Spring Mass System Input Form
Example Data Table
| Case | Mass kg | Stiffness N/m | Damping N·s/m | Force N | Forcing Hz | Use Case |
|---|---|---|---|---|---|---|
| Light mechanism | 5 | 1200 | 18 | 8 | 1.5 | Small moving part |
| Machine mount | 25 | 8500 | 140 | 65 | 3.0 | Equipment isolation |
| Vehicle suspension | 320 | 18000 | 1600 | 350 | 1.2 | Ride response study |
| Precision table | 80 | 45000 | 620 | 40 | 4.0 | Vibration control |
Formula Used
Natural angular frequency: ωn = √(k / m)
Natural frequency: fn = ωn / 2π
Critical damping: cc = 2√(km)
Damping ratio: ζ = c / cc
Damped angular frequency: ωd = ωn√(1 - ζ²), for underdamped systems.
Underdamped free response: x(t)=e^(-ζωnt)[x₀cos(ωdt)+((v₀+ζωnx₀)/ωd)sin(ωdt)]
Forced amplitude: X = (F₀/k) / √((1-r²)² + (2ζr)²)
Frequency ratio: r = ω / ωn
Phase angle: φ = atan2(2ζr, 1-r²)
Transmissibility: T = √(1+(2ζr)²) / √((1-r²)²+(2ζr)²)
How to Use This Calculator
- Enter the mass in kilograms.
- Enter spring stiffness in newtons per meter.
- Enter damping coefficient in newton seconds per meter.
- Add starting displacement and starting velocity.
- Enter force amplitude and forcing frequency for forced vibration.
- Set graph duration and sample count.
- Click the calculate button.
- Review results above the form.
- Download CSV or PDF for reporting.
Mechanical Vibrations Spring Mass System Guide
Why This Model Matters
A spring mass system is a basic model for mechanical vibration. It helps describe machines, vehicles, tools, floors, and parts that move after a disturbance. The model uses mass, spring stiffness, damping, starting displacement, and starting velocity. These values describe how fast the body moves and how quickly the motion decays.
Frequency and Damping
Natural frequency is the main property. It shows the rate at which the system wants to vibrate without outside forcing. A larger stiffness raises this frequency. A larger mass lowers it. Damping reduces the motion and changes the visible response. Low damping allows oscillation for a long time. High damping can stop oscillation and return the system slowly.
Forced Vibration and Resonance
This calculator also checks forced vibration. A periodic force can create large motion near resonance. Resonance happens when the forcing frequency is close to natural frequency. Damping can limit the peak, but it cannot remove the need for safe design. Designers often compare frequency ratio, phase angle, transmissibility, and steady amplitude before choosing supports or isolators.
Reports and Graphs
The result cards help users review the most important values quickly. The graph shows free motion, forced motion, and an estimated combined response. The table gives sample points, so the trend is easy to inspect. CSV export is useful for spreadsheets. PDF export is useful for reports and project records.
Input Accuracy
Use consistent units for every field. Enter mass in kilograms, stiffness in newtons per meter, damping in newton seconds per meter, displacement in meters, velocity in meters per second, and force in newtons. Start with measured data when possible. If the data is estimated, test several values. Small changes in damping or stiffness can change the response near resonance.
Engineering Limits
The calculator is for study, planning, and early design checks. It does not replace laboratory testing or code based engineering review. Real machines can include nonlinear springs, friction, looseness, impacts, and changing loads. Still, the model is valuable. It gives a clear first view of vibration behavior, resonance risk, and damping needs.
Compare Several Cases
For best results, compare several load cases. Study the unloaded case, normal operating case, and worst expected case. This reveals whether motion remains controlled across the full working range and expected service conditions safely.
Frequently Asked Questions
What is a spring mass system?
It is a simple vibration model with a mass attached to a spring. A damper and external force can also be added. The model predicts motion, frequency, damping behavior, and possible resonance.
What is natural frequency?
Natural frequency is the rate at which the system vibrates freely after a disturbance. It depends mainly on spring stiffness and mass. Higher stiffness raises it, while higher mass lowers it.
What does damping ratio mean?
Damping ratio compares actual damping with critical damping. It shows whether the system is underdamped, critically damped, or overdamped. This affects oscillation size and decay speed.
When does resonance occur?
Resonance occurs when forcing frequency is close to natural frequency. Motion can become large in this region. Damping reduces the peak, but safe design checks are still needed.
What is critical damping?
Critical damping is the damping level that returns the system to equilibrium quickly without oscillation. It is calculated from mass and stiffness using the critical damping formula.
Can this calculator handle forced vibration?
Yes. It estimates steady forced amplitude, phase angle, frequency ratio, and transmissibility. It also plots free, forced, and combined motion for the selected time duration.
Which units should I use?
Use kilograms, newtons per meter, newton seconds per meter, meters, meters per second, newtons, hertz, and seconds. Keep all inputs in consistent metric units.
Is this suitable for final engineering design?
It is suitable for learning, planning, and early checks. Final engineering design should include testing, safety factors, standards, and review by a qualified engineer.