1 / 36

ME 322: Instrumentation Lecture 30

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

vivi
Download Presentation

ME 322: Instrumentation Lecture 30

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. ME 322: InstrumentationLecture 30 April 7, 2014 Professor Miles Greiner

  2. 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.

  3. 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

  4. 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

  5. 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

  6. 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

  7. Response • Undamped • t +Dcost , • oscillatory • Underdamped • , • damped sinusoid • Critically-damped , and Over-damped • not oscillatory

  8. 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?

  9. 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

  10. Solution • yP(t) = Bsin+Ccos • ; • For not damping (l = 0), AP for • For :

  11. Compare to Quasi-Steady Solution • Undamped Natural Frequency ; Damping ratio: ; • (want this to be close to 1) • with ,

  12. 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%.

  13. Problem 11.35 (page 421) • Solution: • ? • Find f =?

  14. 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

  15. 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?

  16. Expectation • How to find equivalent (or effective) mass MEQ, damping coefficient lEQ, and spring constant kEQ for the weighted and damped cantilever beam?

  17. 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

  18. Intermediate Mass • How to find uncertainty in MEQ? • Power Product or Linear Sum? • Power product or linear sum? • Power product or linear sum?

  19. Midterm II Scores • Mean 77 • Standard Deviation = 15

  20. Uncertainty Calculation

  21. Dynamic (high speed) Accelerometer Response y(t) y0 + y(t) s(t) z(t) = s(t) + y(t) + y0

  22. Accelerometer Moving damped mass/spring system. + y0 For an accelerometer

  23. For steady or “quasi-steady” a(t). Step Response Characteristic Equation b

  24. Undamped𝜆=0 Under damped Critically damped ζ = 1 • Over damped ζ > 1

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

  26. Now, sinusoidal acceleration: Find y(t) (for all ) 0 Find A & B

  27. Lab 10 Vibration of a weighted cantilever Beam

  28. 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

  29. Lab 10 Prediction: What are the effective values of m, k,  ? Equivalent Point end Mass

  30. 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

  31. Lab 10 Modulus from Lab 4 E, WE Power Product

  32. Uncertainty

  33. Predicted Undamped Frequency

  34. Predicted Damped Frequency 𝜆 = ? = f(Frictional Heating, Fluid Mechanics, Acustics) • Hard to predict, but we can measure it.

  35. Predicted Damped Frequency If then

More Related