Natural Frequency Calculator

Related Calculators

Natural frequency: The rate a system tends to vibrate at after a small disturbance. In this calculator it is shown as Natural frequency in Hz.

Angular natural frequency: The same vibration rate written in radians per second instead of cycles per second. It is shown as Angular natural frequency.

Period: The time for one full vibration cycle. It is shown as Period and equals 1 divided by frequency.

Stiffness: How strongly a system resists being stretched or bent. In the mass-spring case, stiffness is the Spring constant (N/m).

Mass per unit length: How much mass the beam has per meter of length. It appears as Mass per unit length (kg/m) or is computed from density and area.

Area moment of inertia: A cross-section property that shows how strongly the beam shape resists bending about the chosen axis. It is entered as Area moment of inertia (m^4).

Mode: A vibration shape pattern. Mode number 1 is the lowest and most common natural frequency, while higher modes vibrate faster.

Mistake: Leaving System type on Mass-spring while entering beam values.
Fix: Switch System type to Cantilever beam or Simply supported beam before using beam inputs.

Mistake: Entering total beam mass into Mass per unit length (kg/m) .
Fix: Enter mass per meter in Mass per unit length (kg/m) , or change Beam mass input method to density-area and use Density (kg/m^3) plus Cross-sectional area (m^2) .

Mistake: Using a section property with the wrong units in Area moment of inertia (m^4) .
Fix: Convert your value to meters to the fourth power before calculating. A value in mm^4 must be converted to m^4 first.

Mistake: Typing 200 instead of 200000000000 in Young's modulus (Pa) for steel.
Fix: Enter Young's modulus (Pa) in pascals, not GPa. If your data is in GPa, convert it to Pa first.

Mistake: Forgetting that Mode number changes the beam result.
Fix: Use Mode number 1 for the fundamental frequency unless your class or problem specifically asks for mode 2 or 3.

Mistake: Reading Angular natural frequency as if it were the same as Natural frequency .
Fix: Use Natural frequency for cycles per second in Hz, and use Angular natural frequency for rad/s in physics equations.

• Results are for undamped natural frequency only. The calculator does not include damping, forcing, resonance response, or safety checks.
• Beam formulas assume a uniform Euler-Bernoulli beam, meaning constant material and cross section along the length with small deflection behavior.
• Beam results can be less accurate for thick beams, strong shear deformation, rotary inertia effects, large attached tip masses, axial loads, or non-uniform geometry.
• The beam formulas use the selected Mode number only for modes 1, 2, and 3.
• Inputs must be positive and in the stated units. Wrong units, especially for Young's modulus (Pa) and Area moment of inertia (m^4), can change results by a very large amount.
• For beam cases, only the visible mass method is used. Hidden beam mass fields are ignored so they do not affect the result.
• The calculator reports ideal mathematical values. Real structures may differ because of supports, joints, added hardware, cracks, temperature, and manufacturing tolerances.

Formulas used

For a simple mass-spring system, the calculator uses the standard undamped single-degree-of-freedom relationship [1].

ωn = sqrt(k / m)

fn = ωn / (2π)

T = 1 / fn

Here, k is Spring constant (N/m), m is Mass (kg), ωn is angular natural frequency in rad/s, fn is natural frequency in Hz, and T is period in seconds.

For beam cases, the calculator uses the Euler-Bernoulli beam form for a uniform beam.

omega_n = (beta_n * L)^2 * sqrt(E * I / (mu * L^4))

fn = ωn / (2π)

Here, E is Young's modulus (Pa), I is Area moment of inertia (m^4), μ is beam mass per unit length, and L is Beam length (m). The mode constant depends on support condition and mode.

Cantilever: βnL = 1.875104, 4.694091, 7.854757 for modes 1, 2, 3

Simply supported: βnL = nπ for modes 1, 2, 3

The equation uses the dimensionless mode constant beta_n * L directly; it must not be divided by L again. "Mode constant used" shows that dimensionless value. For the same material, cross section and mass per length, doubling the beam length divides the frequency by four.

Beam mass handling

If you choose direct beam mass entry, the calculator uses Mass per unit length (kg/m) as μ. If you choose density-area input, it first computes beam mass per length with:

μ = ρA

Here, ρ is Density (kg/m^3) and A is Cross-sectional area (m^2). The resulting value is shown as Beam mass per unit length used.

Worked mini-example

Mass-spring example using the default values: Mass (kg) = 10 and Spring constant (N/m) = 1000.

ωn = sqrt(1000 / 10) = sqrt(100) = 10 rad/s

fn = 10 / (2π) = 1.5915 Hz

T = 1 / 1.5915 = 0.6283 s

That means the system would complete about 1.59 cycles each second, and one cycle would take about 0.63 seconds.

Interpretation tips

A higher result means faster vibration. In general, increasing stiffness raises frequency, while increasing mass lowers frequency. For beams, making the beam longer lowers frequency strongly because length appears to the fourth power in the denominator.

Validation and assumptions

The calculator requires all active inputs for the chosen case to be greater than zero and ignores hidden inputs completely. It blocks undefined results such as division by zero or non-numeric entries. Beam results assume uniform Euler-Bernoulli behavior and can differ from real structures when support conditions, geometry, or added masses are not ideal [2].