July 2025 8 min read
Resonance Calculator
Determine whether your mechanical system operates near resonance by comparing the forcing frequency to the natural frequency. Essential for vibration isolation design and avoiding damaging vibrations.
Enter Your Values
Mass
kg
Stiffness
N/m
Forcing Frequency
Hz
Resonance Formula
The frequency ratio determines whether a system is in resonance (r ≈ 1), below resonance (r < 1), or above resonance (r > 1). The natural frequency is first calculated from mass and stiffness, then compared to the forcing frequency.
r = f / fₙ = ω / ωₙ
Alternative Form
fₙ = (1 / 2π) × √(k / m)
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| r | Frequency Ratio | - | Ratio of forcing frequency to natural frequency (ω/ωₙ) |
| f | Forcing Frequency | Hz | Frequency of the external excitation |
| fₙ | Natural Frequency | Hz | Undamped natural frequency of the system |
| k | Stiffness | N/m | Spring stiffness of the system |
| m | Mass | kg | Mass of the vibrating body |
Explanation
When the frequency ratio r = 1, the system is at resonance and experiences maximum vibration amplitude. The amplification factor at resonance depends on damping: D_max = 1/(2ζ). For effective vibration isolation, design for r > √2 (r > 1.414), where transmissibility drops below 1.
Derivation
The natural frequency of a spring-mass system is fₙ = (1/2π)√(k/m). The frequency ratio r = f/fₙ. At r = 1, the system is in resonance. The amplification factor D = 1/√((1−r²)²+(2ζr)²) peaks at r = √(1−2ζ²) with maximum value D_max = 1/(2ζ√(1−ζ²)).
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Step 1
Calculate natural frequency from mass and stiffness.
fₙ = (1/2π) × √(k/m)
Mass:100 kg
Stiffness:1000 N/m
Intermediate: (1/2π) × √(1000/100)
Result: 0.5033 Hz
Step 2
Compute frequency ratio to determine resonance proximity.
r = f / fₙ
Forcing Frequency:0.5 Hz
Natural Frequency:0.5033 Hz
Intermediate: 0.5 / 0.5033
Result: r = 0.993 — NEAR RESONANCE (r ≈ 1)
Step 3
Assess resonance condition and amplification.
D ≈ 1/(2ζ) at resonance
r:0.993
Result: Resonance condition detected — amplification factor D > 10
Final Answer
r = 0.00 — Safe
Worked Examples
See how the calculator works with different scenarios.
Scenario
A fan operating at 1800 RPM (30 Hz) mounted on a 200 kg base with stiffness 500000 N/m.
Inputs
Mass:200 kg
Stiffness:500000 N/m
Forcing Freq:30 Hz
Calculation
fₙ = (1/2π) × √(500000/200) = 7.96 Hz, r = 30/7.96 = 3.77
Result
fₙ = 7.96 Hz, r = 3.77 — No resonance
Interpretation
The operating speed (30 Hz) is well above the natural frequency (7.96 Hz). No resonance occurs during normal operation, but the system passes through resonance during startup and coast-down.
What Is Resonance?
Resonance is the condition where an external periodic force applied to a mechanical system matches one of the system's natural frequencies. The resulting vibration amplitudes can be many times larger than the static deflection — often leading to noise, accelerated wear, fatigue failure, or catastrophic structural collapse.
Resonance Behavior
Resonance is characterized by: (1) Peak amplitude near the natural frequency. (2) A 90° phase shift at resonance. (3) Amplification inversely proportional to damping. (4) Width of resonance peak proportional to damping (half-power bandwidth). The chart above shows how different damping ratios affect the amplification near resonance.
Applications
Resonance analysis is used in: machine design (avoiding resonance), vibration isolation (operating in r > √2 region), structural dynamics (earthquake response), automotive NVH (noise path analysis), aerospace (flutter prevention), and energy harvesting (intentional resonance).
Best Practices
- Maintain at least 20% separation margin (ISO 10816 recommendation)
- Design for r > √2 when vibration isolation is the goal
- Always consider startup and coast-down resonance passage
- Add damping when resonance passage is unavoidable
- Use tuned mass dampers for problematic resonances that cannot be shifted
Common Misconceptions
- "Resonance always causes failure" — Many systems operate safely at resonance with proper damping and stress analysis. Controlled resonance powers ultrasonic welders, compactors, and conveyors.
- "Only the fundamental mode matters" — Higher harmonics (2×, 3×, etc.) can also excite natural frequencies. Check all excitation frequencies.
Practical Tips
- •Design for r > √2 (r > 1.414) to achieve effective vibration isolation (transmissibility < 1). For r = 3, only 12.5% of force is transmitted.
- •Always check both startup and coast-down conditions — rotating machinery must pass through resonance when changing speed, even if normal operation is safe.
- •Use damping to limit amplification at resonance. Even ζ = 0.1 reduces peak amplification from infinite to 5×. Add dampers if resonance passage is unavoidable.
- •Consider the entire system stiffness — the foundation, mounting bolts, and adjacent structures all contribute. The softest element in the load path dominates.
- •For variable-speed machinery, perform a Campbell diagram analysis to ensure no operating speed coincides with any natural frequency.
- •Avoid resonance in piping systems by changing support spacing or adding braces to shift natural frequencies away from pump or compressor operating speeds.
Common Mistakes
Checking resonance at operating speed only, ignoring startup and coast-down.
Rotating machinery passes through all frequencies from 0 to operating speed during startup. Calculate critical speeds within the entire operating range, not just at nominal speed.
Assuming stiffness is constant across all load conditions.
Stiffness may change with preload, temperature, and amplitude. Rubber mounts stiffen under compression; gas springs have nonlinear stiffness. Verify stiffness at actual operating conditions.
Confusing frequency ratio r with amplification factor D.
r = ω/ωₙ is the frequency ratio input. D = 1/√((1−r²)² + (2ζr)²) is the amplification output. At r = 1 (resonance), D ≈ 1/(2ζ). They are related but distinct quantities.
Neglecting damping when assessing resonance severity.
Without damping, amplification at resonance is theoretically infinite. Real systems always have some damping. Calculate expected amplification using the damping ratio to assess actual risk.
Assumptions
This calculator makes the following assumptions to provide estimates:
Linear System: Assumes linear stiffness and damping behavior. Large amplitudes at resonance may excite nonlinearities not captured by the model.
Single-DOF Approximation: Reduces the system to a single degree of freedom. Real structures have multiple modes that may interact near resonance.
Steady-State Condition: Resonance assessment assumes steady-state harmonic excitation. Transient events and shock loads require different analysis methods.
Negligible Cross-Coupling: Assumes vibration in one direction does not excite modes in other directions. Real systems may have coupled modes.
Limitations
Be aware of these limitations when using this calculator:
Requires Damping Data: The amplification factor at resonance cannot be calculated without knowing the damping ratio. Use the Damping Ratio Calculator if ζ is unknown.
Narrow Focus: This calculator assesses resonance from a single forcing frequency. Multiple excitation sources may produce complex interactions.
Installation-Dependent: Actual resonance behavior depends on installation conditions, adjacent equipment, and foundation characteristics not captured by the model.
Related Calculators
Educational Guides
Glossary
Technical terms used in mortgage calculations explained in plain language.
- Resonance
- The condition where an external forcing frequency coincides with a natural frequency of the system, causing large amplitude oscillations.
- Frequency Ratio (r)
- The ratio of forcing frequency to natural frequency (r = ω/ωₙ). r = 1 indicates perfect resonance; r > √2 indicates vibration isolation.
- Amplification Factor (D)
- The ratio of dynamic response amplitude to static deflection under the same force. Maximum at resonance, decreasing as r moves away from 1.
- Transmissibility
- The fraction of vibration force transmitted from the machine to the foundation. TR < 1 indicates isolation; TR > 1 indicates amplification.
- Critical Speed
- A rotational speed at which a natural frequency is excited, causing large vibrations. Operating at critical speeds must be avoided or quickly traversed.
Authoritative References
Frequently Asked Questions
Answers to common questions about this calculator.
What is resonance in mechanical systems?
Resonance occurs when an external forcing frequency matches or is close to a natural frequency of the system. The vibration amplitude is amplified significantly — potentially 10-50× the static deflection — leading to excessive noise, accelerated wear, and structural failure.
How is resonance detected?
Resonance is detected by measuring vibration amplitude while varying the excitation frequency (sweep test). A sharp peak in the frequency response function (FRF) indicates a resonance. Sudden increase in vibration at a specific speed is a practical indicator.
What is the frequency ratio r?
The frequency ratio r = f/fₙ = ω/ωₙ compares the forcing frequency to the natural frequency. r < 1: below resonance (stiffness-dominated). r = 1: resonance. r > 1: above resonance (mass-dominated). r > √2: isolation region.
What happens at resonance?
At resonance, the system vibrates at maximum amplitude. The amplification factor D ≈ 1/(2ζ) determines the peak amplitude. A damping ratio of ζ = 0.05 gives D ≈ 10 — ten times the static deflection.
How do I avoid resonance in machine design?
Ensure operating speeds are at least 20% away from any natural frequency (30% for critical equipment). Use Campbell diagrams to visualize frequency separation. Add mass or stiffness to shift natural frequencies, or add damping to limit amplification.
What is the difference between resonance and natural frequency?
Natural frequency is a property of the system — it exists whether or not the system is being excited. Resonance is a condition that occurs when the excitation frequency equals a natural frequency.
Can resonance be beneficial?
Yes, resonance is intentionally used in: vibrating conveyors, compactors, vibratory feeders, ultrasonic cleaners, musical instruments, and tuned mass dampers. The key is controlled, purposeful resonance versus uncontrolled, damaging resonance.
What is the phase change at resonance?
At resonance, the phase angle between the forcing function and the response is 90°. Below resonance, response is in phase with force (0°). Above resonance, response lags by 180°. This phase shift is a key indicator of resonance.
How does damping reduce resonance effects?
Damping dissipates vibrational energy, limiting the peak amplitude at resonance. The peak amplification is approximately 1/(2ζ). Increasing ζ from 0.01 to 0.1 reduces amplification from 50× to 5×.
What is a Campbell diagram?
A Campbell diagram plots natural frequencies and harmonic excitation frequencies (1×, 2×, 3× of rotational speed) on the same axes. Intersections indicate resonance conditions at specific RPM. Essential for rotating machinery design.
Why does startup cause resonance?
During startup, a machine accelerates from 0 to operating speed, passing through every frequency in between. If any natural frequency lies below operating speed, resonance occurs at that RPM during startup. This is called critical speed passage.
What is the isolation region (r > √2)?
When the frequency ratio r exceeds √2 ≈ 1.414, the transmissibility drops below 1, meaning the vibration transmitted to the foundation is less than the input. This is the operating region for vibration isolators.
How do I calculate the required natural frequency for isolation?
For a desired transmissibility TR, the required natural frequency is fₙ = f / √(1 + 1/TR). For 90% isolation (TR = 0.1), fₙ = f / √(11) ≈ f/3.32.
What is a mode shape at resonance?
The mode shape describes how the system deforms at a resonant frequency. Each natural frequency has an associated mode shape. At resonance, the system vibrates in that specific deformation pattern with amplified amplitude.
Can resonance cause structural failure?
Yes. Notable examples: Tacoma Narrows Bridge (1940) collapsed due to wind-induced resonance. Flutter in aircraft wings. Resonance continues to cause failures in machinery, piping, and structures when not properly addressed.
What is self-excited vibration?
Self-excited vibration (chatter, flutter, squeal) occurs when the vibration itself modulates the exciting force, sustaining oscillation at a natural frequency. Unlike forced resonance, it can occur without external periodic excitation.
How do accelerometers detect resonance?
Accelerometers measure vibration amplitude across a frequency range. At resonance, the amplitude shows a distinct peak. Modern FFT analyzers display the frequency spectrum, making resonance identification straightforward.
What is the half-power bandwidth method?
The half-power bandwidth measures the width of the resonance peak at 1/√2 (70.7%) of the peak amplitude. The damping ratio ζ = (f₂ − f₁) / (2fₙ) where f₁ and f₂ are the half-power frequencies.
How does added damping affect resonance frequency?
Damping slightly reduces the resonant frequency. The damped natural frequency is ω_d = ωₙ√(1−ζ²). For ζ < 0.2, the shift is less than 2%. For ζ = 0.5, the shift is about 13%.
What is the difference between series and parallel resonance?
In mechanical systems, series resonance occurs in compliant elements (low impedance path to ground). Parallel resonance (antiresonance) occurs in stiff elements and produces a notch in the FRF where motion is minimal.
How do I identify resonance in operating machinery?
Signs include: sudden increase in vibration amplitude with small speed change, 90° phase shift, dominant frequency equal to a natural frequency, and elliptical or whirling orbits in proximity probe measurements.
What is a resonance test (sweep test)?
A resonance test slowly varies the excitation frequency while measuring response. The frequency response function (FRF) reveals natural frequencies as peaks. This can be done with impact hammers or shakers.
Can resonance occur in fluid systems?
Yes. Acoustic resonance in pipes, water hammer, and pump pulsation can excite fluid column natural frequencies. This causes pressure fluctuations, noise, and pipe vibration.
What is torsional resonance?
Torsional resonance occurs when the torsional natural frequency (of shafts, gear trains) coincides with torque pulses from engines, pumps, or compressors. Causes gear wear, coupling fatigue, and shaft failures.
How does temperature affect resonance conditions?
Temperature changes material properties (E decreases with temperature), alters clearances in bearings, and changes fluid viscosity. This shifts natural frequencies, potentially creating or removing resonance conditions.
What is a tuned mass damper (TMD)?
A TMD is a secondary mass-spring system tuned to the same frequency as the primary system. At resonance, the TMD vibrates out of phase, canceling the primary system motion. Used in skyscrapers, bridges, and machinery.
How do I calculate the margin between operating speed and natural frequency?
Separation margin = |f_op − fₙ| / fₙ × 100%. ISO 10816 recommends > 20% for general machinery, > 50% for critical equipment. This applies to all harmonics (1×, 2×, 3×, etc.) of operating speed.
What is the effect of wear on resonance?
Wear in bearings, clearances, and joints changes system stiffness and damping over time. This shifts natural frequencies and alters resonance behavior. Regular vibration monitoring detects these changes.
Can resonance occur at sub-harmonic frequencies?
Yes, sub-harmonic resonance (at fₙ/2, fₙ/3, etc.) can occur in nonlinear systems. This is common in loose or cracked structures, impacting machinery, and systems with clearance nonlinearities.
What is the LAL (Lord Aerospace Limited) method for resonance?
The LAL method uses multiple shakers to excite a structure and identify modal parameters. It is an advanced experimental modal analysis technique for complex aerospace and automotive structures.
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Page Information
Last Updated:
July 2025
Version:
2.0.0
References:
License:
Free to use. Results for educational purposes only.