Base Excitation Response Analysis
Base Excitation Response Analysis: Theoretical Foundations
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.
Computational Methods for Base Excitation Response Analysis
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.