基礎加振応答解析
Theory and Physics
What is Base Excitation?
Professor, what is "base excitation"?
It's a problem where vibration is input to the support part (base) of a structure. Building vibration due to earthquakes, vibration transmission from a vehicle body to equipment, and excitation by a vibration test bench are typical examples.
So the structure vibrates because "the floor shakes," not because a load acts directly on the structure.
Correct. The equation of motion:
$\ddot{u}_g$ is the base acceleration input. The right-hand side acts as an inertial force.
Absolute Response and Relative Response
Two definitions of response:
- Absolute Response — Displacement/acceleration of the structure in a stationary coordinate system
- Relative Response — Displacement/acceleration of the structure relative to the base
Which one is important for design?
It depends on the application:
- Stress evaluation → Relative response (stress is calculated from relative displacement)
- Acceleration limits → Absolute response (impact on the human body is based on absolute acceleration)
- Displacement limits → Relative response (clearance with adjacent structures)
Setting in FEM
Methods for inputting base excitation:
1. Acceleration Input at Support Points
Apply acceleration to support points (SPC points). Input via the large mass method or enforced displacement using SPC.
2. Inertial Force Input
Apply $\{F\} = -[M]\{1\} \ddot{u}_g$ as a load. In modal methods, effective mass is used to calculate input to each mode.
Nastran
```
SOL 111
CEND
DLOAD = 100
BEGIN BULK
RLOAD2, 100, 200, , , 1.
TABLED1, 200, ...
$ Acceleration input at support points
SPCD, ...
```
Abaqus
```
*STEP
*STEADY STATE DYNAMICS
1., 100., 100, 1.
*BASE MOTION, DOF=2, AMPLITUDE=accel_input
*END STEP
```
Abaqus's *BASE MOTION seems the most straightforward.
*BASE MOTION can directly input base excitation. You just specify the direction (DOF) and the input waveform (AMPLITUDE).
Summary
Let me organize the base excitation response.
Key points:
- The base shakes and the structure responds — Earthquakes, vibration tests, vehicle body vibration
- $F = -M \ddot{u}_g$ — Input as inertial force
- Absolute response vs. relative response — Use appropriately depending on the application
- Abaqus *BASE MOTION is the most intuitive — Just specify direction and waveform
The Principle of Seismometers is Forced Vibration Itself
The modern seismometer invented by John Milne (UK) in 1880 is a device that records the relative displacement of a mass-spring-damper system subjected to base excitation. The design, which sets a long natural period to make it sensitive to ground acceleration, is a direct application of forced vibration theory. Current broadband seismometers (e.g., Streckeisen STS-2) also operate on the same principle, covering 0.008–50 Hz.
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, which assumes "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's 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" is incorrect. 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 $f_s$ (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 common pitfall here: getting the load direction wrong. Intending "tension" but applying "compression"—sounds like a joke, but it actually happens when coordinate systems are rotated 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 vibrational 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 vibrational energy to improve ride comfort. What if damping were zero? Buildings would keep 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: Geometric nonlinearity is required for large deformation/large rotation problems. Constitutive law extension is needed for nonlinear material behavior like plasticity and creep
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the 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 as N in mm system, N in m system |
Numerical Methods and Implementation
Frequency Response of Base Excitation
How do you calculate the frequency response for base excitation?
Give the base acceleration input as a function of frequency $\ddot{U}_g(\omega)$ and calculate the structural response at each frequency. It is often expressed as transmissibility.
Large Mass Method
A method that adds a very large mass (about $10^6$ times the total structural mass) to the support point and applies a force to that mass to equivalently realize acceleration input. Widely used in Nastran.
Why is such a large mass needed?
If $M_{large}$ is orders of magnitude larger than the structural mass, the input acceleration is hardly affected by the structural response (becomes a "stiff" input). This allows simulating base excitation with a normal load input.
Vibration Test Simulation
Simulate sine sweep excitation from MIL-STD-810 (military vibration test) or IEC 60068 (environmental test) in FEM. Use frequency response analysis to calculate response at each frequency and check if it's within the standard's limits.
Summary
Let me organize the numerical methods for base excitation.
Key points:
- Large Mass Method — Equivalently realizes acceleration input using a large mass
- Abaqus *BASE MOTION — Most direct input method
- Vibration Test Simulation — Pre-evaluation for MIL-STD-810, IEC 60068
- Evaluate with Transmissibility — Ratio of structural response to base acceleration
Large-Scale Seismic Isolation Table Excitation Experiments Date Back to 1958
The world's largest facility for base excitation experiments is the E-Defense at the National Research Institute for Earth Science and Disaster Resilience in Hyogo Prefecture (opened 2005), with a maximum excitation force of 18,000 kN and 3D 6 degrees of freedom. The concept that established its predecessor was formulated in 1958 by MIT's Professor J. M. Biggs, who published the frequency response function (FRF) formulation for base input motion. There has been 60 years of evolution from theory to experimental facilities.
Linear Elements (1st 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 important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately.
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 each iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using 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 $\epsilon = 10^{-3}$ to $10^{-6}$). 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 curve.
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
1st order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, per element,
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