Simple Evaluation Using Miles' Equation
Theory and Physics
What is the Miles Equation?
Professor, what is the Miles equation?
It is a simplified formula for random vibration of a single-degree-of-freedom system derived by John Miles (1954). When the input PSD is constant (white noise approximation), the RMS value of the response can be obtained with a single-line equation.
Or acceleration RMS:
Here $Q = 1/(2\zeta)$ is the quality factor of resonance, and $S_{\ddot{u}}(f_n)$ is the input acceleration PSD at the natural frequency.
It can be calculated in one line without FEM!
The Miles equation is ideal for screening evaluation. It provides a rough estimate before performing FEM PSD analysis to grasp the order of magnitude.
Assumptions and Limitations
Assumptions of the Miles equation:
1. Single-degree-of-freedom system — Cannot be directly applied to multi-degree-of-freedom systems
2. Input PSD is constant near the natural frequency — White noise approximation
3. Small damping — Approximately $\zeta < 0.1$
What happens if the input PSD is not constant?
Using the input PSD value at the natural frequency $f_n$ yields accuracy within 10-20% in many cases. It becomes inaccurate when the input PSD changes rapidly near $f_n$.
Practical Usage
Practical steps for using the Miles equation:
1. Estimate the equipment's first natural frequency $f_n$
2. Read the PSD value $S(f_n)$ from the vibration environment specification
3. Assume a damping ratio $\zeta$ (typically $\zeta = 0.02 \sim 0.05$)
4. Calculate the RMS response
5. Estimate the maximum response using 3σ (3×RMS)
6. Compare with allowable values
So there are situations where it can be used instead of FEM.
It is sufficient for screening during the conceptual design phase. For detailed design, proceed to FEM PSD analysis.
Summary
Key points:
- $a_{rms} = \sqrt{\pi f_n Q S(f_n) / 2}$ — A single-line formula
- Ideal for screening evaluation — Rough estimate before FEM
- White noise approximation — Input PSD is constant near $f_n$
- Use 3σ for maximum response — Can be used as a design value
- Not directly applicable to multi-degree-of-freedom systems — Can be applied individually to each mode
The Secret Story of the Miles Equation's Birth
The origin is the paper "On Structural Fatigue Under Random Loading" published by John W. Miles in 1954 in the Journal of the Aeronautical Sciences. Against the backdrop of the era when the U.S. Air Force struggled to predict aircraft fatigue failure, this groundbreaking formula was born under the bold assumptions of white noise approximation and a single-degree-of-freedom system, enabling the calculation of response RMS with just three parameters.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted in impact load or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That is Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (e.g., pressure, contact force). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A typical mistake here: getting the load direction wrong. Intending "tension" but ending up with "compression"—sounds like a joke, but it actually happens when coordinate systems rotate in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only equilibrium between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Note unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify to N in mm system, N in m system |
Numerical Methods and Implementation
Miles Equation Calculation Example
Please show me a specific calculation example.
Vibration evaluation of an electronic device's printed circuit board (PCB):
- $f_n = 200$ Hz (PCB's first natural frequency)
- $S_{\ddot{u}} = 0.04$ g²/Hz (Input PSD from MIL-STD-810)
- $Q = 20$ ($\zeta = 2.5\%$)
67 G acceleration! I'm worried if the BGA solder on the PCB can withstand that.
Typical shock resistance acceleration for BGA is 50-100 G. If the Miles equation gives 67 G, detailed evaluation with FEM is necessary. Consider PCB reinforcement or mounting changes.
Extension to Multi-Mode Systems
For multi-degree-of-freedom systems, apply Miles to each mode and combine using SRSS (Square Root of Sum of Squares):
Combine contributions from each mode using the square root of the sum of squares. Same as SRSS in response spectrum method, right?
SRSS combination is accurate if modes are sufficiently separated (approximately $f_{i+1}/f_i > 1.2$). For closely spaced modes, CQC (Complete Quadratic Combination) is required.
Summary
3-Step Calculation Method for Miles Equation
The application procedure for the Miles equation is 3 steps: ① Confirm natural frequency fn (Hz), ② Read the input PSD value G²/Hz (W(fn)) at fn, ③ Calculate response RMS = √(π/2 × fn × Q × W(fn)). The Q value (≈1/(2ζ)) is determined from the structural damping ratio ζ, typically assumed as Q=10 (ζ=5%) for aerospace structures. The calculation can be completed in 10 seconds even in Excel.
Linear Elements (First-Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-Side Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix every iteration. Achieves quadratic convergence within convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix at initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$〜$10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Instead of applying full load at once, apply in small increments. The arc-length method (Riks method) can trace beyond extremum points on the load-displacement relationship.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
First-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. Quadratic elements are like "flexible curves"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on overall cost-effectiveness.
Practical Guide
Miles Equation in Practice
Widely used for screening in random vibration evaluation of electronic equipment, space equipment, and military equipment.
Sensitivity Parameters
Parameters that most affect response in the Miles equation:
| Parameter | Effect on Response | Notes |
|---|---|---|
| $Q$ (Quality Factor) | $a_{rms} \propto \sqrt{Q}$ | If $Q$ doubles → RMS increases by $\sqrt{2}$ times |
| $f_n$ (Natural Frequency) | $a_{rms} \propto \sqrt{f_n}$ | Higher $f_n$ → Larger RMS |
| $S(f_n)$ (Input PSD) | $a_{rms} \propto \sqrt{S}$ | If input doubles → RMS increases by $\sqrt{2}$ times |
$Q$ (inverse of damping) is the most uncertain parameter, right?
Response changes by $\sqrt{5} \approx 2.2$ times between $Q = 10$ and $Q = 50$. The accuracy of damping estimation governs the accuracy of the Miles equation.
Practical Checklist
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