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(#51) Shock and Vibration Analysis · Mechanical Engineering Made Simple

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  Welcome back design junkies, to mechanical engineering Made
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  simple. Today we are strapping ourselves
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  into one of the densest and most critical fields in engineering
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  the world of mechanical shock and vibration.
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  This really is the domain that separates theory from reality.
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  We're taking a deep dive into some extensive excerpts from
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  engineering handbooks. You know the kind.
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  You usually need a decade of experience just to navigate to
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  give you a shortcut. So our mission today is to get
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  you informed and quickly on everything that shakes, rattles
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  or rolls. We're going to move from the
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  foundational theoretical analysis, through all the
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  cutting edge measurement tech and finally into the practical
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  control of these problems. The scope is pretty intentional.
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  We start with the analysis fundamentals, you know, covering
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  the contrast between simplified rigid bodies and these complex
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  continuous structures. Then we'll shift gears entirely
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  into instrumentation, the transducers, the signal
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  processing, all the data cleanup techniques that let us actually
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  see the problem. And then we'll wrap up with the
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  applications. Right.
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  Critical applications, things like machine monitoring,
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  specialized testing like the shock response spectrum, and
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  even how we model the human body under impact.
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  That model treating the human spine like a spring, that's
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  where it gets really interesting later.
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  But OK, let's start at the beginning analysis.
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  When engineers look at a vibrating system, they
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  immediately categorize it right. Exactly.
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  You've got 2 massive categories. First, systems defined by what
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  we call lunked parameters. Think of a heavy machine sitting
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  on big rubber isolation pads. You can simplify that.
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  Treat the machine as one rigid mass and the pads as separate
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  springs and dampers. So all the important properties
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  are in discrete or lumped places.
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  Precisely, And even if you have several modes of vibration
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  coupled together, the math is still manageable because the
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  parameters mass, stiffness, damping are all just discrete
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  values. But then you have the other kind
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  of structure where the mass and elasticity aren't so
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  conveniently located. And that brings us to
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  distributed parameters. This is where the mass and the
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  stiffness are spread throughout the entire volume of the
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  material. Think beams, plates, shells.
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  OK, so you can't just use a single equation of motion
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  anymore. Not at all.
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  You have to start solving partial differential equations.
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  The whole analysis fundamentally changes.
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  You move from algebra to continuous calculus to describe
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  the motion. And as soon as you start
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  modeling these complex systems, especially under forcing, you
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  run out of that comfort zone of linearity pretty fast.
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  Oh, nonlinearity is the standard in the real world.
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  I mean a key distinguishing feature of non linear free
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  vibration, and it's a huge red flag, is that the period of the
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  motion actually depends on the amplitude.
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  So if I have a simple linear pendulum, it doesn't matter if I
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  swing it a little or a lot, the time it takes for one full swing
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  stays the same but for a non linear system.
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  Push it harder and the duration of the swing changes.
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  It's that relationship between amplitude and frequency that's
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  so key, and that spills right into forced vibration.
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  If you plot the response curve for a system with non linear
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  restoring forces, maybe a structure that stiffens or
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  hardens the resonance peak doesn't just shoot straight up.
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  It bends. It bends Oregon get swept over.
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  In a hardening system, the peak leans to the right toward higher
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  frequencies. In a softening system, it leans
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  left and damping still does its job.
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  You know it limits the amplitude at resonance, but the geometry
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  of these curves is what makes them so unique.
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  And as long as we're talking about these continuous systems,
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  we have to look at rotation, torsional vibration, that
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  twisting motion is just fundamental to things like
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  engine crankshafts. Torsional issues are their own
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  specialty, but they're built on the same foundations.
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  When you model a rotating system, you have to account for
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  how the mass is distributed, and that means calculating the polar
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  moment of inertia J. Which is J calculated as the
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  integral of R-squared DM. It's a measure of how far the
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  mass is from the axis of rotation.
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  Exactly. And if you get that wrong in a
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  reciprocating engine, you're looking at catastrophic shaft
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  failure, and very quickly. It connects the math directly to
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  durability. So we've laid out the theory.
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  Now how do we actually measure these complex motions?
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  Let's move to the art of measurement.
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  The primary tool here is the accelerometer, isn't it?
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  Absolutely. For acceleration the
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  accelerometer is king. But you know there are major
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  trade-offs and how they're built.
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  Depending on what you need. We mostly use two types,
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  piezoelectric and piezoresistive.
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  OK, tell us about piezoelectric first.
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  Those are the self generating ones.
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  Right. They rely on the direct
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  piezoelectric effect. You stress the material, it
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  generates a charge. They're real workhorses,
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  especially in extreme environments using materials
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  like tourmaline and lithium niobate.
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  We've developed centers with incredible thermal stability.
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  We're talking reliable measurements from -425°F up 2 +
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  1400. That is an insane range.
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  Why is that so important? Because even a tiny change in
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  the material properties can throw off the sensitivity.
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  But beyond temperature, there's a huge practical advantage in
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  the signal conditioning. We prefer charge sensing because
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  the sensor sensitivity becomes independent of your cable
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  capacitance. Well, up to a limit anyway.
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  That's the real world insight. If you're running a long cable
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  through a factory, you don't want the cable itself screwing
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  up your signal. Exactly.
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  It makes deployment so much more robust.
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  Now you contrast that with piezo resistive accelerometers.
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  They are not self generating. How do they?
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  Work then. They use semiconductor
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  materials, usually silicon, where the electrical resistivity
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  changes dramatically with strain.
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  They need external power, but they offer fantastic sensitivity
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  even down to DC. The trick is in the circuitry.
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  The Wheatstone Bridge. Precisely, you arrange the
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  sensing elements in a Wheatstone bridge, often with four active
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  elements. This is 2 brilliant things.
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  One, it dramatically increases the sensitivity, and two, it
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  cancels out errors caused by temperature changes.
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  That cancellation is such an elegant piece of engineering.
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  Now every sensor has its limits based on that ratio of the
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  vibration frequency to its own natural frequency, Omega over
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  Omega N I was surprised just how restrictive an undamped
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  accelerometer. Is it's severely limited.
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  An accelerometer with 0 damping is only accurate up to about 20%
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  of its own natural frequency. Any higher and the errors just
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  explode. So you're forced to pick a
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  sensor with a ridiculously high natural frequency just to
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  measure something in the mid range.
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  How do you extend that usable range?
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  With damping. But not just any amount.
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  If you increase the fraction of critical damping to about .65,
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  you get this mathematically optimized flat response.
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  That one change extends the measurable range all the way up
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  to about 60% of the natural frequency.
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  It's a huge improvement. OK, so we've measured the
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  motion. We have a signal, but we live in
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  a digital world, so that signal needs to be sampled.
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  And this brings us to that first major data problem, aliasing.
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  Aliasing is data corruption. I mean pure and simple.
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  When you digitally sample a signal, if you sample 2 slowly
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  slower than twice the highest frequency in your signal, the
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  spectrum overlaps on itself. And the high frequencies get
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  falsely mapped onto the low frequencies.
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  It just looks like noise, but it's actually corrupted data.
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  Exactly, and you can't undo it. That's why the anti Dali sync
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  filter is so essential. It's only job is to make sure
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  before the signal gets digitized that no frequency components
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  exist above half the sampling frequency.
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  These filters need to be incredibly steep.
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  We're talking 120 decibels per octave to cut things off
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  cleanly. It's the digital gatekeeper.
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  So let's say we're monitoring a machine, maybe a gearbox, and we
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  have a good signal, but it's buried in all this factory
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  noise. We can use a technique called
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  synchronous averaging. A very powerful technique.
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  The key is synchronization. We use a repetitive pulse, maybe
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  from a sensor that triggers once per revolution of a shaft, to
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  start recording our data. So the specific vibration
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  signature you're looking for is always recorded starting at the
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  exact same point in time. Precisely.
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  And when you average all those recordings, the real repetitive
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  signal from that gear fall say adds up directly proportional to
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  end number of averages. But all the random background
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  noise, all the non synchronous stuff, it only adds up like
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  random noise proportional to the square root of N.
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  So you're improving the signal relative to the noise.
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  Yes, the overall improvement in the signal to noise ratio is
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  proportional to the square root of N So you average 100 times
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  you get a 10 fold improvement in clarity.
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  It's invaluable for pulling a tiny signal out of a mountain of
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  noise. OK, that gives us a clean
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  signal. Let's move to testing and
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  control, starting with shock. When we test something against
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  an impact, say dropping a phone, we use the shock response
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  spectrum, or SRS. The SRS is, well, it's the
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  fundamental way to characterize a shock in terms of its
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  destructive potential. It's a plot that shows the
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  maximum response, acceleration, velocity, whatever that a simple
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  single degree of freedom system would experience as a function
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  of its own natural frequency. So if I have a component that
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  naturally vibrates at 100 Hertz, the SRS tells me exactly how
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  hard that 100 Hertz component was hit by the shock.
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  That's it exactly. And while you can plot it
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  different ways, velocity is often the preferred parameter
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  when you're thinking about damage potential.
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  Why velocity? Because the maximum stress
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  inside a structure is directly proportional to the modal or the
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  relative response velocity. So if you're designing for
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  material strength, the velocity SRS gives you the most direct
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  prediction of failure. That's fascinating.
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  Let's swing back to condition monitoring for machinery.
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  If a local fault like a tiny crack on a gear tooth starts to
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  develop, how does that show U in the data we just cleaned U?
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  It shows U in a very reliable pattern.
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  You'll see the main vibration eak of the gear, but it'll be
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  surrounded by a family of smaller Eaks called sidebands,
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  and the spacing between those side bands is the key.
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  It's exactly equal to the rotation speed of the faulty
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  gear. Because that cracked tooth hits
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  the mesh once per revolution precisely.
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  And we use specialized techniques like cepstrum
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  analysis and envelope detection to reliably diagnose and track
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  these defects before they cause a catastrophic failure.
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  Now for active control, if we have a machine that vibrates at
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  one specific frequency, the classic countermeasure is the
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  auxiliary mask system, the tuned damper.
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  The tuned damper is a mechanical masterstroke.
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  It's just a small secondary system, a spring and an
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  auxiliary mask that you tune to. Its own natural frequency is the
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  same as the disturbing frequency of the main structure.
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  When the machine starts to vibrate, the damper starts
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  vibrating 180° out of phase and it applies an equal and opposite
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  force. Effectively neutralizing the
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  vibration. Right at that one specific
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  frequency. We've seen them used
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  successfully on machine tools, for example, where a plastic
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  ring element can combine both elasticity and the damping into
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  one single unit. Which is different from a system
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  where the mass is coupled only by damping, like with viscous
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  fluid. Yes, a purely damp system is
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  focused on absorbing and dissipating energy across a
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  wider band, not neutralizing a single frequency peak like a
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  tuned damper does. Let's shift our focus to extreme
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  environments. Structures don't just respond to
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  their own forces, they respond to the whirlwind.
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  Fluid could flow, high intensity sound.
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  High intensity acoustic fields are a huge deal in aerospace.
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  The sound pressure from a rocket engine alone can cause fatigue
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  failure, but maybe the most dramatic fluid interaction is
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  vortex induced oscillation. The classic example being a
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  cylinder and a cross flow like a bridge piling.
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  Right as fluid flows past, it creates these alternating
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  swirling vortices. The von Karman vortex St.
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  The frequency of that shedding F sub S is proportional to the
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  flow speed. And the trouble starts when F
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  sub S approaches the structure's natural frequency F sub N.
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  That's the critical point. As the flow increases and F sub
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  S gets close to F sub N, you get this phenomena called lock in.
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  The shedding frequency stops increasing and instead locks
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  onto the structure's natural frequency.
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  This causes huge vibrations and the maximum amplitude typically
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  happens around a flow velocity ratio U divided by FD of about
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  5.5. And finally, let's consider the
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  most complex system we analyze under shock the human body.
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  For scenarios like a car crash or an aircraft ejection, we use
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  something called the Dynamic Response Index, or DRI.
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  It's a tool for estimating the potential for spinal injury.
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  It seems almost reductive to model the spine that way, but
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  you have to start somewhere. You do.
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  The DRI is based on a highly simplified single degree of
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  freedom model of the spine. The index itself is defined as
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  Omega n ^2 times delta Max all divided by G.
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  And are there specific parameters for this analytical
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  spine? Yes, based on tons of
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  experimental data, the model uses a natural frequency Omega
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  sub N of 52.9 radians per second and a damping ratio Zeta of
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  .224. Those specific numbers define
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  the model used for safety standards globally.
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  And those numbers lead directly to safety criteria.
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  They do. For instance, ADRI value of 18
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  suggests about a 5% probability of spinal injury based on
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  historical data. It's an incredibly practical
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  metric, even though the model itself is a massive
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  simplification. It's really remarkable.
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  We've gone from the abstract math of distributed systems
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  through the physics of piezoelectric materials all the
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  way to a specific number like 52.9 radians per second used to
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  predict human safety. We've covered the entire
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  mechanical landscape. Well, what's fascinating is the
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  tension between these practical linear models we use and the the
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  underlying reality of the systems.
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  For the DRI, we treat the body as a linear passive system,
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  which is OK for small amplitudes, but soft tissues
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  like muscle and tendon. They aren't simple springs.
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  They're more like viscoelastic elastomers, and they show
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  considerable nonlinearity, even well below the point of injury.
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  That nonlinearity seems to be the ultimate challenge.
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  We use linear analysis to get our heads around the basics, but
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  whether it's a hardening spring, turbulent flow, or the tissues
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  in your own body, the world around us is defined by that
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  elusive, difficult to model nonlinear behavior.
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  Thank you for guiding us through this.
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  Always a pleasure.