What is the relationship between static deflection and natural frequency?

Under gravity loading, natural frequency can be approximated directly from static deflection (δst in mm) via fn ≈ 15.76 / √(δst) or fn ≈ 5 / √(δst_cm).

How does beam self-mass affect the natural frequency?

Because beam mass is distributed rather than lumped at the center, Rayleigh energy methods apply an effective mass factor: 23.6% of beam mass for cantilevers, 48.6% for simply supported beams, and 37.1% for fixed-fixed beams.

When does vibration isolation actually work?

Vibration isolation only occurs when the frequency ratio r = f_excitation / fn is greater than √2 (approx 1.414). For r < 1.414, dynamic forces transmitted to the foundation are amplified, not isolated.

Structural & Mechanical Dynamics

Natural Frequency & Vibration Calculator

Calculate undamped and damped natural frequencies, oscillation periods, Rayleigh beam distributed mass corrections, resonance danger bands, and dynamic transmissibility.

Dynamic Vibration Configuration

k = 48EI/L³ + Rayleigh beam mass

Structural System Model

Select physical model to compute equivalent stiffness

Support / Geometry Mode

Discrete Lumped Mass (M)

Motor, pump, or applied central mass.

Beam Mechanical Properties

Elastic Modulus (E)

GPa

Steel: 200–210 GPa, Aluminum: 70 GPa.

Area Moment of Inertia (I)

mm⁴

Bending axis second moment of area.

Beam Span Length (L)

Distance between support centerlines.

Beam Self Mass (m_beam)

Rayleigh dynamic mass correction.

Damping Ratio (ζ)

Steel: 0.01–0.03, Rubber: 0.10–0.20.

Excitation Forcing Speed (RPM)

RPM

Operating frequency to verify resonance safety.

Fundamental Natural Frequency (fn)

24.47Hz

Angular Frequency (ωn): 153.72 rad/s (1,468 CPM)

Equivalent k

7,777.8 N/mm

Effective Mass

329.1 kg

Static Deflection (δst)

0.415 mm

Forcing (f_exc)

24.67 Hz

Dynamic Magnification (DMF)

15.95×

Transmissibility (TR)

15.98

CRITICAL RESONANCE HAZARD — Excitation in Modal Band

Forcing frequency (24.67 Hz) sits INSIDE the ±15% resonance band (20.80 Hz – 28.14 Hz).

Comprehensive Modal Response Parameters

Damped Frequency (fd)

24.45 Hz

Critical Damping (c_cr)

101,193 N·s/m

Log Decrement (δ)

0.189

Quality Factor (Q)

16.7

Isolation Efficiency

0.0%

Time Period (Tn)

0.0409 s

Governing Dynamic Formulas

fn = (1 / 2π) · √(k_eq / m_eff) | TR = √[1 + (2ζr)²] / √[(1 - r²)² + (2ζr)²]

Engineering Code

Export calculation logic to Python or MATLAB.

Python

# Natural Frequency & SDOF Vibration Analysis
import math

def calculate_vibration(stiffness_N_m, mass_kg, zeta=0.03, excitation_rpm=1480):
    omega_n = math.sqrt(stiffness_N_m / mass_kg)
    fn_Hz = omega_n / (2 * math.pi)
    period_s = 1.0 / fn_Hz
    static_deflection_mm = (mass_kg * 9.80665 / stiffness_N_m) * 1000
    
    f_exc_Hz = excitation_rpm / 60.0
    r = f_exc_Hz / fn_Hz
    
    # Transmissibility
    denom = math.sqrt((1 - r**2)**2 + (2*zeta*r)**2)
    DMF = 1.0 / denom if denom > 0 else 999
    TR = math.sqrt(1 + (2*zeta*r)**2) / denom if denom > 0 else 999
    
    return {
        "fn_Hz": fn_Hz,
        "omega_n_rad_s": omega_n,
        "period_s": period_s,
        "static_deflection_mm": static_deflection_mm,
        "DMF": DMF,
        "TR": TR
    }

# System Inputs
k_eq = 7777777.777777777 # N/m
m_eff = 329.142 # kg
zeta = 0.03
rpm = 1480

res = calculate_vibration(k_eq, m_eff, zeta, rpm)
print(f"Natural Frequency: {res['fn_Hz']:.2f} Hz ({res['omega_n_rad_s']:.2f} rad/s)")
print(f"Static Deflection: {res['static_deflection_mm']:.3f} mm")
print(f"Dynamic Magnification Factor (DMF): {res['DMF']:.2f}")
print(f"Transmissibility (TR): {res['TR']:.2f}")

MATLAB

% SDOF Vibration & Resonance Script
k = 7777777.777777777;
m = 329.142;
zeta = 0.03;
f_exc = 24.666666666666668;

omega_n = sqrt(k / m);
fn = omega_n / (2*pi);
r = f_exc / fn;

denom = sqrt((1 - r^2)^2 + (2*zeta*r)^2);
DMF = 1 / denom;
TR = sqrt(1 + (2*zeta*r)^2) / denom;

fprintf('Natural Frequency: %.2f Hz\n', fn);
fprintf('Transmissibility: %.2f\n', TR);

Excel Formula

=(1/(2*PI()))*SQRT(k_eq / m_eff)

Example Calculation: Simply Supported Beam Skid

A 300 kg motor sits at the center of a steel beam span (E = 210 GPa, I = 4,500,000 mm⁴, L = 1,800 mm). The beam itself has a distributed mass of 60 kg, with a damping ratio ζ = 0.03:

k = 48 × (210×10⁹) × (4.5×10⁻⁶) / (1.8)³ = 7,777,778 N/m (7,778 N/mm)

m_eff = 300 kg + (0.4857 × 60 kg) = 329.14 kg | Static Deflection δ_st = 0.415 mm

fn = 24.47 Hz (153.74 rad/s) | Resonance Danger Zone: 20.80 Hz – 28.14 Hz

Technical Explanation: Structural Dynamics & Vibration Isolation

Every elastic mechanical structure possesses characteristic natural frequencies where inertia and elastic restoring forces balance. If operating machinery operates at frequencies close to these modes, harmonic forces produce destructive resonance.

Rayleigh Distributed Mass Method

Real beams do not have zero mass. Because sections near supports move with smaller amplitudes than the midspan, their kinetic energy contribution is scaled via Rayleigh shape functions:

Cantilever End Load: m_eff = M_tip + 0.2357 · m_beam (k = 3EI / L³)
Simply Supported Center Load: m_eff = M_center + 0.4857 · m_beam (k = 48EI / L³)
Fixed-Fixed Center Load: m_eff = M_center + 0.3714 · m_beam (k = 192EI / L³)

The Physics of Vibration Isolation (Transmissibility)

Mounting machines on elastic isolators only isolates dynamic forces when the frequency ratio r = f / fn exceeds √2 (1.414). Below r = 1.414, the spring actually amplifies transmitted dynamic force into the floor structure. At r = 1.0 (resonance), transmitted amplitude reaches the Q-factor peak: Q = 1 / (2ζ).

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Real-World Engineering Cases

Resonance Disaster in Chilled Water Pump Skids

A hospital installed four 1,480 RPM (24.6 Hz) variable-speed secondary pumps on steel channel skids. The structural natural frequency of the skid was 25.1 Hz. When the VFD swept through 98% speed, massive resonant vibration sheared mounting bolts and cracked the pump suction flanges.

Engineering Lesson

Always design base structures so their fundamental modal frequency sits at least ±20% away from any steady or transient operating speed of variable frequency drives.

Faulty Isolator Selection Causing Force Amplification

A reciprocating compressor operating at 12 Hz was mounted on stiff rubber pads designed with a natural frequency of 10.5 Hz (r = 1.14). Rather than isolating vibration, the isolator operated directly on the dynamic magnification slope, transmitting 350% of the dynamic shaker force into building columns.

Engineering Lesson

Effective isolation requires soft mounts where fn < f_operating / 1.414. For a 12 Hz source, target a mounting natural frequency under 4 to 5 Hz.

Frequently Asked Questions

What is the quick static deflection formula for natural frequency?

For systems where mass directly produces the static deflection under gravity, fn ≈ 15.76 / √(δ_st in mm) or fn ≈ 5 / √(δ_st in cm). A 1 mm static deflection corresponds to approx 15.8 Hz natural frequency.

What does the Quality Factor (Q) indicate?

Quality Factor Q = 1 / (2ζ) measures resonance peak sharpness. A lightly damped steel frame with ζ = 0.02 has Q = 25, meaning dynamic resonant stresses are 25 times higher than static equivalent loads.

How do multi-degree of freedom systems differ?

Continuous beams and multi-mass systems possess multiple natural frequencies (Mode 1, Mode 2, Mode 3...). This tool computes the dominant fundamental (lowest) mode, which typically carries 80-90% of dynamic vibration energy.