# Estimation of Fundamental Natural Frequency, Damping Ratio

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Estimation of Fundamental Natural Frequency, Damping Ratio and Equivalent Mass 421L/521L (Lab 8)

Single DOF Modeling accelerometer E, I, L, ρ Cantilever c k E, I, L, ρ Fixed-Fixed E: Young’s modulus I: Moment of inertia L: length ρ: mass per unit length mx” cx’ kx f(t) M x k, stiffness, N/m m, mass, kg c, damping coefficient, N/(m/s) x(t) Aexp(-ξωnt)COS(ωnsqrt(1-ξ2)t- ψ) Bsin(ωt) Time response Transient response Forced response(sinusoidal) Where, ωn sqrt(k/m), undamped natural frequency, rad/s ξ c/sqrt(2mk), damping ratio ωd ωnsqrt(1-ξ2), damped natural frequency, rad/s

Visualization of responses Sinusoidal part Exponential part 1 0 Transient response -1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 1 0 -1 5 Forced response (Sinusoidal input) 0 -5 5 Transient response 0 Forced response -5

Experiment Identify the fundamental mode characteristics using logarithmic decrement Mount Accelerometer onto beam – End for cantilever beam – Center for fixed-fixed beam Excite beam by applying ‘impulse’ or initial displacement – Observe transient response (No forced response) Collect time response Pick two peaks and measure amplitude and period Find natural frequency, damping ratio Find equivalent mass from beam equation Find damping coefficient and stiffness

? Equivalent mass and natural frequency estimation by Rayleigh method (See the handout) – Cantilever Beam meq 0.2235ρ L ωn 3.6639sqrt(EI/(ρL4)) rad/s – Fixed-Fixed Beam meq 0.3836ρ L ωn 22.373sqrt(EI/(ρL4)) rad/s Does your measurement match to your estimation? – Show your measurement and measured value What if you count the mass of the accelerometer?

Experimental setup: Cantilever Beam Aluminum Beam – Thickness 4.84mm – Width 19.09mm – Length 640mm Accelerometer is mounted at the end of the beam Mass of accelerometer 7.83 gram

Cantilever Beam NOTE: X1,2 time in s, y1,2 acceleration in g, (m ‘mili’)

Work Sheet: Cantilever Beam # A B C D E F G Item Time @ peak #1 Time @ peak #2 Amplitude @ peak #1 Amplitude @ peak #2 Time between A and B Number of periods between A and B Period of oscillation, E/F Unit s s g Value # Item H Damped natural frequency, wd Unit rad/s I Natural frequency, wn rad/s s J zeta K Equivalent mass, meq s L Stiffness, k N/m M Damping, c N/(m/s) N Natural frequency rad/s estimation by Rayleigh method g kg Value

Experimental setup: Fixed-Fixed Beam Aluminum – Thickness 4.84 mm – Width 19.09 mm – Length 640 mm Accelerometer is mounted at the center Mass of accelerometer 7 .83 gram

Fixed-Fixed Beam NOTE: X1,2 time in s, y1,2 acceleration in g, (m ‘mili’)

Work Sheet: Fixed-Fixed Beam # A B C D E F G Item Time @ peak #1 Time @ peak #2 Amplitude @ peak #1 Amplitude @ peak #2 Time between A and B Number of periods between A and B Period of oscillation, E/F Unit s s g Value # Item H Damped natural frequency, wd Unit rad/s I Natural frequency, wn rad/s s J zeta K Equivalent mass, meq s L Stiffness, k N/m M Damping, c N/(m/s) N Natural frequency rad/s estimation by Rayleigh method g kg Value

Different material? Repeat the experiment with Steel and any nonmetal material Compare the result