at the KMODDL.
If everything seems under control, you're just not going fast enough
Mario Andretti