Resonance Calculator
Calculate natural frequency, resonance, and response of spring-mass-damper systems. Analyze forced vibration parameters including damping ratio and dynamic amplification.
Natural Frequency
— Hz
Damping Ratio
—
Damped Natural Freq.
— Hz
Resonant Frequency
— Hz
Natural Period
— s
Damped Natural Period
— s
Amplitude Magnification Factor
—
Phase Angle
—°
Critical Damping Coefficient
— N·s/m
Damping Case
—
About Spring-Mass-Damper Systems
A spring-mass-damper system is a mechanical system that exhibits vibration. It consists of three elements:
The spring provides a restoring force proportional to displacement (F = -kx). The spring constant (k) represents the stiffness of the spring, measured in units of force per distance (N/m).
The mass (m) provides inertia to the system. According to Newton's second law, force equals mass times acceleration (F = ma).
The damper (viscous damping) provides a force proportional to velocity (F = -cv), dissipating energy from the system. The damping coefficient (c) is measured in units of force per velocity (N·s/m).
System Classification
Based on Damping Ratio (ζ)
- Undamped (ζ = 0): System oscillates indefinitely without energy loss.
- Underdamped (0 < ζ < 1): System oscillates with decreasing amplitude. Most real-world systems fall into this category.
- Critically Damped (ζ = 1): System returns to equilibrium as quickly as possible without oscillation. Used in applications requiring fast response without overshoot.
- Overdamped (ζ > 1): System returns to equilibrium without oscillation, but more slowly than critically damped.
Equation of Motion
The governing differential equation for a spring-mass-damper system is:
m·ẍ + c·ẋ + k·x = F(t)
Where:
- m = mass
- c = damping coefficient
- k = spring stiffness
- F(t) = forcing function
- x = displacement
- ẋ = velocity (first derivative of x)
- ẍ = acceleration (second derivative of x)
Response Types and Time Domain Solutions
Exhibits decaying oscillations:
x(t) = A·e-ζωn·t·sin(ωd·t + φ)
Where:
- ωd = ωn·√(1-ζ²) = damped natural frequency
- A = amplitude
- φ = phase angle
Period of oscillation increases with damping:
τd = 2π/ωd
Returns to equilibrium without oscillation:
x(t) = (A + Bt)·e-ωn·t
Where:
- A = initial displacement (x₀)
- B = v₀ + x₀·ωn (v₀ = initial velocity)
This response provides the fastest return to equilibrium without oscillation.
Exhibits non-oscillatory decay:
x(t) = A₁·er₁t + A₂·er₂t
Where r₁ and r₂ are negative real roots of the characteristic equation.
When forcing frequency approaches natural frequency:
- Amplitude magnification factor reaches maximum
- Maximum response occurs at resonant frequency: ωr = ωn·√(1-2ζ²) for ζ < 0.707
- At resonance, response lags input by 90°
- For very small damping, resonant frequency ≈ natural frequency
Important Formulas
Natural Frequency
fn = (1/2π) × √(k/m)
Where k is spring stiffness and m is mass
Damping Ratio
ζ = c / (2√(km))
Where c is damping coefficient
Critical Damping Coefficient
cc = 2√(km)
Damping value that produces critical damping (ζ=1)
Damped Natural Frequency
fd = fn × √(1-ζ²)
Only applicable for underdamped systems (ζ < 1)
Amplitude Magnification Factor
AMF = 1 / √[(1-r²)² + (2ζr)²]
Where r is frequency ratio (forcing/natural)
Phase Angle
φ = tan⁻¹(2ζr / (1-r²))
Angle by which response lags behind forcing
Applications
Engineering Applications
- Vibration isolation in machinery
- Automotive suspension systems
- Seismic protection of structures
- Tuned mass dampers in tall buildings
- Electronic oscillators and filters
Design Considerations
- Avoid operating near resonant frequencies
- Add damping to reduce resonance effects
- Use isolation mounts to prevent vibration transmission
- Consider critically damped systems for positioning applications
Related Calculator
For time domain analysis and damping behavior, check our Damping Calculator, which provides detailed insights into system response and transient behavior.