Engineering
Vibration Damping Ratio (zeta), Damped Frequency & Logarithmic Decrement
Classify mass-spring-damper dynamics into underdamped, critically damped, and overdamped regimes via zeta = c / (2 * SQRT(k * m)) and compute omega_d and logarithmic decrement.
Free vibration of an ideal linear mass-spring-damper
Assume one degree of freedom, constant positive mass and spring stiffness, nonnegative linear viscous damping and no external forcing. These expressions do not model dry friction, nonlinear springs or a complete vehicle suspension.
The damping ratio is dimensionless; natural and damped angular frequencies are in rad/s. Divide angular frequency by 2*pi for Hz. For 0<zeta<1 the motion oscillates with a decaying envelope; zeta=0 is the separate undamped case.
Critical and overdamped solutions are non-oscillatory, but an arbitrary initial velocity can still cause an equilibrium crossing. Critical damping is not a universal optimum for every design or initial condition. The real underdamped-frequency and successive-peak decrement formulas are not applicable when zeta>=1.
| Mechanical Vibration Metric | Formula | m=25 kg, k=3,600 N/m, c=120 N*s/m | c = 600 / c = 750 N*s/m |
|---|---|---|---|
| Critical Damping c_c | 2 * SQRT(B3 * B2) | 600 N*s/m | 600 N*s/m |
| Damping Ratio zeta | B4 / (2 * SQRT(B3 * B2)) | 0.20 (Underdamped) | 1.00 (Critical) / 1.25 (Over) |
| Damped Freq omega_d (rad/s) | SQRT(B3/B2) * SQRT(1 - zeta^2) | 11.7576 rad/s (vs 12.00 omega_n) | No oscillatory frequency |
| Logarithmic Decrement delta | 2*PI()*zeta / SQRT(1 - zeta^2) | 1.2825 | N/A for zeta >= 1 |
Dimensionless Damping Ratio zeta = c / (2 * SQRT(k * m))
Compares actual dashpot coefficient B4 against critical damping 2 * SQRT(k * m).
=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0),"Invalid inputs",ROUND(B4 / (2 * SQRT(B3 * B2)), 4)))Damped angular frequency in the oscillatory regime
Compare the unrounded damping ratio with one. Return a descriptive non-oscillatory result at and above critical damping instead of presenting zero as an oscillation frequency.
=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0),"Invalid inputs",IF((B4/(2*SQRT(B3*B2)))>=1,"Non-oscillatory",ROUND(SQRT(B3/B2)*SQRT(1-(B4/(2*SQRT(B3*B2)))^2),4))))Logarithmic decrement for same-sign peaks
The decrement is the natural logarithm of the ratio of successive same-sign peak amplitudes one full cycle apart in a free decay. It is dimensionless. Zero damping gives zero decrement. Critical and overdamped motion has no such cycle.
=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0),"Invalid inputs",IF((B4/(2*SQRT(B3*B2)))>=1,"Not applicable",ROUND(2*PI()*(B4/(2*SQRT(B3*B2)))/SQRT(1-(B4/(2*SQRT(B3*B2)))^2),4))))Classify the unrounded damping ratio
The exact comparison is useful for these constructed inputs. A displayed ratio rounded to 1.0000 can still be slightly below or above one; measured coefficients have uncertainty and do not establish exact critical damping.
=IF(COUNT(B2:B4)<>3,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<0),"Invalid inputs",IF(B4=0,"Undamped",IFS((B4/(2*SQRT(B3*B2)))<1,"Underdamped",(B4/(2*SQRT(B3*B2)))=1,"Critically Damped",TRUE,"Overdamped"))))Try the example
Use the original values or change them to test your understanding.