Vibration and Its Measurement

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ME 144L – Prof. R.G. LongoriaDynamic Systems and Controls Laboratory

Department of Mechanical EngineeringThe University of Texas at Austin

Vibration and Its Measurement

Prof. R.G. Longoria

Updated Fall 2010





This week’s goals• Continue learning how to capture transient electrical signals

using both DAQ (with LabVIEW) and a digital scope

• Review model for mass-spring-damper and develop working

knowledge of parameters that characterize unforced and forced

vibration response

• Study unforced vibration of a mass-spring and mass-beamsystem in the laboratory

• Learn how to use accelerometers to make vibration

measurements





Two Basic ModelsThis week’s laboratory study focuses on mechanical motion and its measurement,

and the mass-spring-damper system is a central theme:

mass-spring-damper systems are ubiquitous in engineering, and understanding

their natural (unforced) and forced response lends insight into system dynamicsand provides tools to aid design of physical experiments and sensors.

Practical problems arise involving two different configurations:

k b

mFixed-base – studyresponse x to

forces on mass

This models many

simple vibration

problems

k b

m

( ) y t

x

Base-excited –response of x subject to forces

induced by motion of

base

This is similar to a

vehicle suspension, andalso models seismic

sensors

x

This case forms the basis for the simple

experiments we will conduct in lab.

This case forms the basis for understanding the

frequency response of seismic sensors, and

particularly accelerometers.





Laboratory ExercisesIn lab, we’ll study two simple realizations of the fixed-based mass-spring-damper

system.

1. We’ll hang a mass by a spring from a table to study the natural response of the mass-spring system (which has low damping). An accelerometer will

measure acceleration of mass.

2. We’ll attach an accelerometer to one end of a cantilevered aluminum beam.

The mass-beam system will be modeled as a simple mass-spring system. Inthis case there may be more damping than in case 1, and the damping

changes depending on the length of the scale left overhanging the table.

We need to have a working knowledge of basic mass-spring-damper response in

order to interpret the results from these experiments.

This insight can be used to configure basic experiments that might be used in

engineering design and analysis practice.





Fixed-Base Mass-Spring-Damper

k b

m

For a full discussion of all cases of the mass-spring-damper system

response, refer to Ogata handout. From that reading, you should

become familiar with:

1. How natural frequency (ω n) and damping ratio (ζ ) are definedfrom the basic 2nd order differential equation,

2. How the mass, stiffness and damping influence ω n and ζ ,3. How the response in each case depends on ω n , ζ , and the initial

conditions (initial position and initial velocity), and that these

are closed-form solutions you can use for basic design andpredictive calculations.

x

2

0

0

2n n

mx bx kx

b k x x xm m

ζω ω

+ + =

+ + =

ɺɺ ɺ

ɺɺ ɺ

≜ ≜

0 undamped

0 1 0 under-damped1 critically-damped

1 over-damped

ζ

ζ ζ

ζ

= ⇒

< < = ⇒= ⇒

> ⇒

Unforced case, F(t) = 0





Undamped Harmonic Motion

For additional information, see Ogata

handout.

0mx kx+ =ɺɺBasic model: Initial conditions: (0)

(0) 0

o x x

x

=

=ɺResponse: cos( )

'natural frequency'

2'natural period'

o n

n

n

n

x x t

k m

T

ω

ω π

ω

=

= =

= =

x+

Vibration relations:

2

Velocity

sin( )

Accelerationcos( )

o n n

o n n

v x x t

a x x t

ω ω

ω ω

= = −

= = −

ɺ

ɺɺ

Useful measures:

2

peak displacement

peak velocity

peak acceleration

o

o n

o n

x

x

x

ω

ω

=

=

=



ME 144L – Prof. R.G. LongoriaDynamic Systems and Controls Laboratory Department of Mechanical EngineeringThe University of Texas at Austin

Logarithmic DecrementThe logarithmic decrement refers to the relationship between the

amplitude of the peaks in the response of an under-damped

system versus the cycle of oscillation.

This is a specific analysis of the response for the case where an

under-damped system is given an initial condition set:

The response data allows you to determine the damping ratio, ζ,

without any other information about the system.

See the Ogata handout and/or the handout in Exercise 2 of the

lab description for more details.

(0)

(0) 0

o x x

x

=

=ɺ




Note on Logarithmic Decrement• The logarithmic decrement helps you find the

damping ratioby measuring the

slope of a lineformed by the natural log of the amplituderatios plotted against cycle number.

• If you plot this data, and it does not form astraight line, we usually interpret this to meanthat the decay is NOT exponential. This meansthat the assumption that the damping in thesystem is linear is NOT valid – i.e., dampingmust be nonlinear.




Using logarithmic decrement to identify

dominant damping




Unforced vs. Forced Response• We learn about motion measurement using

accelerometers and seismic devices in general,and recognize they are undergoing forced

response.

• Design is based on 2nd order system

• We learn about frequency response and how to

interpret the forced response of a seismic device




Use an accelerometer in a frequency

range where you don’t excite its

dynamics.

To understand what determines thisrange, we study the model.




Motion Sensor Dynamics• Sensors used to measure motion, relying on base-

excited mkb system configuration.

• A seismic mass is used and a displacement sensingmechanism monitors the relative position betweenthe seismic mass and the housing.

k b

m z

( ) y t

relative position z =

Seismic

mass

Most sensing mechanisms

either detect or respond to z.

Sensing mechanisms are

discussed in the Appendix.




Mathematical model2 2sin ( )

( ) sin( )

z x y

mz bz kz m Y t m Y t

z t Z t

ω ω ω

ω φ

= −

+ + = =

= +

ɺɺ ɺ

2

22 2

1 2

n

n n

Z Y

ω

ω

ω ω ζ

ω ω

= − +

Amplitude response

2

2

tan

1

n

n

ω ζ

ω φ ω

ω

=

−

phase response

These relations are derived in detail in the pre-lab handout

(ref. Thomson (1993)), and plotted in the graph on the

following slide.

Forcing function

If Y(t) is a sinusoid,

the response is a

sinusoid, but the

amplitude is

different and thereis a phase




The frequency response

function can be

derived by:

1. Converting ODE to

s-domain

2. Letting s = jω3. Deriving the

magnitude and phase

functions*

*These are functions of

frequency, ω




Frequency Response Function

Magnitude: |Z/Y|

From Thomson (1993)

Seismometers

operate in thisregion

Accelerometers

operate in thisregion




‘Seismometer’ Case2

22 2

( )( )

1 2

n

n n

Z Y

ω

ω ω ω

ω ω ζ

ω ω

= − +

This ratio is the ‘sensitivity’ – basically,

how much does the spring element

compress for a given displacement input.

Remember, the spring element represents a

sensing element of some type. for 1n Z Y

ω

ω →≫

Frequency response of Z to Y (displacement) input

i.e., mass does not move!




‘Accelerometer’ CaseFrequency response of Z to Y (acceleration) input

2

2for 1

n n

Y Z

ω ω

ω ω

→ ≪

2

22 2

( )( )

1 2

n

n n

Z Y

ω

ω ω ω

ω ω ζ

ω ω

= − +

This indicates that for this frequency range, Z (which is the sensed

variable) is proportional to acceleration of Y.

2

( ) sin( )( ) cos( )

( ) sin( )

Y t Y t Y t Y t

Y t Y t

ω ω ω

ω ω

==

= −

ɺ

ɺɺ

First, note:

Then from thefrequency

response:

For frequencies well below natural frequency of the sensor:




Frequency (rad/sec)

P

h a s e ( d e g ) ; M a g n i t u d e ( d B )

Bode Diagrams

-40

-30

-20

-10

0

10From: U(1)

10-1 100 101-200

-150

-100

-50

0

T o : Y ( 1 )

Accelerometer – another viewpoint

A magnitude plot helps us to

understand a critical specification

for any sensor :

useful frequency range = bandwidth.

The ‘flat region’ of theresponse is where we

want to operate.

2

22 2

( ) 1 1

( )

1 2

y n

n n

Z

A

ω

ω ω ω ω

ζ ω ω

=

− +

This is now a magnitude function

for a transfer function between the

output X and input acceleration.

Bandwidth

Phase response

Magnitude response




Calibration sheet for a Sensotec (Honeywell) JFT flat pack accelerometer

This is a piezoresistive-type accelerometer



ME 144L – Prof. R.G. LongoriaDynamic Systems and Controls Laboratory Department of Mechanical Engineering

The University of Texas at Austin

Sensors avoid the dynamics• In both cases, the device is designed to respond

to forcing in a frequency range well away fromthe natural frequency.

• If we force it close to the natural frequency, we

induce ‘significant dynamics’ in the sensor.This is generally not a good thing.





Summary• The mass-spring-damper system model and its

unforced and forced response should be working

knowledge.• The mass-spring-damper concept is useful in

experiment design.

• You can understand the underlying design of manytypes of sensors such as accelerometers byunderstanding 2nd order dynamics.

• The frequency response function for a sensor basicallyshows you the sensitivity as a function of the inputfrequency.





Appendix• Types of accelerometers and specifications

• Discussion of some sensing mechanism:– Capacitive

– Piezoresistive

– Piezoelectric





Types of Accelerometers• A short note on accelerometers is provided in the

laboratory web documents.

• There are several types of accelerometers

distinguished by the type of sensing element used to

monitor displacement of the seismic mass.

• The type used in this lab will either be a capacitive or

piezoresistive accelerometer.

• These types give reasonably good low frequencyresponse, and both are made using micro-

electromechanical devices (MEMS).





Capacitive Sensing Mechanism• The measurand directly or indirectly causes a change in the capacitance.

• The easiest conceptualization is to imagine parallel plates.

C

dx•

v

v

q•

F

x•

Energy is stored by virtue o

changes in q and x.

q

where ε is the permittivity, A is thearea, and d is the distance between

the plates.

•Typical scenarios leading to change in C:

–changing the distance between capacitor plates

–changes in the dielectric constant (e.g., due to humidity)

–changes in the area (e.g., a variable capacitor)

AC

d

ε =





Some C Sensors

fluid level

h

ho

H

1 2

insulating material

pressure

deflected diaphragm

dielectric“fixed plate”

mass

“fixed plate”

insulating material

dielectric anddamping

flexible/support beam

motion of

case

chromium layer

Polymerdielectric

Tantulum layerglass

substrate

Humidity

Pressure

Level

Acceleration





ADXL05 (Capacitive) AccelerometerNote: the construction is

basically a mass-spring-damper

system, where the beam and

spring elements deflecthorizontally, and their position is

sensed by the capacitor plates.

However, it is not a simple

‘passive’ system, because there

is feedback in the operation.





ADXL05 Operation

Commonly used with other types of

sensing/actuation

You actively ‘null’ the output, then measure the

voltage or current required. Contrast with how

a Wheatstone bridge works.





ADXL05 AccelerometerThis accelerometer has the frequency response shown below.

This region defines the bandwidth of

this accelerometer. Strictly speaking,

the bandwidth is defined by the

frequency range for which the deviation

is 3 decibels from 0 dB.

This would dictate that you can use this

accelerometer to measure signals with

frequencies out to about 1000 Hz.




Piezoresistive AccelerometerThese devices rely on strain gauges that are

typically solid-state and directly

manufactured into the deflecting beam.

The basic design still relies on a seismic

mass (here labeled inertial mass).

The gauges monitor strain induced by

deflection during acceleration.

The calibration sheet for a piezoresistive

accelerometer from Honeywell (Sensotec)

is shown on the next slide.




Calibration sheet for a Sensotec (Honeywell) JFT flat pack accelerometer

This is a piezoresistive-type accelerometer




On Sensitivity of AccelerometersWe saw that the amplitude function for an accelerometer relates the

displacement response (Z) to the input.

If the displacement response represents the deflection of capacitor

plates or the bending of a beam with strain gauges, you can see

how the amplitude response is related to the sensor output,

typically in voltage. Hence, sensitivity is usually specified as theratio voltage/acceleration. Typical units are mV/g.

Further, the frequency response curve should give you a ‘picture’

of how this sensitivity varies with frequency, and as such helpsdefine the bandwidth by some appropriate measure (e.g., the 3 dB

point).



ME 144L – Prof. R.G. Longoria

Dynamic Systems and Controls Laboratory Department of Mechanical EngineeringThe University of Texas at Austin

Piezoelectric AccelerometersMany high grade accelerometers use piezoelectric material in shear

(left) and the other uses it in compression to form the sensing

element. (Diagram from Bruel & Kjaer). Can you see how

these are basic seismic devices in accelerometer form?





Bruel & Kjaer PZT AccelerometerThis particular specification is for a B&K

accelerometer used for structural response

studies.





“Home-made” SolutionsCourtesy of F. Mims, “Sensor Projects” Mini-Notebook

Using a piezo-electric buzzer element, you can build your own vibration sensor.

Since the PZ material is self-generating you

will get “some” signal to drive the diode.

Mims claims that this setup

detected a train that was 1mile away.

Documents

Vibration and Its Measurement