Single DOF Forced Vibration Response Calculator

Calculator overview

Inputs and outputs

This summary comes from the calculator's published input and output contract.

Inputs

VR Stiffness: About this input

Positive effective linear restoring stiffness on the modeled displacement coordinate.

Unit N/m Default 1283 Range At least 0
VR Mass: About this input

Positive effective modal or physical mass represented by the single degree of freedom.

Unit kg Default 7.3 Range At least 0
VR Viscous Damping Coefficient: About this input

Nonnegative linear viscous coefficient c; zero is explicitly supported away from resonance.

Unit N*s/m Default 18.7 Range At least 0
VR Unit System: About this input

Select SI metric or US customary entry/display units; both paths use the same SI base equations.

Default SI metric Allowed SI metric, US customary
VR Excitation Frequency: About this input

Nonnegative cyclic frequency of the active sinusoidal force or base motion.

Unit Hz Default 2.37 Range At least 0
VR Base Displacement Amplitude Conditional: About this input

Nonnegative peak prescribed base displacement; active only in base-displacement mode.

Unit mm Default 4.63 Range At least 0
VR Force Amplitude Conditional: About this input

Nonnegative peak force applied directly to the mass; active only in harmonic-force mode.

Unit N Default 43.7 Range At least 0
VR Excitation Mode: About this input

Choose direct sinusoidal force on the mass or prescribed sinusoidal support/base displacement.

Default Harmonic force Allowed Harmonic force, Base displacement

Outputs

Static force deflection amplitude: About this output

Force mode: F0/k static displacement reference. Base mode: entered base displacement amplitude.

Unit mm
VR Peak Velocity Amplitude: About this output

Excitation angular frequency multiplied by response displacement amplitude.

Unit m/s
VR Peak Acceleration Amplitude: About this output

Excitation angular frequency squared multiplied by response displacement amplitude.

Unit m/s^2
VR Response Regime: About this output

Classifies the excitation frequency as below, at, or above the undamped natural frequency using a 1e-9 display classification tolerance.

No unit declared
VR Response Phase Lag: About this output

Force mode: displacement lag relative to applied force. Base mode: absolute mass-displacement lag relative to base displacement.

Unit deg
VR Response Displacement Amplitude: About this output

Peak steady-state absolute displacement amplitude of the mass.

Unit mm
VR Critical Damping Coefficient: About this output

Critical viscous coefficient 2*sqrt(k*m) on the entered displacement coordinate.

Unit N*s/m
Dynamic magnification factor: About this output

Force mode: displacement magnification relative to F0/k. Base mode: absolute mass-displacement transmissibility relative to base amplitude.

Unit ratio
Model Status: About this output

Returns actionable NOT VALID or CHECK text. OK means the stated linear steady-state equations evaluated; it is not a vibration qualification, allowable, or safety approval.

No unit declared
VR Natural Frequency: About this output

Undamped cyclic natural frequency sqrt(k/m)/(2*pi).

Unit Hz
VR Frequency Ratio: About this output

Excitation angular frequency divided by undamped natural angular frequency.

Unit ratio
VR Damping Ratio: About this output

Entered viscous damping coefficient divided by the critical damping coefficient.

Unit ratio

What it is

The Single Degree of Freedom Forced Vibration Response Calculator works out the steady-state response of one lumped mass on one linear spring and one linear viscous damper to a steady sinusoidal excitation. It runs in two modes: a harmonic force applied directly to the mass, and a harmonic displacement prescribed at the base.

It reports the undamped natural frequency, the critical damping coefficient, the damping ratio, the ratio of excitation frequency to natural frequency, the dynamic magnification factor or the absolute displacement transmissibility, the steady response amplitude, the phase lag, and the peak velocity and acceleration. It also states whether the excitation sits below, at, or above the natural frequency.

Entry is in SI metric or US customary, and both paths run the same SI base equations, so the same physical system entered either way returns the same natural frequency and damping ratio. Changing the selector reinterprets whatever is already in the fields rather than converting it. Displacement amplitudes are in millimetres or inches, frequency in hertz, phase in degrees, and velocity and acceleration in metres or inches per second and per second squared.

The single degree of freedom is the limitation to keep in front of you. One mode, one concentrated mass, one linear viscous damper, and the steady state only. A real structure has distributed mass and many modes, and the mode this model describes may not be the one that governs. It is a response screen, not a vibration acceptability determination.

Methodology

Purpose and model boundary

This model calculates the linear steady-state sinusoidal response of one mass-spring-viscous-damper degree of freedom. It supports direct harmonic force and prescribed base-displacement excitation and reports absolute mass-displacement response. It is a deterministic single-frequency response model, not a transient, modal, fatigue or vibration-acceptability analysis.

The spreadsheet remains the calculation authority. The page submits the named inputs through the calculation service and displays the workbook's results, comparison chart and status; the response equations are not reimplemented in browser code.

Inputs and units

Input group	Values used by the model
Unit basis	SI metric or US customary; the workbook converts to SI internally and converts displayed results back.
System	Effective mass `m`, stiffness `k`, and viscous damping coefficient `c`.
Excitation	Harmonic force or base displacement, excitation frequency `f`, and the active force amplitude `F0` or base amplitude `Y`.

Mass and stiffness must be positive. Damping, frequency and the active excitation amplitude may be zero. The inactive amplitude is hidden and not used.

Governing relationships

The natural circular frequency, natural frequency, critical damping, damping ratio and frequency ratio are

ωn = √(k/m) and fn = ωn/(2π);
cc = 2√(km) and ζ = c/cc;
ω = 2πf and r = ω/ωn.

The common response denominator is

D = √((1 − r²)² + (2ζr)²).

For harmonic force, reference displacement is F0/k, amplitude ratio is 1/D, and phase lag follows the complex force-response denominator. For base displacement, reference displacement is Y, absolute displacement transmissibility is √(1 + (2ζr)²)/D, and phase follows the corresponding absolute-motion transfer function. Response amplitude is reference amplitude times the active ratio. Peak velocity and acceleration are ωX and ω²X.

Calculation sequence

Validate units and excitation mode, positive mass and stiffness, nonnegative damping and frequency, and nonnegative active excitation amplitude.
Convert the selected units to the SI calculation basis.
Calculate natural frequency, critical damping, damping ratio and frequency ratio.
Evaluate the mode-specific numerator, common denominator, amplitude ratio and reference displacement.
Calculate absolute displacement, phase lag, peak velocity and peak acceleration.
Reject the exact undamped-resonance singularity, verify all public and chart values are finite and consistent, and return the ordered status.

Outputs and interpretation

Primary outputs are response displacement amplitude, amplitude ratio, natural frequency, frequency ratio and damping ratio. Details expose reference displacement, critical damping, phase lag, peak velocity, peak acceleration and the below/at/above-natural-frequency regime. The chart compares the active excitation reference amplitude with absolute steady response.

In base-excitation mode, the response is absolute mass motion, not relative suspension deflection or transmitted force. Those quantities require different equations.

Validation and status logic

The workbook applies these conditions in order:

Condition	Returned status
A listed mode is not selected, or active mass, stiffness, damping, frequency, force or base-displacement input is outside its domain	`NOT VALID: choose listed modes and correct active mass, stiffness, damping, frequency, force, or base-displacement inputs`
Damping is zero and `\|r − 1\| ≤ 10⁻¹²`	`NOT VALID: undamped resonance has unbounded steady-state response`
A required converted, response or chart value is not finite or fails the protected numeric relationships	`NOT VALID: derived result exceeds the supported calculation range`
Damping is zero away from the singular point	`CHECK: undamped model; steady-state response is singular at resonance`
The active reference excitation amplitude is zero	`CHECK: zero excitation amplitude gives zero steady response`
None of the preceding conditions applies	`OK`

The undamped warning takes precedence over the zero-excitation warning away from resonance. At natural frequency the regime label uses |r − 1| ≤ 10⁻⁹, while the non-computing undamped singularity uses the tighter 10⁻¹² tolerance.

Assumptions and limitations

The system is linear, time-invariant and represented by one concentrated mass, one linear spring and one linear viscous damper.
Only steady-state sinusoidal response is reported; free transients and initial conditions have decayed.
System properties are effective user inputs and are not inferred from geometry or a material/component database.
Shock, random vibration, response spectra, fatigue, modal superposition and multiple degrees of freedom are outside scope.
Nonlinear stiffness, friction, hysteretic or frequency-dependent damping, backlash, clearance, impact and amplitude-dependent properties are not modeled.
Near resonance, small uncertainty in m, k, c or f can dominate response. No proprietary severity zone, machine class, isolation criterion or allowable is embedded.

Restrictions and non-computing states

The exact undamped-resonance state has an unbounded steady response and is deliberately non-computing. Zero damping away from resonance and zero excitation amplitude are computing boundary states with CHECK messages. Any NOT VALID state supersedes displayed residue. A finite response is not a qualification or safety approval.

Errors and warnings

A rejected entry means the submitted values did not satisfy the published input rules. NOT VALID means the workbook refused a singular, unsupported or malformed state. CHECK identifies a mathematically defined boundary needing interpretation. Calculation-service failures are availability errors and must not be confused with zero response.

References

No isolator catalogue, licensed damping dataset, vibration severity table or machine class is embedded. Mass, stiffness and the viscous damping coefficient are entered by you, and the shipped values are synthetic demonstration numbers describing no real machine, mount or structure. The workbook implements the ordinary textbook form of the linear single-degree-of-freedom force and base excitation relations, independently coded, rather than any one publication's derivation. The sources cited for those relations and the unit identities are below.

NASA. Frequency Response of Single Degree-of-Freedom Systems, for the force-excited equation of motion, critical damping, damping ratio, natural frequency and dynamic magnification. https://ntrs.nasa.gov/api/citations/20190033330/downloads/20190033330.pdf
NASA. Vibration and Shock Isolation, for the distinction between applied force and prescribed base motion, transmissibility, phase and the undamped resonance singularity. https://ntrs.nasa.gov/api/citations/19810017924/downloads/19810017924.pdf
NIST. Guide to the SI, Appendix B: Conversion Factors, for the exact inch, standard gravity and pound-force identities on the US path. https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors
NIST. Handbook 133 (2014), Appendix E, for the exact avoirdupois pound identity used by the mass conversion. https://www.nist.gov/document/14-app-e-14-hb133-finalpdf

Citing these sources implies no endorsement by the agencies named. Allowable vibration levels, machine severity classes, isolator selection data, modal survey results and fatigue assessment are not supplied here. Take them from the governing standard, the equipment manufacturer, and measurement on the actual installation.

Additional source notes migrated from Methodology

Project-specific vibration criteria, validated system properties and qualified dynamics review govern real equipment or structural decisions.

Frequently asked questions

My excitation is above the natural frequency, so why is the motion getting worse?

Because isolation does not begin above the natural frequency. It begins above the square root of two times it, a frequency ratio of about 1.414, where the transmissibility is exactly 1 for every damping ratio. Between 1 and 1.414 the mounted mass still moves further than its support. The shipped default sits at 1.1232 and returns a transmissibility of 3.0098 in base mode, so the mass moves three times as far as the base. If you are choosing mounts, work out where they put the frequency ratio first.

Does adding damping always reduce the motion?

Not in base displacement mode. In harmonic force mode more damping always reduces the response. In base mode the damping term appears in the numerator of the transmissibility as well as the denominator, and above a frequency ratio of 1.414 the numerator wins, so more damping transmits more motion. A mount well isolated in steady running wants light damping, while surviving run-up through resonance wants heavy damping. Those requirements conflict, and this calculator does not resolve them.

What happens if I enter zero damping?

Away from resonance it computes normally, with a CHECK status reminding you that the undamped solution is singular at resonance. On the shipped defaults, taking the damping to zero raises the response from 100.184 to 130.157 millimetres and puts the phase at exactly 180 degrees. At a frequency ratio within 1e-12 of one the amplitude is unbounded, and the model returns NOT VALID with every output at zero rather than print an enormous number. That is a refusal, not a failure.

How much can I trust the amplitude near resonance?

Less than the digits suggest. At a frequency ratio of one the magnification is one divided by twice the damping ratio, so it doubles whenever your damping estimate halves: about 5.18 at the shipped damping ratio of 0.0966, about 10.4 at half of it, and nothing at all at zero. Damping is also the property you are least likely to know, rarely measured and usually assumed. Near resonance the uncertainty in the answer is essentially the uncertainty in the damping.

Is the reported amplitude the absolute motion or the motion across the mount?

In base displacement mode it is the absolute displacement of the mass, and the ratio reported is absolute transmissibility. Relative motion between mass and base is a different quantity and is not on this page. If you are sizing a clearance, a flexible connection or a snubber, relative motion is what you need. In force mode the question does not arise.

Why did every output come back as zero?

Something was rejected. A zero mass or a zero stiffness returns NOT VALID and zeroes the whole output block, as does any input whose intermediate products fall outside the supported floating-point range, including a positive value too small to carry. This is not what zero damping, zero frequency or zero excitation amplitude do: those are accepted and answered. Read the status line first, and again when the numbers look reasonable: CHECK states return numbers too.

Can I use this for rotating unbalance?

Only with care. Unbalance produces a force that grows with the square of running speed, while this calculator holds the force amplitude constant as you change the excitation frequency. Sweep the frequency to imitate a run-up and the curve comes out the wrong shape, because the real force would be rising throughout. Re-enter the force at each speed; the model will not do it for you.

When is one degree of freedom not enough?

Often. It is a reasonable model when one mode dominates and the excitation is near it, which is the usual case for a rigid machine on soft mounts. It is a poor model for a structure with several modes close together, for a distributed system such as a pipe run or a slender beam where mass and stiffness are spread along its length, for anything sweeping across more than one resonance, and wherever the mode shape decides which part moves. The calculator cannot tell you which situation you are in, and that judgement has to be made before you use the number.

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This page is provided by LogicCommons for informational purposes only. Results are analysis outputs computed from the inputs you supply and are not engineering advice, a design, or a substitute for review by a licensed professional under the codes adopted where the work is built. Verify all inputs and results independently.

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