Force scaling curve with current model pointLogarithmic axes
Unit Converters
Force Scale Calculator
Evaluate force similitude under an explicit power law. Compare linear, area, volume, and user-defined exponents, apply density and gravity ratios where appropriate, and test whether model force fits rig capacity and measurement resolution.
Input evidence: declare the similarity law, geometric scale, material-density ratio, gravity environment, dynamic basis, model capacity, and measurement resolution before interpreting scaled force.
Scaled model force-
Dynamic test force-
Capacity utilization-
Force scale ratio-
Resolution share-
Test status-
Similarity-law response
See how force changes across geometric scale and governing exponent
The log response distinguishes an area law from volume, inertia, and user-defined similarity rules.
| Scaling law | Exponent | Density/gravity applied | Predicted model force | Capacity utilization | Appropriate question |
|---|
How to use
Scale force only after stating the similitude law
- Define prototype and model lengths at homologous locations and enter the model-to-prototype ratio.
- Select the force exponent from the governing physics—area, volume, inertia, or a validated empirical relationship.
- Enter density and gravity ratios only when those quantities belong in the chosen similitude law.
- Apply dynamic amplification from the test event, then compare model force with rig capacity and measurement resolution.
- Review alternative exponent curves and document which phenomena are preserved, distorted, or left unscaled.
Similarity fundamentals
Geometric resemblance is not complete physical similarity
A scale model can preserve geometry while failing to preserve inertia, stiffness, gravity, fluid effects, or time response. The exponent and correction ratios declare which relationship the force estimate represents.
Law-selection rule: an exponent is an engineering assumption that needs a derivation or validated reference; it is not chosen to make the model fit the rig.
Calculation method
Apply one declared power law and keep correction ratios visible
The geometric ratio is raised to the force exponent, then multiplied by density and gravity ratios. Prototype force is scaled by that ratio and the dynamic factor creates the test demand.
λ = Lₘ ÷ Lₚ; φ = λᵉρᵣgᵣ; Fₘ = Fₚφ; Fₜ = FₘDCapacity utilization equals dynamic test force divided by model rig capacity.
- Area-governed forces commonly scale with λ² when other ratios remain one.
- Volume or weight effects commonly introduce λ³ and may also depend on density and gravity.
- Dynamic similarity can require velocity, time, or fluid-property ratios not represented here.
- Use unrounded scale ratios for inverse reconciliation.
Choosing an exponent
Area, volume, inertia, and empirical response scale differently
Derive the exponent from the governing equation or dimensional analysis.
- Surface pressure integrated over area suggests an area relationship.
- Self-weight suggests a volume relationship with density and gravity.
- Structural response may require stiffness and section-property scaling.
- Empirical exponents need a validated range and uncertainty statement.
Similarity distortion
One physical model rarely preserves every phenomenon
Reynolds, Froude, Mach, Cauchy, and stiffness relationships can demand incompatible scale choices.
- Name the dimensionless groups that govern the prototype.
- Prioritize the phenomenon tied to the test decision.
- Quantify correction or uncertainty for distorted effects.
- Do not extrapolate outside the validated scale range.
Decision interpretation
A feasible rig point is not proof that the scale law is valid
Scaled model force and dynamic test force describe demand under the chosen law. Capacity utilization addresses overload; resolution share addresses whether the signal can be distinguished. Both can pass while the governing similitude argument is wrong.
How to read the log-scale response
The horizontal axis is geometric length ratio and each curve shows force predicted by a different exponent; the current model point marks the entered law. Editing exponent changes curve slope, while density, gravity, and dynamic factors shift demand. The visual can mislead when a single power law is used across regime changes.
Test-system suitability
Capacity and resolution constrain opposite ends of the experiment
A rig must tolerate the dynamic maximum and still resolve the smallest meaningful signal. Increasing capacity can reduce sensitivity if the measurement chain's range and resolution worsen.
- Check fixture, actuator, transducer, and data-acquisition limits separately.
- Include preload, tare, off-axis force, and transient overshoot.
- Set a minimum number of resolvable increments for the decision.
- Use a multi-range system when one transducer cannot cover both ends credibly.
Detailed calculation process
Apply the scale law and reconcile its inverse
1. Geometric ratio: λ = Lₘ ÷ Lₚ.
2. Force ratio: φ = λᵉ × ρᵣ × gᵣ.
3. Model force: Fₘ = Fₚ × φ.
4. Dynamic test force: Fₜ = Fₘ × D.
5. Utilization = Fₜ ÷ capacity; resolution share = resolution ÷ Fₘ.
- Lₘ, Lₚ
- model and prototype homologous lengths
- λ
- model-to-prototype length ratio; dimensionless
- e
- force scaling exponent; dimensionless
- ρᵣ
- model-to-prototype density ratio; dimensionless
- gᵣ
- model-to-prototype gravity ratio; dimensionless
- Fₚ, Fₘ
- prototype and scaled model force; N
- D
- dynamic amplification factor; dimensionless
Default substitution and reconciliation
λ = 0.25 and φ = 0.25² × 1 × 1 = 0.0625. Fₘ = 12,000 × 0.0625 = 750 N, and Fₜ = 750 × 1.15 = 862.5 N. Dividing 750 N by 0.0625 returns the 12,000 N prototype force, reconciling the scale law.
Evidence requirements
Document the similitude argument
The record must let another engineer reproduce the chosen scale law and its validity boundary.
- Homologous dimensions, measurement basis, and geometric scale
- Governing equations, exponent derivation, and dimensionless groups
- Density, gravity, material, boundary, and dynamic conditions
- Rig capacity, resolution, calibration, and validation-scale evidence
Scope and limitations
What this single-law model excludes
- Multiple coupled exponents and regime-dependent behavior
- Time, velocity, stiffness, damping, viscosity, and compressibility scaling
- Boundary-condition distortion, manufacturing tolerances, and nonlinear materials
- Uncertainty propagation or validation outside the tested scale range
Key terminology
Similarity glossary
- Prototype
- Full-scale system represented by the model.
- Geometric scale
- Ratio between homologous model and prototype lengths.
- Similitude
- Preservation of governing relationships across scale.
- Exponent
- Power relating geometric scale to a response quantity.
- Dimensionless group
- Unit-free combination governing a physical regime.
- Distorted model
- Model that intentionally does not preserve every scale relationship.
Practical examples
Two model-test decisions
Area-loaded fixture
A quarter-scale panel uses an area-based exponent. The predicted force fits the rig, but the team separately verifies that stiffness and boundary conditions do not invalidate the response comparison.
Centrifuge gravity model
A geotechnical model uses a non-unity gravity ratio. The analyst includes density and gravity explicitly rather than applying a geometric exponent alone.
Important note
Before relying on this result
The power-law model excludes coupled dimensionless groups, nonlinear materials, turbulence, fracture, boundary distortion, and uncertainty.
Additional Force Scale Calculator questions
Which exponent should be used?
Use the exponent derived from the governing physical mechanism.
Does every force scale with area?
No. Self-weight and dynamic phenomena can scale differently.
Why include density and gravity ratios?
Body-force laws depend on both.
Can one model preserve every phenomenon?
Usually not; scale effects must be assessed.