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Unit 18 Vibrationdata Force Vibration Response Spectrum 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 Governing equation of motion m x cx kx f (t ) 3 Rayleigh Peak Response Formula Vibrationdata Consider a single-degree-of-freedom system with the index n. The maximum response can be estimated by the following equations. cn 2 ln fn T Cn cn Maximum Peak fn T ln n 0.5772 cn Cn n is the natural frequency is the duration is the natural logarithm function is the standard deviation of the oscillator response 4 Vibrationdata Steady-State Response to Sine Force The normalized displacement is kx F 1 2 2 1 2 2 where F is the applied force magnitude The natural frequency fn is fn 1 2 k m f / fn f is the applied force frequency fn is the natural frequency 5 Steady-State Response to Sine Force (cont) Vibrationdata The transmitted force to ground ratio is Ft F 1 2 2 1 2 2 2 2 , f / fn where Ft is the transmitted force magnitude F is the applied force magnitude The transmitted force ratio is the same as that for the acceleration response to base excitation. 6 SDOF STEADY-STATE RESPONSE TO APPLIED SINUSOIDAL FORCE DISPLACEMENT MAGNITUDE [ k x / F ] 20 Q = 10 Q=2 Q=1 Vibrationdata 10 1 0.1 0.01 0.1 1 10 FREQUENCY ( f / fn ) Control by Frequency Domain Low Freq Resonance High Freq Stiffness Damping Mass 7 SDOF STEADY-STATE TRANSMITTED FORCE 20 Q = 10 Q=2 Q=1 10 TRANS FORCE MAG | Ft / F | Vibrationdata 1 0.1 0.01 0.1 1 10 FREQUENCY ( f / fn ) 8 Exercise Vibrationdata vibrationdata > Miscellaneous Functions > SDOF Response: Steady-State Sine Force or Acceleration Input Practice some sample calculations for applied force using your own parameters. Try resonant excitation and then +/- one octave separation between the excitation and natural frequencies. 9 SDOF Response to Force PSD, Miles Equation Vibrationdata The overall displacement x is 1/ 2 1/ 4 1 3 / 4 A 1 x RMS m k 8 where m is the mass k is the stiffness is viscous damping ratio is the amplitude of the force PSD in dimensions of [force^2 / Hz] at the natural frequency A Miles equation assumes that the PSD is white noise from 0 to infinity Hz. 10 Miles Equation, Velocity & Acceleration The overall velocity is Vibrationdata x RMS n x RMS • An accelerance FRF curve is shown for a sample system in the next slide • The normalized accelerance converges to 1 as the excitation frequency becomes much larger than the natural frequency • The acceleration response would be infinitely high for a white noise force excitation which extended up to an infinitely high frequency • A Miles equation for the acceleration response to a white noise applied force cannot be derived 11 Miles Equation, Acceleration Vibrationdata ACCELERANCE MAGNITUDE ( ACCELERATION / FORCE ) SDOF SYSTEM: mass= 1 kg fn = 100 Hz Damp = 0.05 10 2 ACCELERANCE ( m/sec / N ) 100 1 0.1 0.01 0.001 1 10 100 1000 EXCITATION FREQUENCY (Hz) 12 SDOF Response to Force PSD, General Method Vibrationdata Displacement 1 x RMSf n , k N 1 2 i 1 1 i 2 2 i 2 F PSD(fi ) fi , i f i / f n Velocity 2 x RMSf n , k N fi 2 2 i 1 1 i 2 2 i 2 F PSD(f i ) f i 13 SDOF Response to Force PSD, General Method Vibrationdata Acceleration x RMSf n , 4 2 k N fi 4 2 i 1 1 i 2 2 i 2 F PSD(f i ) f i , i f i / f n Transmitted Force Ft RMSf n , N 1 2 i 2 2 i 1 1 i 2 2 i 2 F PSD(f i ) f i 14 Force PSD Vibrationdata Frequency (Hz) Force (lbf^2/Hz) 10 0.1 1000 0.1 Duration = 60 sec The same PSD was used for the time domain calculation in Webinar 17. 15 Vibrationdata SDOF Example Apply the Force PSD on the previous slide to the SDOF system. Duration = 60 seconds (but only affects peak value) Mass = 20 lbm, Q=10, Natural Frequency = independent variable 16 SDOF Response to Force PSD, Acceleration Vibrationdata Response at 400 Hz agrees with time domain result in previous webinar unit. fn (Hz) Accel (GRMS) 100 0.80 200 1.0 400 1.3 vibrationdata > Power Spectral Density > Force > SDOF Response to Force PSD 17 SDOF Response to Force PSD, Transmitted Force Vibrationdata 18 Acceleration VRS Vibrationdata fn (Hz) Accel (GRMS) 100 0.80 200 1.0 400 1.3 vibrationdata > Power Spectral Density > Force > Vibration Response Spectrum (VRS) 19 Velocity VRS Vibrationdata 20 Displacement VRS Vibrationdata 21 Transmitted Force VRS Vibrationdata 22 Homework Vibrationdata Repeat the examples in the presentation using the Matlab scripts 23