Latest / Mechanical Engineering Made Simple / (#105) Coriolis Force Catastrophe: How Cylindrical Coordinates and Angular Momentum Unmask the Side Load in Spinning Systems

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(#105) Coriolis Force Catastrophe: How Cylindrical Coordinates and Angular Momentum Unmask the Side Load in Spinning Systems · Mechanical Engineering Made Simple

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  Welcome back design junkies to Mechanical Engineering Made
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  Simple. We're embarking on a deep dive
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  today into really the foundations of motion analysis,
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  the kinematics and dynamics of a particle.
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  If you've ever wondered how engineers track objects like
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  missiles, or maybe design high speed centrifuges, or you know,
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  model systems where something spins and slides at the same
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  time, this is where that critical analysis really starts.
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  Our source material takes us straight into advanced dynamics,
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  focusing on how we apply geometry and Oculus when motion
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  is genuinely curved and, well, three-dimensional.
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  Right. And the mission here really is
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  to navigate these coordinate systems that actually move with
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  the object. We need to go far beyond the
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  standard fixed XY and Z axes that, you know, we all learned.
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  First, we need to simplify the crucial and, yeah, often
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  intimidating equations for velocity and acceleration in
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  these complex curvilinear systems.
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  By the end of this deep dive, you'll understand why picking
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  the right set of coordinates is, well, often half the battle in
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  solving a complex engineering problem.
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  And we'll prove it with a really powerful practical example
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  involving a spinning rod later on.
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  OK, sounds good. Let's unpack this then, starting
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  with the absolute fundamental building block for this whole
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  branch of mechanics, the particle.
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  In dynamics, how do we actually define a particle?
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  It's defined conceptually as a material point without
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  dimensions, but containing A definite quantity of matter.
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  Yeah. And what's fascinating here is
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  why engineers rely on this idea. I mean, it's a physically
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  impossible dimensionless point, right?
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  We use it because when the size of a massive body, think a
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  planet or a star, is extremely small compared with its overall
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  range of motion, its trajectory, we can actually model it very
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  accurately as a particle. I don't think about astronomical
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  mechanics. A planet is huge, sure, but
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  relative to its orbit around a star or its path through the
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  Galaxy, it's tiny. Modeling it as a particle is
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  precise and efficient. This simplification lets us
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  tackle these immensely complex trajectories.
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  OK, so it's a simplification tool and the standard approach.
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  The baseline for describing motion, especially straight line
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  motion, is simple Cartesian coordinates XYZ.
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  We all know Cartesian works great for straight lines, mainly
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  because it's unit vectors AJ and K are fixed in space, they don't
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  change direction. Exactly.
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  But the moment the path curves, those fixed axes kind of fail
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  us, or at least they become computationally messy because
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  the particles movement is constantly misaligned with them.
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  If we want to analyze motion that involves rotation or radial
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  change, like that block sliding on a spinning disk you
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  mentioned, we really need a system where the axes themselves
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  move with the particle. OK, that makes sense.
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  Which brings us pretty directly to our most useful curvilinear
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  system. For 3D analysis, I'd say
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  cylindrical coordinates. This system seems perfect for
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  motion that mostly happens in a plane, but might also move
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  vertically, like maybe a satellite in a decaying orbit or
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  some manufacturing process feeding material outward while
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  spinning. That's right, this system is
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  defined by three components. We use row, which is a radial
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  projection onto the XY plane, sort of the distance from the Z
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  axis if you look from above. Then there's Phi, the angle in
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  that XY plane measured from the X axis usually, and finally Z,
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  which is just a standard height component, same as in Cartesian.
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  So the position vector is written as R equals rho arrow
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  plus ZK. But here's the crucial part.
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  The unit vectors arrow and Fe are no longer fixed like I and J
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  were. They actually change direction
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  as the particle moves and rotates.
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  That's the source of all the interesting complexity,
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  mathematically speaking. Right, that dynamic change in
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  the unit vectors. It means we can't just take
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  simple time derivatives like we did with Cartesian motion to get
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  velocity and acceleration. So when we calculate velocity,
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  we end up with a vector expression.
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  Let's break that down conceptually.
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  What are the physical components of the velocity vector V in this
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  cylindrical system? OK, velocity in the system
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  splits nicely into 3 pretty easily recognizable physical
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  actions. First, you have the radial
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  velocity component. That's simply the speed at which
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  the particle is moving directly away from, or maybe toward the
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  central Z axis. We can describe this as the rate
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  of change of the radial distance D or O over DT.
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  OK, outward or inward speed? Then there's the rotation part.
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  Yes, that's the tangential speed, purely due to the
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  rotation. It's calculated as rho
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  multiplied by the rate of change of the angle Phi D Phi over DT,
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  which is simply the radius times the angular speed Omega makes
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  sense. And finally, the third component
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  is just the vertical speed DZ over DT, same as before.
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  So velocity seems pretty straightforward once you
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  understand the physical meaning of each term.
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  Dr. row D set, DDZEDD, radial, tangential, vertical.
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  Got it. But you mentioned the complexity
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  comes in with acceleration. This is where the physics can
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  surprise you, right when we take the second derivative.
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  This is where those changing unit vectors really make their
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  presence felt, creating components that just don't exist
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  in simple straight line motion. Indeed, this is where it gets
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  really interesting. In the final full acceleration
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  expression, we see a radial component, the part along Airho
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  that actually has 2 terms. The first term is what you might
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  expect the second time derivative of rho d ^2, rho ODT
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  squared. That's just change in radial
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  velocity, but the second term is minus rho multiplied by the
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  angular velocity squared, so minus rho times D Phi DT
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  squared. Hold on, I recognize that second
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  term minus rho Omega squared. That looks like the inward
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  centripetal acceleration, doesn't it?
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  That's the term responsible for keeping the particle curving
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  inward instead of just flying off in a straight line
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  tangential to the curve. Precisely.
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  That's the pull toward the center needed for circular
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  motion or any curved motion in that plane.
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  The tangential direction, the E Phi direction, also contains 2
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  terms, and one of them is hugely significant, especially in
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  engineering design. It's called the Coriolis term.
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  This term is calculated as 2 multiplied by the radial speed
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  DRODT multiplied by the angular velocity DFIDT.
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  So 2 * d ROD t * d. Wait a second.
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  So if I'm, say, building a robot arm that's extending outward
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  while its base is rotating, I have to account for a side
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  acceleration and therefore a side force.
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  A force that doesn't feel intuitive because it relies on
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  the multiplication of 2 completely independent motions
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  moving outwards and spinning. That is probably the single
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  biggest design risk captured by this acceleration equation that
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  engineers sometimes overlook, especially early on.
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  The Coriolis acceleration exists purely because you have motion
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  in the radial direction and rotation happening
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  simultaneously. It's an interaction term.
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  If either the radial speed is 0 or the angular speed is 0, that
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  Coriolis term just vanishes. Poof.
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  But it's the interaction between the two that generates this
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  often unexpected side force. If you're designing high speed
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  machinery like turbines or pumps, or even certain types of
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  amusement park rides, this interaction force can be
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  absolutely catastrophic if it isn't properly accounted for in
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  the design. Wow, OK, that's a perfect
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  illustration of why just sticking with Cartesian
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  coordinates wouldn't work here. You'd completely miss that
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  effect unless you did some very complicated projections.
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  Now, just briefly, we should probably mention the the big
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  sibling of cylindrical coordinates, spherical
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  coordinates. This system uses the total
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  radial distance R from the origin, then an angle Theta
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  measured down from the positive Z axis, and finally the same
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  angle Phi in the XI plane as before.
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  The position vector looks deceptively simple here, doesn't
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  it? It's just R = R * R.
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  It looks simple, yeah, but don't let that fool you, because now
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  you're tracking motion in three dimensions using 3 unit vectors
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  air, A, Theta and MP, and all three of those unit vectors
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  change direction as the article moves.
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  The result is that the full acceleration equation in
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  sherical coordinates has an incredible 9 separate terms 9.
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  It includes things like centripetal terms, Coriolis
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  terms related to Theta changes, terms related to fee changes.
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  It gets complex fast. This is exactly why engineers
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  always try to use the simplest coordinate system that fits the
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  problem. The complexity really explodes
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  when you track full 3D motion using two changing angles and a
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  changing radius simultaneously. 9 terms.
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  OK message received. Stick to cylindrical if you can.
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  All right, now that we have the kinematics, the description of
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  motion, let's jump into the kinetics.
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  This is where forces come into the picture.
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  The underlying principle for all particle dynamics is pretty
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  simple. Fundamentally, it's Newton's
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  second law, FMA force equals mass times acceleration.
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  We can use this, along with what we just learned about
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  cylindrical coordinates to solve a real world problem involving
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  that tricky Coriolis effect. Yes, let's set up the scenario
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  properly. Imagine a small block.
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  OK, it's sliding. Let's say frictionlessly for
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  simplicity, along a straight rod.
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  But here's the catch. That rod is itself spinning,
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  rotating in the XY plane around the Z axis like a propeller
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  blade. We need to figure out the side
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  force, let's call it F, that the rod has to exert on the block
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  just to keep it moving along the straight path of the rod while
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  the whole thing spins. It's the force perpendicular to
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  the rod. OK.
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  I can visualize that and we're given some key parameters from
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  our source material to work with the mass of the block and our
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  mayor is 130 slug. Remember, slug is the imperial
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  unit of mass. The block is sliding outward
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  along the rod at a constant radial velocity, so that's a
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  dray of 30 feet per second, and the rod itself is spinning with
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  a constant angular velocity Omega, which is de effort of 50
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  radians per second constant speeds both radially and
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  angularly. Right now we can try to solve
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  this using fuel MA directly we plug in the full cylindrical
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  acceleration vector. We discussed with this
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  centripetal and Coriolis terms, but honestly that gets pretty
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  messy with substituting all the components.
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  There's often a much cleaner approach, especially for
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  problems involving rotation, using the angular momentum
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  principle. Ah, excellent idea at the
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  angular momentum principle. This states that the net moment
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  applied to the particle M must equal the rate of change of its
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  angular momentum H over time. So M equals DHDT and angular
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  momentum H itself is defined as the cross product of the
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  position vector R and the linear momentum which is M * v SH was R
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  cross MV. Exactly, and in our scenario,
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  this block sliding on a rotating rod, things simplify nicely in
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  cylindrical coordinates. The side force F exerted by the
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  rod on the block creates A rotational moment M acting
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  perpendicular to the rod in the Z direction or K direction.
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  The magnitude is just r * F when we calculate the angular
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  momentum H of the block about the origin, also in cylindrical
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  coordinates. It simplifies beautifully
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  because the velocity components work out just right.
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  H ends up being M times the radial distance R-squared
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  multiplied by the angular velocity Omega.
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  And this angular momentum vector H also points purely in the K
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  direction perpendicular to the plane of rotation, so HEM
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  R-squared Omega K. OK, so the moment M is RFK and
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  the angular momentum H is Mr. Squared Omega K.
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  Now we need the rate of change of angular momentum DHDT to use
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  M equals DHDT. Since the mass M and the angular
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  velocity Omega are constant in this problem, we only need to
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  differentiate the R-squared term with respect to time using the
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  chain rule. That's right, the time
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  derivative of R-squared is 2 * R times DRDT.
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  So DHDT becomes M * 2 RDRDT times Omega all times K We're
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  rearranging slightly. DHTT equals 2m RDRDT Omega K.
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  Perfect. Now we equate the moment M to
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  the rate of change of angular momentum DHDT.
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  So RFK equals 2m RDRDT Omega K. We can immediately cancel the
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  unit vector K from both sides and look at the radial distance.
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  R also cancels out from both sides.
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  This leaves us with a really powerful, clean final
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  relationship for that required side force F.
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  It's just F = 2 * M * D RDT times Omega.
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  And look at that equation F = 2m DRDT Omega.
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  This simple L get result confirms exactly what we
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  suspected from the acceleration equation.
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  The side force needed to constrain the block to keep it
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  on the rod is entirely driven by the interaction between the
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  radial speed DRDT and the angular speed Omega.
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  This force is the macroscopic measurable force resulting
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  directly from that microscopic Coriolis acceleration term we
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  identified earlier, 2D rotor DDDDDOD.
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  It's the same thing, just expressed as a force via FMI.
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  Amazing how it connects like that.
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  OK, now for the calculation. Let's plug in the numbers we
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  were given. Mass m = 130 slug, radial
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  velocity 330 feet per second. Angular velocity Omega equals 50
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  radians per second, so F = 2 * 130 slug times 30 feet times 50
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  rats. And the math is pretty easy
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  here. The 130 and the 30 cancel each
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  other out perfectly, leaving just F = 2 * 50.
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  The result is F100 and the units work out to pounds force pound
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  BF. 100 lbs force for a block with a mass of only 130 slug.
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  That's that's a tiny fraction of a pound of mass, maybe around
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  half an ounce. Yet it requires a massive side
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  force of £100 just to stay on that spinning rod while sliding
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  outward. Wow.
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  Imagine if an engineer designed the system and say, only
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  calculated the centripetal force needed to keep it moving in a
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  circle at a fixed radius. They would completely miss this
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  huge 100 LB Coriolis driven side load.
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  The rod connection or whatever mechanism holds the block would
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  probably just shear off instantly.
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  This analysis. This is literally the difference
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  between a stable design and catastrophic failure in many
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  high speed applications. It's absolutely not a trivial
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  force, and the design take away is crystal clear in any system
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  where you have a body moving radially or changing its
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  distance from the center of rotation while it is rotating
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  you. Absolutely must account for this
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  coupling term. Whether you find it using the
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  full acceleration vector or, like we did, using the angular
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  momentum shortcut, doesn't matter.
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  This rotational dynamics effect, this Coriolis effect, is not
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  just abstract theory, it's a critical physical constraint
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  that has to be managed in the real world.
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  OK, let's just quickly recap or unpack what we covered.
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  We started with the basic concept of a particle in
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  dynamics move through. Why we sometimes need
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  curvilinear coordinates, specifically cylindrical because
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  their unit vectors change direction, unlike Cartesian.
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  Saw how this leads to extra terms and velocity and
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  especially acceleration, most notably that inward centripetal
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  acceleration and the the really interesting Coriolis
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  acceleration term to D rote D fit city.
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  Then we applied the principle of angular momentum M equals DHDT
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  to a practical example, a block sliding on a rotating rod, and
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  we calculated a very real, surprisingly large £100 side
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  force needed just to keep that tiny block on the rod,
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  confirming how vital that Coriolis interaction term is in
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  actual engineering design. Right.
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  And if we maybe connect this back to the bigger picture of
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  mechanics, this whole kinetic analysis using forces and
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  momentum, it really rests on understanding rates of change.
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  We found that the side force F depended directly on the product
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  of the angular speed Omega and the radial speed GRTT.
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  Which brings up an interesting point.
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  We assumed constant angular velocity of maybe 50 rads and
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  constant radial velocity drared AT it was 30 feets in our
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  example. But what if those weren't
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  constant? What if the rods rotation was
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  accelerating or the block speed along the rod was changing?
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  How would the dynamics change then?
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  What additional forces or moments would we need to
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  consider to maintain or control the blocks motion?
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  That's maybe something you think about.
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  Until the next equation Design junkies TuneIn every morning at
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  11 before 8:00 AM.