l5a accelerometers in nutshell - mechanical...

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

Department of Mechanical EngineeringThe University of Texas at Austin

Accelerometers in a Nutshell

Prof. R.G. Longoria

Summer 2014



Overview

• Accelerometers are mass-spring-damper

systems used to measure vibration level (e.g., in

units of ‘g’).

• The sensitivity of accelerometers will vary with

the frequency of the vibration.

• Sensitivity is typically provided in

voltage/acceleration, which is assumed to be

constant over a useful range of frequencies (the

bandwidth).



Motion sensors like accelerometers

are base-excited systems

Typically, a seismic mass, m, is

restrained by a sensing element,

represented here by a spring with

stiffness, k.



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.



Some accelerometers rely on

capacitive sensing elements• Motion causes a change in capacitance.

• A common configuration uses parallel plates, where capacitance is,

with ε the permittivity, A the area, and d 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 common C sensors

fluid level

h

ho

H

1 2

insulating material

pressure

deflected diaphragm

dielectric“fixed plate”

mass

“fixed plate”

insulating material

dielectric and

damping

flexible/support beam

motion of

case

chromium layer

Polymer

dielectric

Tantulum layerglass

substrate

Humidity

PressureLevel

Acceleration



ADXL05 (capacitive) accelerometerThe construction is basically a

mass-spring-damper system,

where the beam and spring

elements deflect, and their

position is sensed by the

capacitor plates.



gravitational field is a constant acceleration at level ‘g’



Another common type of

accelerometer is piezoelectricPiezoelectric material is put into shear (left) or compression induce

changes in charge distribution. (Diagram from Bruel & Kjaer).



“Home-made” accelerometersFrom 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 1

mile away.



www.sparkfun.com

You can find cheap accelerometers



To choose an accelerometer, you need to know

how it responds to vibration

Say the ground motion is

sinusoidal, ( ) siny t Y tω=

Y is the amplitude of the input motion (a displacement) and ω is

the forcing frequency in rad/s. Remember, 2 2f Tω π π= =



2

22 2

( )

1 2

n

Y

n n

ZG

Y

ω

ωω

ω ωζ

ω ω

= =

− +

2

2

tan

1

n

n

ωζ

ωφ

ω

ω

=

−

Magnitude response

Sensitivity (G) relates the deflection (Z) of the

sensing element to the input amplitude (Y).

( )ω= ⋅Z G Y

Frequency response functions (FRFs)

Phase response



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

How the

frequency

response function

(FRF) is derived



The magnitude and phase FRFs allow

you to calculate the output amplitude

(z) given the input amplitude (Y) at

any value of the forcing frequency.( ) sin( ) ( ) sin( ( ))

( ) magnitude FRF

( )= phase FRF

input amplitude

= ( ) output amplitude

Y

Y

Y

Y

z t Z t G Y t

G

Y

Z G Y

ω φ ω ω φ ω

ω

φ ω

ω

= + = ⋅ ⋅ +

=

=

⋅ =



FRFs for motion sensors

From Thomson (1993)

Seismometers

operate in this

region

Accelerometers

operate in this

region

( )Y

ZG

Yω= ( )

Yφ ω



Seismometers monitor input

displacement

2

22 2

( )

1 2

n

Y

n n

G

ω

ωω

ω ωζ

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 1

n

Z Yω

ω→ ≫

Frequency response of Z to Y (displacement) input

i.e., mass does not move!



2

22 2

( )( )

( )

1 2

n

Y

n n

ZG

Y

ω

ωωω

ωω ω

ζω ω

= =

− +

2

2 22 2

( ) 1 1

( )

1 2

n

n n

Z

Y

ω

ω ω ωω ω

ζω ω

=

− +

That’s acceleration amplitude

2

22 2

( ) 1 1( )

( )

1 2

A

y n

n n

ZG

A

ωω

ω ωω ω

ζω ω

= =

− +

Accelerometer FRF

Move ω over:

Tells us how

Z responds to

acceleration

as input

First:



Frequency (rad/sec)

Phase (

deg);

Magnitude (

dB

)

Bode Diagrams

-40

-30

-20

-10

0

10From: U(1)

10-1 100 101-200

-150

-100

-50

0

To: Y

(1)

Accelerometer sensitivity

useful frequency range = bandwidth.The ‘flat region’ of the

response is where we

want to operate.

2

22 2

1 1( )

1 2

A

n

n n

G ωω

ω ωζ

ω ω

=

− +

Bandwidth

Phase response

Magnitude response

We want to use the sensor in a region

of frequencies where the sensitivity is

essentially constant.



Example: 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.



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

This is a piezoresistive-type accelerometer

SENSITIVITY



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 the

ratio 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 helps define

the bandwidth by some appropriate measure (e.g., the 3 dB point).



(Good) Sensors avoid the dynamics

• A suitable sensor has a bandwidth broad

enough so that the natural frequency is not

excited.

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

induce ‘dynamics’ in the sensor. This is

generally not a good thing. You want to

‘operate in the flat region’.



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

accelerometer used for structural response

studies.



Summary

• Motion sensors take advantage of the basic mass-

spring-damper system.

• A frequency response function for a sensor basically

shows you how the sensitivity is a function of the

input (forcing) frequency.

• We would like sensitivity to be effectively constant

over a useful frequency range, and we define this as

the bandwidth.

• It is helpful to understand how an accelerometer

responds to both amplitude and frequency of input.

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