Types of mechanical impedance 2

[NS4U]

Hi,

I am looking for a text or website that talks about different types of mechanical impedances. Mainly as it relates to vibration transmission in buildings.

I am primarily concerned with a discussion about real vs imaginary parts. And how they affect resonant frequencies, attenuation etc...

Thanks!

 

[sreid]

I don't know where to find exactly what you are looking for but this may help If you drive some mechanical device with different frequency sine waves, measured at some other point the sine waves will change in amplitude and phase. Amplitude and phase can also be expressed as real and imaginary parts. An FFT converts vibration data to frequency first as real and imaginary parts (but it is usually displayed as magnitude and phase).

 

[BobM3]

Start with wikipedia and search for "vibration". That will get you started.

 

[NS4U]

I'm not new to vibrations.

Rather, I am interested in why a purely real impedance is an "energy sink," or what does a high real part compared to a low imaginary part tell us about the impedance of the component or why an impedance anti resonance is "mass controlled" and resonance is "stiffness controlled"

I find that most texts do an adequate job of teaching the techniques to develop impedance expressions, but do not do a good job elaborating on what it means physically.

 

[GregLocock]

Ah, now you've asked sensible questions.

"why a purely real impedance is an "energy sink,""

Depends on exactly what domain you are talking about. I'm guessing you are using something akin to voltage and current, or force and velocity. In that case, real work can only be done if the two vectors are not in quadrature.

"what does a high real part compared to a low imaginary part tell us about the impedance of the component"

It tells you it is a resistor, rather than a capacitor or inductance. The high vs low bit depends on which way up you've written your equations.


"why an impedance anti resonance is "mass controlled" and resonance is "stiffness controlled"
"

That would only be so for a particular part of a given response. It is very specific to the setup of the system - for instance, the response of a of a ship floating in the sea, horizontally, is mass controlled at 0 hz, whereas a mass hanging on a spring, vertically, is stiffness controlled at 0 Hz.

I've never seen a textbook that really delves into this stuff in detail, you just have to sit down and do the maths.

First and foremost you need to define the variables, and the form of your characteristic equation (ie are you discussing compliances or stiffnesses, if it is amechanical system)





Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[electricpete]

There is a very tight analogy between discrete M/K/C mechanical systems and discrete C/L/R electrical systems.

Assume sinusoidal steady state.

Force plays the role of current.

Velocity (difference between velocity at two locations) plays the role of voltage.


Mechanical impedance = Force/Velocity unfortunately plays the role of electrical admittance which is the inverse of electrical impedance (leave it to the mech E's to mess that one up ;-)

A spring is a two-terminal element with mechanical impedance Z=K/(jw) = -K j / w
It plays the role of an inductor. (admittance of a capacitor is Y = 1/[jwL])

A mass has one terminal with mechanical impedance Z = j w m and the other terminal is always connected to ground.
It plays the role of a capacitor. (admittance of a capacitor is Y = jwC)

A damping element has mechanical impedance Z=C. It plays the role of a resistor. (admittance of a resistor is 1/R).

In the mechanical world, average power is force times velocity times cos of the angle between them. In the electrical world, average power ic current times voltage times cos of the angle between them (we give it a special name... power factor).

The imaginary impedance elements are mechanical M and K. M stores and returns kinetic energy to the system, but dissipates no energy. K stores and returns potential energy to the system, but dissipates no energy. These are analogous to reactive power (associated with inductors and capacitors).

The real impedance elemtn is C which dissipates energy. The electrical analogy is resistance.


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[NS4U]

Excellent. Thank you for the replies! Sorry this wasn't clear on the first go round.

for example here is a impedance plot of a column excited at the base. F (input force) and V (response velocity) are at the same location (driving point impedance)

Here's what the magnitude looks like:

The real part:

The Imaginary part:

I'm interested in trying to meaningfully interpret what this graph is telling me, aside from here's a resonance, here is an anti-resonance, etc...



It looks like the imaginary part is small at resonances, but at anti resonances (~2 kHz) there's some strange jump, why is that?

In the low freq range (<600 Hz) the impedance is stiffness controlled, and the imaginary part dominates that section of data, as electricpete said it should.

What else can be interpreted here from the impedance? When is it sucking energy out, when is it putting energy in? when is it a wash? How can you tell from looking at the real and imaginary parts?

 

[GregLocock]

I don't usually look at inverse mobility plots!

N/mm/s is a very strange thing to look at - what you are calling a resonance is what I'd call an antiresonance - the structure is not moving, much, for a given excitation.

So, before I spend much time thinking about it can you confirm that the units are correct, and that you really mean to use a low mobility as your definition of resonance?

The weird behaviour at the peaks will be explained if you look in the Nyquist domain.

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[NS4U]

yes mobility plots are more intuitive...

the Units of the plot is Force over velocity, Newtons per (meter per second)

If impedance is Force/Vel then an antiresonance would be one where the response (the velocity) of the structure is very small, thus high impedance. Hence I am calling anti resonances those peaks at 700 Hz, 2kHz, 3.5 kHz, etc.

I am calling the resonances the "dips" where the response (velocity) would be high, thus low impedance.

If this was a mobility plot, anti resonances would be the dips, and the resonances would be the peaks.

 

[electricpete]

Yes, the key is to stand on your head when viewing the graph. Or else stop calling f the input and v the output Just felt like whining ;-)

One observation is that the resonance behavior makes sense. At resonance the imaginary component of mobility gets very large (mass and spring effects cancel) and imaginary component of impedance gets very small, while the real damping component remains fairly flat.







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[electricpete]

I have a suspicion that those peaks in the damping curve are ficticious. When a very high impedance drives the velocity to 0, you can't tell the difference between the real and imaginary parts of the impedance. I suspect that those zero's in the velocity response (peaks in the impedance) would be driven more by the mass spring system (imaginary impedance) than by the damping (real impedance). Just a guess from someone who doens't work with these types of plots.

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[GregLocock]

What's happening in the Nyquist plot near the odd bits is that the circle is slightly displaced from the origin, to the other side from the centre of the circle.

Your damping is ridiculous for a real strucure unless it is welded steel or equivalent.

The gently rising trend from 10-300 Hz is mass controlled, odd for a structure.

Can you plot the phase, and the Nyquist plot?









Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[NS4U]

here's what the phase looks like:

sorry but I never actually generated a Nyquist plot... nor do I really even know what it is...

In regards to your other points, maybe I'm wrong, but the +ive slope (10-300 Hz) is stiffness controlled is it not? Also, damping in this structure- zeta=0.002, this is a 6 ft, 1" x 1" aluminum column.

 

[sreid]

1) I would expect a solid metal piece to have low losses. Q's in the tens of thousands are not uncommon.
2) Sudden phase shifts of 180 deg are typical when going through resonances/antiresonances.

 

[GregLocock]

Nyquist plot=argand diagram=xy plot of real vs imaginary.

It might be a good idea to invert the transfer function before plotting it, I'm getting confused thinking about your upside down plots as it is.


pure mass

f=m.a

a=v.w

therefore

f/v=m.w

that is in your upside down plot a mass controlled line will rise proportional to frequency, ie 20 dB/decade.

pure stiffness

f=k.x

x=v/w

f=k.v/w

f/v=k/w

stiffness controlled line will fall at 20 dB/decade, on your upside down plot.



Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.

 

[spongebob007]

Consider this:

The mechanical impedance of a 1 DOF system is:

Z(i*omega)=-m*omega^2+i*c*omega+K

Let's look at this equation in the frequency limits.

As omega ->0, Z(i*omega)=k Here your structure is stifness dominant.

As omega goes to infinity, the -m*omega^2 term gets much larger than the other terms, so the structure is mass dominant.

A more intuitive approach is to look at the impedances of the individual elements of an undapamped spring-mass.

Impedance is simply Force/Velocity.

For the spring: Z_spring: k*x/omega*x= k/omega

For the mass: Z_mass: -m*omega^2*x/omega*x=-m*omega

as omega ->0 Z_spring goes to infinity and Z_mass goes to 0. This makes physical sense. The spring is "infinitely stiff" This is equivalent to rigid body motion.

as omega -> infinity, Z_spring goes to 0, and Z_mass goes to infinity, thus the mass appears "infinitely massive"

If you were to construct a physical system consisting of a spring and a mass and connect it to a variable frequency drive motor and watch the mass you would see what happens. We did this in college. At very low frequencies the spring won't compress. The motor just pushes the spring in mass. When the motor is tuned closer to resonance the mass really starts moving. Once you pass through resonance you can actually see the spring compressing and extending very fast, but the mass will hardly move at all!!! This is the concept of isolation. The spring is almost infinitely compliant, but the mass sees very little motion transferred from the motor.

When you tune your structure so that the natural frequency is low compared to the driving frequency, this system is mass dominant-isolated. When you tune your system so that the natural frequency is significantly higher than the driving frequency, you "stiffen" the system, hence stifness controled. In either case, there is no amplification response, but in the stiffened case, all the input force is transmitted directly through the structure. In the mass controlled case, the response is LESS than the input force.

For a good explination of these concepts, see "Fundamentals of Acoustics" by Kinsler and Frey.