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Episode #3: Definitions for Motion Control
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Let's start with force and torque. Force is a linear topic. Torque is rotational. Otherwise, they're very similar in a lot of ways. Force has its mass. Torque is worried about the moment of inertia, or the mass times the distance cubed. Force is worried about acceleration. Torque is the angular acceleration. With force, we have the common equation of force = ma, mass times acceleration. Whereas torque is the moment of inertia times the rotational acceleration.
Moment load is the moment arm, as shown here about a fixed point, times the force at some distance. The moment is the force times distance and it causes a torque about the fixed point.
Let's talk about inertia for a moment because this is a topic that a lot of people struggle with. Inertia is a property of mass that defines how much it resists a change in velocity. If we look at the solid cylinder, it has an equation of J = mr2/2. That's the mass times the radius squared divided by 2. But, notice that length is not in that equation. If we look at a hollow cylinder, it also takes into account that internal radius, but, yes, this “plus” is accurate, not “minus” like you would think. You'd have to derive the equation in order to fully understand that. But, again, length is not part of that equation.
So, let's compare these two and see what this actually means. Let's say we take that solid cylinder and we put it on an incline starting at zero. We roll it down the incline and we do the same thing with the hollow cylinder at the same time. Which one is going to hit the bottom of the incline first? They had the same diameter. They have the same length. And they have the same mass. So, the density of the one that's hollow must be higher and all that mass has been pushed to the outside. Which one is actually going to hit the bottom first? Well, if you look at the equation, the equation actually tells us that the inertia of the hollow cylinder is actually higher because more mass is pushed out to the outside of it. Because of that location, its mass times the distance, it's pushed out and that inertia is higher for that hollow cylinder. Okay, so, if it's the property of the mass that defines how much it resists the change of motion, the one that has the higher inertia is the one that's going to resist the change in motion more. So, when we let go of the solid one, and it accelerates down, the one that has the higher inertia is going to accelerate at a lower rate. It's going to resist that change in velocity more because it's going to accelerate more slowly at a lower rate. It's not going to get up to speed as much so the one that has the lower inertia is. Or I could say, one is going to accelerate more quickly and get to the bottom first. This may be counterintuitive because a lot of people think about when you drop two different things, they're going to hit the bottom, they're going to hit the floor, first or at the same time. But we're not talking about dropping. We're talking about rolling the two down an incline together. I hope that makes sense.
Let's talk about axes of motion. Here we have a standard actuator. X is typically defined as the travel direction. Y is typically the horizontal perpendicular to the travel and Z is typically the perpendicular vertical to the travel.
Then we have the moments. Roll is around the X-axis. Pitch is typically around the Y-axis and yaw is typically around the Z-axis very much like an airplane. In an airplane, you have the roll about the direction of travel, you have the pitch about the airplane, about the wings of the airplane, and you have the yaw around the Z-axis. An airplane, when it comes in, it actually yaws typically along the runway. So, those are the moments. Now the X, Y and Z, those are just conventions. I've seen the Z-axis be defined as the horizontal direction based upon something like the shooting of a particle down a beam of a linear accelerator.
Let's take a look at how these forces and moments come to play. We have an actuator here on the horizontal and we have a weight of a load on top of it some distance off the carriage. Now, there's some difference between the carriage and where the bearings are so that has to be added to the moment loading in a lot of calculations. But, as we move, we get a dynamic force of the force in the X direction because of what it takes to move that. But, because we have this force times this distance, we actually get a moment about the Y-axis and then we also get a thrust in the X-axis. We have the force normal in the Z direction, so there's the normal force. Then we get a force X in the thrust direction and then we get a moment load as well if the load is off to the side. We don't only have the normal force, but we're going to have different moment forces again. I have some moment around it in the direction around the Z-axis as well as a moment about the X-axis because it's hanging off to the side. Then, if the weight is in line with the actuator, but off the carriage here, we're going to get a moment about the Y. Now, if the actuator is in the vertical direction, we're going to get some other moments. If it's in the vertical direction, but the weight is way off to the side, we're going to get some more moments around the Y-axis or the Y-direction. And then we have to look at this one as well. So, each one of these has their own combination of forces and moments that have to be taken into account when sizing and selecting actuators.
Here's a typical chart of different connections of X and Y and X, Y and Z axes. For each one of these, the moment loading, and the loads, are going to be seen on the actuators a little bit differently. So, we have to understand this when we're taking into account all these loads and sizing mechanics. If someone tells me they want an XY system, well, is it an XY like in Figure 1? Or an XY like in Figure 5? If it's an XYZ, is it XYZ like in Figure 6? Or is it XYZ like in Figure 8? The calculations, and therefore the effects on the actuators, are different. So, there we have it.
We've covered some force, torque, moment, inertia, and axes of motion. I hope that helps.
Contact Valin today for more information at (855) 737-4716, or fill out our online form.
The Motion Control Show
Before we dive into the meat of the matter of sizing and selecting mechanics, there are some basic concepts we need to understand: namely the force, torque, moment, inertia, and axes of motion. We're going to talk about those topics for a bit. I'm Corey foster of Valin Corporation.Let's start with force and torque. Force is a linear topic. Torque is rotational. Otherwise, they're very similar in a lot of ways. Force has its mass. Torque is worried about the moment of inertia, or the mass times the distance cubed. Force is worried about acceleration. Torque is the angular acceleration. With force, we have the common equation of force = ma, mass times acceleration. Whereas torque is the moment of inertia times the rotational acceleration.
Moment load is the moment arm, as shown here about a fixed point, times the force at some distance. The moment is the force times distance and it causes a torque about the fixed point.
Let's talk about inertia for a moment because this is a topic that a lot of people struggle with. Inertia is a property of mass that defines how much it resists a change in velocity. If we look at the solid cylinder, it has an equation of J = mr2/2. That's the mass times the radius squared divided by 2. But, notice that length is not in that equation. If we look at a hollow cylinder, it also takes into account that internal radius, but, yes, this “plus” is accurate, not “minus” like you would think. You'd have to derive the equation in order to fully understand that. But, again, length is not part of that equation.
So, let's compare these two and see what this actually means. Let's say we take that solid cylinder and we put it on an incline starting at zero. We roll it down the incline and we do the same thing with the hollow cylinder at the same time. Which one is going to hit the bottom of the incline first? They had the same diameter. They have the same length. And they have the same mass. So, the density of the one that's hollow must be higher and all that mass has been pushed to the outside. Which one is actually going to hit the bottom first? Well, if you look at the equation, the equation actually tells us that the inertia of the hollow cylinder is actually higher because more mass is pushed out to the outside of it. Because of that location, its mass times the distance, it's pushed out and that inertia is higher for that hollow cylinder. Okay, so, if it's the property of the mass that defines how much it resists the change of motion, the one that has the higher inertia is the one that's going to resist the change in motion more. So, when we let go of the solid one, and it accelerates down, the one that has the higher inertia is going to accelerate at a lower rate. It's going to resist that change in velocity more because it's going to accelerate more slowly at a lower rate. It's not going to get up to speed as much so the one that has the lower inertia is. Or I could say, one is going to accelerate more quickly and get to the bottom first. This may be counterintuitive because a lot of people think about when you drop two different things, they're going to hit the bottom, they're going to hit the floor, first or at the same time. But we're not talking about dropping. We're talking about rolling the two down an incline together. I hope that makes sense.
Let's talk about axes of motion. Here we have a standard actuator. X is typically defined as the travel direction. Y is typically the horizontal perpendicular to the travel and Z is typically the perpendicular vertical to the travel.
Then we have the moments. Roll is around the X-axis. Pitch is typically around the Y-axis and yaw is typically around the Z-axis very much like an airplane. In an airplane, you have the roll about the direction of travel, you have the pitch about the airplane, about the wings of the airplane, and you have the yaw around the Z-axis. An airplane, when it comes in, it actually yaws typically along the runway. So, those are the moments. Now the X, Y and Z, those are just conventions. I've seen the Z-axis be defined as the horizontal direction based upon something like the shooting of a particle down a beam of a linear accelerator.
Let's take a look at how these forces and moments come to play. We have an actuator here on the horizontal and we have a weight of a load on top of it some distance off the carriage. Now, there's some difference between the carriage and where the bearings are so that has to be added to the moment loading in a lot of calculations. But, as we move, we get a dynamic force of the force in the X direction because of what it takes to move that. But, because we have this force times this distance, we actually get a moment about the Y-axis and then we also get a thrust in the X-axis. We have the force normal in the Z direction, so there's the normal force. Then we get a force X in the thrust direction and then we get a moment load as well if the load is off to the side. We don't only have the normal force, but we're going to have different moment forces again. I have some moment around it in the direction around the Z-axis as well as a moment about the X-axis because it's hanging off to the side. Then, if the weight is in line with the actuator, but off the carriage here, we're going to get a moment about the Y. Now, if the actuator is in the vertical direction, we're going to get some other moments. If it's in the vertical direction, but the weight is way off to the side, we're going to get some more moments around the Y-axis or the Y-direction. And then we have to look at this one as well. So, each one of these has their own combination of forces and moments that have to be taken into account when sizing and selecting actuators.
Here's a typical chart of different connections of X and Y and X, Y and Z axes. For each one of these, the moment loading, and the loads, are going to be seen on the actuators a little bit differently. So, we have to understand this when we're taking into account all these loads and sizing mechanics. If someone tells me they want an XY system, well, is it an XY like in Figure 1? Or an XY like in Figure 5? If it's an XYZ, is it XYZ like in Figure 6? Or is it XYZ like in Figure 8? The calculations, and therefore the effects on the actuators, are different. So, there we have it.
We've covered some force, torque, moment, inertia, and axes of motion. I hope that helps.
Contact Valin today for more information at (855) 737-4716, or fill out our online form.
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