ME 322: Instrumentation Lecture 31 April 6, 2016 Professor Miles Greiner
advertisement
ME 322: Instrumentation Lecture 31 April 6, 2016 Professor Miles Greiner Lab 10 calculations Announcements/Reminders • HW 10 Due Friday – Add problem X2 – Tutorial tomorrow at 7 PM, SEM 321 • ME Dept. Graduate/Undergraduate Poster Session – Friday, 3 PM, HREL 109 • Elective Courses for Fall – MSE 465 Fundamentals of Nuclear Power • Go to MSE Dept. Headquarter to get prerequisite override form – ME 493-1003 Energy Engineering Lab 10 Vibration of Weighted Steel and Aluminum Cantilever Beams Clamp LE LB MW W T E (Lab 5) LT MT • Accelerometer Calibration Data – http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentat ion/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*1000/C • Measure: E, W, T, LB, LE, LT, MT, MW – Estimate uncertainties of each • Use accelerometer/LabVIEW to measure a(t) Accelerometer Figure 2 VI Block Diagram • Starting point VI • Very similar to Lab 5 • Add Formula Block, Spectral Measurements and Statistics – Use Express Menu – Use Mathematics; Probabilty and Statistics Formula Formula: v*1000/c Statistics Statistics This Express VI produces the following measurements: Time of Maximum Spectral Measurements Selected Measurements: Magnitude (Peak) View Phase: Wrapped and in Radians Windowing: Hanning Averaging: None Figure 1 VI Front Panel Disturb Beam and Measure a(t); 2 lengths • • Use a sufficiently high sampling rate to capture the peaks – fS = ~600 Hz (>> 2fM ) Data looks like π π‘ = π΄π ππ‘ sin(2πππ‘ + π) – For under-damped vibration expect; π π‘ = π΄π −(π – How to predict π = • • 1 2π π π − π 2 ? 2π 2π)π‘ sin(ππ‘ + π) , π = 2ππ = Need π, π πππ π, but π = −(π 2π), so π = Measure f using LabVIEW spectral analysis ( fM ) Use sampling period and frequency of T1 = 10 sec and fS = 600 Hz. – Capable of detecting frequencies between ? and ? Hz, with a resolution of ? Hz. • For sample data, frequencies with the peak oscillatory amplitudes are – For LB = 13 inches, fM = 6.50 ± 0.05 Hz – For LB = 7.5 inches, fM = 18.60 ± 0.05 Hz – Easily detected from this plot. • Find b from exponential fit to acceleration peaks π π − 1 2π π 2 2π π π − π2 Warning: Be careful to check your data before processing • For example, see oscillations between 2 and 4 seconds Time and Frequency Dependent Data • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/ Lab%2010%20Vibrating%20Beam/Lab%20Index.htm • Plot a versus t – Initial time 0, increment Dt = 1/fS • Plot aRMS versus f – Lowest frequency 0, increment Df = 1/T1 • Measured Damped (natural) Frequency, fM – Frequency with peak aRMS 1 2 – Uncertainty π€ππ ππ = Δπ = 1 2π1 • Exponential Decay Constant b (Is it constant?) – Show how to find acceleration peaks versus time • Use AND statements to find accelerations that are larger than the ones before and after it • Use If statements to select those accelerations and times • Sort the results by time – Fit data to y = Aebx to find b Fig. 5 Peak Acceleration versus Time • For sample data, – For LB = 13 inches, fM = 6.50 ± 0.05 Hz, b =0.176 Hz – For LB = 7.5 inches, fM =18.60 ± 0.05 Hz, b =0.192 Hz – Slopes are not exactly constant (but close) Predict damped natural frequency ππ and its uncertainty from mass, dimensions, elastic modulus and decay constant measurements • π = 2πππ = – Since π = – ππ = – π0 = 1 2π π0 2 − π 2 2π π − 2π π0 2 − π 2 (predicted damped frequency) π π = kEQ MEQ (un-damped radial freq.) • How to find equivalent (or effective) mass MEQ, damping coefficient lEQ, and spring constant kEQ for a weighted cantilever beam? Equivalent Endpoint Mass LB Clamp LE MW LT MT Beam Mass MB ME • Beam is not massless, so its mass affects its motion and natural frequency. It can be shown that for a uniform cross-section beam: • ππΈπ = ππΈ + 0.23ππ΅ – ππ΅ = ππ πΏπ΅ πΏπ – ππΈ = ππ + πΏπΈ ππ πΏπ (beam mass) (end mass) • ππ = mass of weight, accelerometer, pin, nut – Weight them together on analytical balance (uncertainty = 0.1 g) Equivelent Mass, ππΈπ • ππΈπ = ππΈ + 0.23ππ΅ • ππΈπ = ππ + πΏπΈ ππ πΏπ + πΏπ΅ 0.23ππ πΏπ • ππΈπ = ππ + πΏπΈ + 0.23πΏπ΅ ππ πΏπ – How to find uncertainty in MEQ? – Power Product or Linear Sum? • ππΈπ = ππ + ππΌ – Power product or linear sum? • ππΌ = πΏπΈ + 0.23πΏπ΅ ππ πΏπ =π – Power product or linear sum? • π = πΏπΈ + 0.23πΏπ΅ – Power product or linear sum? ππ πΏπ Intermediate Mass, ππΌ uncertainty • ππΌ = πΏπΈ + 0.23πΏπ΅ • • π€ππΌ 2 ππΌ π€ππΌ ππΌ 2 = = π€ππ 2 ππ π€ππ ππ 2 + + ππ πΏπ π€πΏπ 2 πΏπ π€πΏπ πΏπ 2 + + π(πΏπΈ +0.23πΏπ΅ ) 2 πΏπΈ +0.23πΏπ΅ π€ πΏπΈ 2 + 0.23π€πΏπ΅ πΏπΈ +0.23πΏπ΅ 2 2 Beam Equivalent Spring Constant, KEQ F LB d • From Solid Mechanics: –πΏ= πΏ3π΅ πΉ 3πΈπΌ –πΌ= ππ 3 12 – E = Elastic modulus measured in Lab 5 • πΎπΈπ = πΉ πΏ = πΉ πΏ3 π΅πΉ 3πΈπΌ = 3πΈ ππ 3 πΏ3π΅ 12 = – Power product or linear sum? πΈπ π 3 4 πΏπ΅ Predicted Frequencies • Undamped – π0π = π0π 2π = πΎπΈπ 1 2π ππΈπ – Power Product? • Damped – ππ = ππ 2π = 1 2π πΎπΈπ ππΈπ − π 2ππΈπ 2 = 1 2π πΎπΈπ ππΈπ − π2 – Power product? – If π2 βͺ πΎπΈπ ππΈπ , then ππ ≈ π0π , and π€ππ ≈ π€π0π Table 1 Measured and Calculated Beam Properties Elastic Modulus, E Beam Width, W Beam Thickness, T Beam Total Length, LT End Length, LE Beam Mass, MT Combined Mass, MW Units Value GPa inch inch inch inch g g 206.3 0.9968 0.1220 23.813 0.469 377.2 333.9 3s Uncertainty 7.5 0.0015 0.0015 0.063 0.063 0.1 0.1 • You will be given the beam you used in Labs 4 and 5 – The value and uncertainty in E were determined in Lab 5 – W and T were measured using micrometers whose uncertainty were determined in Lab 4 • LT, LE, and LB were measured using a tape measure (readability = 1/16 in) • MT and MW measured using an analytical balance (readability = 0.1 g) Table 2 Calculated Values and Uncertainties LB = 7 inch Units Value LB = 13 inch 3s Uncertainty Value 3s Uncertainty 0.1778 0.0016 0.3302 0.0016 Beam Length, LB [m] Intermedate Mass, MI [kg] 0.033 0.001 0.055 0.001 Equivalent Mass, MEQ [kg] 0.367 0.001 0.389 0.001 6912 402 1079 58 [Hz] 21.8 0.6 8.4 0.2 [Hz] 18.60 0.05 7.50 0.05 [1/sec] -0.192 0.1409 21.8 0.6 -0.176 0.1368 8.4 0.2 17% - 12% - Equivalent Beam Spring [N/m] Constant, kEQ Predicted Undamped Frequency, foP Measured Damped Frequency, fM Decay Constant, b Damping Coefficient, l M [Ns/m] Damped Frequency, f p Percent Difference (fP/fM-1)*100% ππ πΏπ π€ππΌ 2 π€ππ 2 2 + 0.23π€πΏπ΅ πΏπΈ +0.23πΏπ΅ 2 2 • LB was measured using a tape measure (readability = 1/16 in) • The equivalent mass is not strongly affected by the intermediate mass • The predicted undamped and damped frequencies, fOP and fP, are essentially the same (frequency is unaffected by damping). • The confidence intervals for the predicted damped frequencies are outside the measure values ππ πΏπ + π€πΏπΈ ππΌ = πΏπΈ + 0.23πΏπ΅ ππΌ + π€πΏπ 2 • ; = [Hz] Midterm 2 • • • • • Average 76, St Dev = 15 Range: 100 to 34 Please pick up exams in Lab now or in your lab period Solutions posted outside PE 213 Will consider re-grading for the next week only Final • Repeat one of the last three labs (10, 11 or 12) – Solo, start to finish • Generate LabVIEW, Excel and PowerPoint during final • Only given instructions, book, and 1 page of notes – No sample lab, partner, or connection to internet • Make sure to prepare yourself during the next three labs Effect of Sampling Rate • If the sampling rate is too slow, then it is likely that the peak accelerations will be missed for most of the oscillations • Can cause a type of aliasing problem Very repeatable frequency for same beam length (7.5 and 15 inch) • 7 inch Measurements and Uncertainties • Lengths – W, T, wW, wT: Lab 4 – LT, LE, LB: Ruler w = 1/16 inch • Masses – MT Total beam mass – MW End components measured together – Uncertainty 0.1 g
