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Unit 17 Vibrationdata SDOF Response to Applied Force Revision A 1 Introduction SDOF systems may be subjected to an applied force Modal testing, impact or steady-state force Wind, fluid, or gas pressure Acoustic pressure field Rotating or reciprocating parts Vibrationdata Rotating imbalance Shaft misalignment Bearings Blade passing frequencies Electromagnetic force, magnetostriction 2 Vibrationdata SDOF System, Applied Force m = mass c = viscous damping coefficient k = stiffness x = displacement of the mass f(t) = applied force 3 Vibrationdata Free Body Diagram Summation of forces f(t) x m kx Fm x m x cx kx f ( t ) cx (c / m) 2n (k / m) n 2 m x cx kx f ( t ) c k 1 x x x f ( t ) m m m x 2n x n2 x 1 f (t) m Solve using Laplace transform. 4 For an arbitrary applied force, the displacement x is Smallwood-type, ramp invariant, digital recursive filtering relationship xi Vibrationdata 2 exp n T cosd T x i 1 exp 2n T x i 2 2exp n T cosd T 1 exp n T n 2 2 1 sin d T n T f i 3 m n T d 1 n 2 2 n T exp n T cosd T 21 exp 2n T 2 2 1 exp n T sin d T f i 1 3 m n T d 1 n 2 2 1 sin d T 2 cosd T f i 2 2 n T exp 2n T exp n T 3 d m n T 1 d n 1 - 2 T = time step 5 Vibrationdata SDOF Acceleration For an arbitrary applied force, the displacement x is x i 2 exp n T cosd T x i 1 exp 2n T x i 2 exp n T sin d T m d T f i 2 f i 1 f i 2 6 Time Domain Calculation for Applied Force Vibrationdata Let fn = 10 Hz Q=10 mass = 20 lbm Calculate response to applied force: F = 4 lbf, freq = 10 Hz, 4 sec duration, 400 samples/sec First: vibrationdata > Generate Signal > Sine Export time history as: sine_force.txt Next: vibrationdata > Select Input Data Type > Force > Select Analysis > SDOF Response to Applied Force 7 Applied Force Time History Vibrationdata 8 Displacement Vibrationdata 9 Transmitted Force Vibrationdata Special case: SDOF driven at resonance Transmitted force = ( Q )( applied force ) 10 Synthesize Time History for Force PSD Vibrationdata Frequency (Hz) Force (lbf^2/Hz) 10 0.1 1000 0.1 Duration = 60 sec Similar process to synthesizing a time history for acceleration PSD. But the integrated force time history does not need to have a mean value of zero. 11 Synthesized Time History for Force PSD Vibrationdata Export as: force_th.txt vibrationdata > Power Spectral Density > Force > Time History Synthesis from White Noise f = 4.26 Hz 12 Histogram of Force Time History Vibrationdata 13 PSD Verification Vibrationdata 14 SDOF Response Vibrationdata Let fn = 400 Hz Q=10 mass = 20 lbm Calculate response to the previous synthesized force time history. vibrationdata > Select Input Data Type > Force > Select Analysis > SDOF Response to Applied Force 15 Displacement Vibrationdata Export: disp_resp_th.txt Overall Level = 7.4e-05 in RMS 16 Velocity Vibrationdata Export array: vel_resp_th.txt Overall Level = 0.18 in/sec RMS 17 Acceleration Vibrationdata Export array: accel_resp_th.txt Overall Level = 1.3 GRMS Crest Factor = 5.0 Theoretical Rayleigh Distribution Crest Factor = 4.6 18 Transmitted Force Vibrationdata Export array: tf_resp_th.txt Overall Level = 24.3 lbf RMS 19 Vibrationdata Frequency Response Function Dimension Displacement/Force Name Admittance, Compliance, Receptance Dimension Force/Displacement Name Dynamic Stiffness Velocity/Force Mobility Force/Velocity Acceleration/Force Accelerance, Inertance Force/Acceleration Mechanical Impedance Apparent Mass, Dynamic Mass 20 FRF Estimators H1 G FX () G FF () Vibrationdata Cross spectrum between force and response divided by autospectrum of force Cross spectrum is complex conjugate of first variable Fourier transform times the second variable Fourier transform. G FX () F * X * Denotes complex conjugate The response can be acceleration, velocity or displacement. 21 FRF Estimators (cont) H 2 G XX () G XF () Vibrationdata Autospectrum of response divided by cross spectrum between response and force Coherence Function is used to assess linearity, measurement, noise, leakage error, etc. Coherence is ideally equal to one. 2 G FX () 2 G XX () G FF () 0 2 1 22 Frequency Response Function Exercise Vibrationdata Calculate mobility function (velocity/force) using: vibrationdata > miscellaneous > modal frf - Two separate Arrays – Ensemble Averaging Arrays: force_th.txt & vel_resp_th.txt df = 3.91 Hz & use Hanning Window Important! Plot H1 Freq & Mag & Phase 23 Vibrationdata 24 Mobility H1 SDOF fn=400 Hz, Q=10 Vibrationdata Save Complex Array: H1_mobility _complex.txt 25 Mobility H2 SDOF fn=400 Hz, Q=10 Vibrationdata 26 Coherence from Mobility Vibrationdata Coherence = 0.98 at 400 Hz 27 Estimate Q from H1 Mobility, Curve-fit Vibrationdata fn=400 Hz Q=10.1 H1_mobility _complex.txt vibrationdata > Damping Functions > Half-power Bandwidth Curve-fit, Modal FRF 28 Homework Vibrationdata Repeat the examples in the presentation using the Matlab scripts Read: • T. Irvine, Machine Mounting for Vibration Attenuation, Rev B, Vibrationdata, 2000 • Bruel & Kjaer Booklets: Mobility Measurement Modal Testing 29