ME 322: InstrumentationLecture 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 • 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 • 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
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 • 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) 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 • Undamped • t +Dcost , • oscillatory • Underdamped • , • damped sinusoid • Critically-damped , and Over-damped • not oscillatory
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 • 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 • yP(t) = Bsin+Ccos • ; • For not damping (l = 0), AP for • For :
Compare to Quasi-Steady Solution • Undamped Natural Frequency ; Damping ratio: ; • (want this to be close to 1) • with ,
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 421) • Solution: • ? • Find f =?
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) • 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 • 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 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 • How to find uncertainty in MEQ? • Power Product or Linear Sum? • Power product or linear sum? • Power product or linear sum?
Midterm II Scores • Mean 77 • Standard Deviation = 15
Dynamic (high speed) Accelerometer Response y(t) y0 + y(t) s(t) z(t) = s(t) + y(t) + y0
Accelerometer Moving damped mass/spring system. + y0 For an accelerometer
For steady or “quasi-steady” a(t). Step Response Characteristic Equation b
Undamped𝜆=0 Under damped Critically damped ζ = 1 • Over damped ζ > 1
Define: • 1)Undamped Natural Frequency • 2) Damping ratio • For steady or “quasi-steady” a(t).
Now, sinusoidal acceleration: Find y(t) (for all ) 0 Find A & B
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 Prediction: What are the effective values of m, k, ? Equivalent Point end Mass
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 10 Modulus from Lab 4 E, WE Power Product
Predicted Damped Frequency 𝜆 = ? = f(Frictional Heating, Fluid Mechanics, Acustics) • Hard to predict, but we can measure it.
Predicted Damped Frequency If then