Calculate damping constant, mass, damping ratio, or natural frequency from any 3 values with unit conversions for kg, Hz, rad/s, and N·s/m.
Damping Constant Formula
The damping constant for a single-degree-of-freedom mass-spring-damper system is based on mass, damping ratio, and natural frequency:
c = 2*m*zeta*omega_n
- c = damping constant, in N·s/m
- m = mass, in kg
- zeta = damping ratio, dimensionless
- omega_n = natural frequency, in rad/s
The calculator can also rearrange the same equation to solve for any missing variable.
m = c/(2*zeta*omega_n)
zeta = c/(2*m*omega_n)
omega_n = c/(2*m*zeta)
- Solving for damping constant: enter mass, damping ratio, and natural frequency.
- Solving for mass: enter damping constant, damping ratio, and natural frequency.
- Solving for damping ratio: enter damping constant, mass, and natural frequency.
- Solving for natural frequency: enter damping constant, mass, and damping ratio.
Internally, values are converted to base SI units before calculation: kg for mass, rad/s for natural frequency, and N·s/m for damping constant. If you enter frequency in Hz, it is converted using 1 Hz = 2π rad/s.
Damping Ratio Interpretation
The damping ratio describes how strongly a system is damped compared with critical damping.
| Damping Ratio ζ | Damping Type | Typical Motion |
|---|---|---|
| 0 | Undamped | Oscillates without decay in the ideal model |
| 0 < ζ < 1 | Underdamped | Oscillates while gradually decreasing in amplitude |
| ζ = 1 | Critically damped | Returns to equilibrium as quickly as possible without oscillating |
| ζ > 1 | Overdamped | Returns to equilibrium slowly without oscillating |
Common Unit Conversions
| Quantity | Conversion | Used For |
|---|---|---|
| Mass | 1 g = 0.001 kg | Converting grams to SI mass |
| Mass | 1 lb = 0.453592 kg | Converting pounds to SI mass |
| Natural frequency | 1 Hz = 6.283185 rad/s | Using cycles per second in the damping equation |
| Damping constant | 1 lbf·s/ft = 14.5939 N·s/m | Converting imperial damping coefficient units |
Example Problems
Example 1: Calculate damping constant
You have a mass of 10 kg, a damping ratio of 0.25, and a natural frequency of 8 rad/s.
c = 2*m*zeta*omega_n
c = 2*10*0.25*8 = 40
The damping constant is 40 N·s/m.
Example 2: Calculate damping ratio
You have a damping constant of 60 N·s/m, a mass of 5 kg, and a natural frequency of 12 rad/s.
zeta = c/(2*m*omega_n)
zeta = 60/(2*5*12) = 0.5
The damping ratio is 0.5, so the system is underdamped.
FAQ
What is the damping constant?
The damping constant, also called the damping coefficient, measures the resistive force caused by damping per unit velocity. In a linear viscous damper, the damping force is commonly written as F = c*v, where c is the damping constant and v is velocity.
What units should I use for the damping constant?
The standard SI unit is N·s/m. The calculator also supports lbf·s/ft. If you use mixed units, the values are converted before the equation is applied.
Is natural frequency the same as frequency in Hz?
Not exactly. Natural frequency in this formula is angular natural frequency, measured in rad/s. Frequency in Hz is cycles per second. To convert Hz to rad/s, multiply by 2π.
