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What are my options for mechatronics modeling?
I like to break modeling methods into two groups: finite element and lumped element. Both seek to predict the behavior of some system based on three inputs: physical parameters defining system components, topology of the component interconnections, and integro-differential equations governing the behavior of the components. Of the two, the finite element method (FEM) is generally more realistic, but requires more computational resources. To see how they differ, consider a system made up of a mass suspended by a rubber band from a rigid support.
The lumped element method (LEM) views the rubber band as one component out of three. The rigid support is another component. It is characterized by having a fixed location. The rubber band is the second component, and it is characterized by its length and its Young’s modulus. The third component is the mass, which has location and mass. The governing equations are Hooke’s Law and Newton’s Laws.
FEM, on the other hand, breaks the system into a great many more “finite elements.” It would, for example, break the rubber band itself into small blocks of material connected at interfaces. Each of these finite elements would have its own mass, position, and momentum. It would interact only with its nearest neighbors through forces applied at the interfaces.
Newton’s second law, for example, would be written for each element, with a net force on the element’s top surface and a net force on the bottom surface. The unbalanced (difference) part would accelerate the element’s center of mass, while the balanced part would deform the element through Hooke’s Law.
The advantage FEM has is that it can look in much more detail. Whereas LEM sees the system as a simple harmonic oscillator, FEM sees a large collection of coupled harmonic oscillators. It can, therefore, model things like tears or holes quite naturally, whereas LEM can’t. It can also see waves propagating through the band that could affect the motion.
Suppose that you want to predict the lift on a wing moving through air. Start by dividing the wing’s surface into a bazillion finite elements. Next apply the Navier-Stokes equations to each finite element to calculate the aerodynamic force on that element. Then, calculate the vector sum of all those forces to get the net lift.
While you’re at it, calculate the vector (cross) product of the element’s position vector and the aerodynamic force on that element to get a torque for that element in that reference frame. Summing over all the torques will then give the net moment on the wing.
Now look at the forces between finite elements to work out a stress field within the wing’s skin (a much more complicated calculation). From that field and some information about the stress/strain properties of the wing, you can work out the wing’s aeroelastic response to the aerodynamic forces.
Before the advent of digital computers, it was seldom possible to use FEM for any but the simplest problems, such as calculating the center of mass of some object. Applied mathematicians spent a great deal of effort finding analytical solutions to particular classes of differential equations, such as Bessel’s equation, to make it possible to get numerical solutions in a reasonable length of time. Most problems, however, could not be solved by FEM.
As digital computers have become more powerful, however, it has become possible to solve FEM simulations involving millions of elements within a reasonable amount of time. We now can make the choice of whether to use FEM or LEM to make the required calculations.
LEM has advantages other than just ease of computation. The torsion-pendulum system described in the July 23, 2007, Ask Charlie [http://www.controleng.com/blog/820000282/post/290012229.html] posting, for example, lent itself nicely to LEM analysis, which provided an analytic solution. Analytical solutions in general provide greater physical insight than numerical solutions, and can be applied to classes of similar problems.
What are my options for mechatronics modeling?
October 1, 2007
I like to break modeling methods into two groups: finite element and lumped element. Both seek to predict the behavior of some system based on three inputs: physical parameters defining system components, topology of the component interconnections, and integro-differential equations governing the behavior of the components. Of the two, the finite element method (FEM) is generally more realistic, but requires more computational resources. To see how they differ, consider a system made up of a mass suspended by a rubber band from a rigid support. The lumped element method (LEM) views the rubber band as one component out of three. The rigid support is another component. It is characterized by having a fixed location. The rubber band is the second component, and it is characterized by its length and its Young’s modulus. The third component is the mass, which has location and mass. The governing equations are Hooke’s Law and Newton’s Laws.
FEM, on the other hand, breaks the system into a great many more “finite elements.” It would, for example, break the rubber band itself into small blocks of material connected at interfaces. Each of these finite elements would have its own mass, position, and momentum. It would interact only with its nearest neighbors through forces applied at the interfaces.
Newton’s second law, for example, would be written for each element, with a net force on the element’s top surface and a net force on the bottom surface. The unbalanced (difference) part would accelerate the element’s center of mass, while the balanced part would deform the element through Hooke’s Law.
The advantage FEM has is that it can look in much more detail. Whereas LEM sees the system as a simple harmonic oscillator, FEM sees a large collection of coupled harmonic oscillators. It can, therefore, model things like tears or holes quite naturally, whereas LEM can’t. It can also see waves propagating through the band that could affect the motion.
Suppose that you want to predict the lift on a wing moving through air. Start by dividing the wing’s surface into a bazillion finite elements. Next apply the Navier-Stokes equations to each finite element to calculate the aerodynamic force on that element. Then, calculate the vector sum of all those forces to get the net lift.
While you’re at it, calculate the vector (cross) product of the element’s position vector and the aerodynamic force on that element to get a torque for that element in that reference frame. Summing over all the torques will then give the net moment on the wing.
Now look at the forces between finite elements to work out a stress field within the wing’s skin (a much more complicated calculation). From that field and some information about the stress/strain properties of the wing, you can work out the wing’s aeroelastic response to the aerodynamic forces.
Before the advent of digital computers, it was seldom possible to use FEM for any but the simplest problems, such as calculating the center of mass of some object. Applied mathematicians spent a great deal of effort finding analytical solutions to particular classes of differential equations, such as Bessel’s equation, to make it possible to get numerical solutions in a reasonable length of time. Most problems, however, could not be solved by FEM.
As digital computers have become more powerful, however, it has become possible to solve FEM simulations involving millions of elements within a reasonable amount of time. We now can make the choice of whether to use FEM or LEM to make the required calculations.
LEM has advantages other than just ease of computation. The torsion-pendulum system described in the July 23, 2007, Ask Charlie [http://www.controleng.com/blog/820000282/post/290012229.html] posting, for example, lent itself nicely to LEM analysis, which provided an analytic solution. Analytical solutions in general provide greater physical insight than numerical solutions, and can be applied to classes of similar problems.
Posted by Charlie Masi on October 1, 2007 | Comments (0)
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