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ME 322: Instrumentation Lecture 30. April 7, 2014 Professor Miles Greiner. Announcements/Reminders. Extra-Credit Opportunities Both 1%-of-grade extra-credit for active participation Open ended Lab 9.1 proposals due now LabVIEW Computer-Based Measurements Hands-On Seminar

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me 322 instrumentation lecture 30

ME 322: InstrumentationLecture 30

April 7, 2014

Professor Miles Greiner

announcements reminders
Announcements/Reminders
  • Extra-Credit Opportunities
        • Both 1%-of-grade extra-credit for active participation
    • Open ended Lab 9.1
      • proposals due now
    • LabVIEW Computer-Based Measurements Hands-On Seminar
      • Friday, April 18, 2014, 2-4 PM, Place TBA
      • Signup on WebCampus
      • If enough interest then we may offer a second session Noon-2
  • HW 10 due Friday
    • I revised the Lab 10 Instructions, so please let me know about mistakes or needed clarifications.
piezoelectric accelerometer
Piezoelectric accelerometer
  • Seismic mass increases/decreases compression of crystal,
    • Compression causes electric charge [coulombs] to accumulate on its sides
    • Changing charge can be measured using a charge amplifier
  • High damping, stiffness and natural frequency
  • But not useful for steady acceleration
accelerometer model

Charge

Q=fn(y)

= fn(a)

Accelerometer Model

y = Reading

  • Un-deformed sensor dimension y0 affected by gravity and sensor size
  • Charge Q is affected by deformation y, which is affected by acceleration a
  • If acceleration is constant or slowly changing, then F = ma = –ky, so
    • yS = (-m/k)a
    • Static transfer function
  • What is the dynamic response of y(t) to a(t)?

y

a

y0

l

[N/(m/s)]

k [N/m]

-m/k

a(t) = Measurand

moving damped mass spring system
Moving Damped Mass/Spring System
  • Want to measure acceleration of object at sensor’s bottom surface
  • Forces on mass,
    • z(t) = s(t) + yo + y(t) (location of mass’s bottom surface)
    • Fspring = -ky, Fdamper = -lv = -l(dy/dt)

z

s(t)

Inertial Frame

response to impulse step change in v
Response to Impulse (Step change in v)

v

a

  • Huge a at t = 0, but a(t) = 0 afterward
    • Ideally: y(t) = -(m/k)a(t)= 0
  • my’’+ ly’ + ky = 0
  • Solution:
    • depend on initial conditions
      • Depends on damping ratio:

t

t

response
Response
  • Undamped
    • t +Dcost ,
    • oscillatory
  • Underdamped
    • ,
    • damped sinusoid
  • Critically-damped , and Over-damped
    • not oscillatory
response to continuous shaking
Response to Continuous “Shaking”
    • A = shaking amplitude
    • = forcing frequency
  • Find response y(t) for all
    • For quasi-steady (slow) shaking,
      • Expect
    • For higher , expect lower amplitude and delayed response
  • my’’+ ly’ + ky = -ma(t) = -m
  • y(t) = yh(t) + yP(t)
    • Homogeneous solutions yh(t) same as response to impulse
    • yh(t) 0 after t  ∞
  • How to find particular solution to whole equation?
particular solution
Particular Solution
  • myP’’+ lyP’ + kyP= -m
  • Assume yP(t) = Bsin+Ccos (from experience)
    • Find B and C
    • yP’ = cosCs
    • yP’’= Bscos
  • m(sCcos)+ l(BcosCs)+ k(Bsin+Ccos) = -m
  • s() = 0
    • Two equations and two unknowns, B and C
solution
Solution
  • yP(t) = Bsin+Ccos
    • ;
    • For not damping (l = 0), AP for
    • For :
compare to quasi steady solution
Compare to Quasi-Steady Solution
    • Undamped Natural Frequency ; Damping ratio: ;
  • (want this to be close to 1)
  • with ,
problem 11 35 page 421
Problem 11.35 (page 421)
  • Consider an accelerometer with a natural frequency of 800 Hz and a damping ratio of 0.6. Determine the vibration frequency above which the amplitude distortion is greater than 0.5%.
problem 11 35 page 4211
Problem 11.35 (page 421)
  • Solution:
  • ?
  • Find f =?
lab 10 vibration of a weighted cantilever beam
Lab 10 Vibration of a Weighted Cantilever Beam

LE

LB

  • Accelerometer Calibration Data
    • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2010%20Vibrating%20Beam/Lab%20Index.htm
    • C = 616.7 mV/g
    • Use calibration constant for the issued accelerometer
    • Inverted Transfer function: a = V/C

Clamp

W

T

Accelerometer

LT MT

disturb beam and measure a t
Disturb Beam and Measure a(t)
  • Use a sufficiently high sampling rate to capture the peaks
    • Find f from spectral analysis
    • Find b from exponential fit to acceleration peaks
  • Can we predictf from mass, dimension and elastic modulus measurements?
expectation
Expectation
  • How to find equivalent (or effective) mass MEQ, damping coefficient lEQ, and spring constant kEQ for the weighted and damped cantilever beam?
equivalent endpoint mass
Equivalent Endpoint Mass

LE

LB

Clamp

  • Beam is not massless, so its mass affects its motion and natural frequency
  • mass of weight, accelerometer, pin, nut
    • Weight them together on analytical balance (uncertainty = 0.1 g)

LT MT

ME

Uniform Beam MB

intermediate mass
Intermediate Mass
  • How to find uncertainty in MEQ?
  • Power Product or Linear Sum?
  • Power product or linear sum?
  • Power product or linear sum?
midterm ii scores
Midterm II Scores
  • Mean 77
  • Standard Deviation = 15
dynamic high speed accelerometer response
Dynamic (high speed) Accelerometer Response

y(t)

y0 +

y(t)

s(t)

z(t) = s(t) + y(t) + y0

accelerometer
Accelerometer

Moving damped mass/spring system.

+ y0

For an accelerometer

slide23

For steady or “quasi-steady” a(t).

Step Response

Characteristic Equation b

slide24

Undamped𝜆=0

Under damped

Critically damped

ζ = 1

  • Over damped

ζ > 1

slide25

Define:

  • 1)Undamped Natural Frequency
  • 2) Damping ratio
  • For steady or “quasi-steady” a(t).
slide26

Now, sinusoidal acceleration:

Find y(t) (for all

)

0

Find A & B

slide29
Measure a(t)Find damping coefficient and damped natural frequency, and compare to predictionsHow to predict?

t (s)

Fit to data: find b and f

lab 10
Lab 10

Prediction:

What are the effective values of m, k,  ?

Equivalent Point end Mass

lab 101
Lab 10

Beam Spring ConstKeq

Beam cross-section moment of Inertia

In Lab 4 measure & estimate uncertainty

Length

W, T, WW, WT

LT, LE, LB - ruler W0 = ±

  • inch

Masses

MT ≡ Beam total mass

MW ≡ End components – Mass end, nut, bolt, accelerometer

lab 102
Lab 10

Modulus from Lab 4 E, WE

Power Product

predicted damped frequency
Predicted Damped Frequency

𝜆 = ? = f(Frictional Heating, Fluid Mechanics, Acustics)

  • Hard to predict, but we can measure it.