Module 13 · Lesson 13.1
The base-excitation equation
Where −m üg comes from, why the equation is written in relative displacement, and the distinction between two answers that are routinely interchanged.
Why this matters
Every loading so far has been a force applied to the structure. An earthquake is not. The ground moves, the foundations move with it, and the structure is dragged along from below.
No force is applied anywhere. Everything that happens comes from the structure's own inertia resisting being moved — which is why a heavier building attracts more earthquake load, and why the intuition built up under wind and gravity has to be set aside.
By the end of this lesson you should be able to
- Derive m ü + c u̇ + k u = −m üg from first principles
- Say why the damping and stiffness terms contain relative motion
- Recover absolute acceleration from the relative solution
- Explain the two limiting cases: rigid and very flexible
What you should already know
- The equation of motion (Module 3)
- D'Alembert's principle (Module 1)
- Harmonic base excitation and transmissibility (Module 5)
From first principles
The base-excitation equation
We want to show: Derive m ü + c u̇ + k u = −m üg for a structure whose support moves, and understand why the equation is naturally written in RELATIVE displacement.
The spring and the damper connect the mass to the ground. They do not care where either of them is in absolute terms — only how far apart they are, and how fast that gap is changing. The mass, on the other hand, does care about absolute motion: Newton's law is written in an inertial frame, so it is the mass's total acceleration that generates the inertia force. Writing down that mismatch is the whole derivation.
Two answers, and they are different questions
The analysis produces a relative displacement history. From it two quite different quantities are extracted, and confusing them is the most common error in reading earthquake results.
Relative displacement, u. This is what deforms the structure. Inter-storey drift, member forces, cladding movement, pounding against a neighbour, the moat gap of an isolated building — all of these depend on relative displacement and none on absolute.
Absolute acceleration, ü^t. This is what the occupants and the contents feel. Equipment qualification, overturning of unrestrained plant, damage to services, human comfort — all of these depend on absolute acceleration.
A rigid structure has almost no relative displacement and feels the full ground acceleration. A very flexible structure has large relative displacement and almost no absolute acceleration. Neither is safe in general — they simply fail differently, one by deformation and the other by inertia forces on its contents.
Ask what the analysis is for before deciding which output matters. A base-isolation design lives or dies on the isolator displacement, and a hospital equipment qualification lives or dies on the floor acceleration. The same analysis produces both, and reporting the wrong one is easy.
Three properties of a record, and they are independent
Amplitude, usually quoted as peak ground acceleration. It is the most cited and the least informative on its own.
Duration. Two records with identical PGA can shake for 5 s or 45 s. For an elastic structure duration matters modestly; for a yielding one it matters enormously, because damage accumulates over cycles.
Frequency content. Where in the spectrum the energy sits. A rock site produces short-period energy; a deep soft-soil basin can concentrate energy near 1–2 s and destroy long-period buildings that a rock record would leave untouched.
The standard summary measures try to capture more than PGA does: Arias intensity integrates a² over the record and measures total energy; significant duration is the time between 5% and 95% of it. Both are computed by the course library and both appear in the ground-motion explorer.
Why every record has been processed
A raw accelerogram integrated twice produces a displacement that wanders off to absurd values, because a tiny constant error in acceleration becomes a quadratic drift in displacement.
So every published record has been baseline-corrected and high-pass filtered. The corner frequency chosen — typically 0.05 to 0.2 Hz — is a processing decision, not a measurement.
The consequence is worth knowing: records from different agencies differ at long period, because they were filtered differently. For an ordinary building this never matters. For a base-isolated structure with a 3 s period, or a long-span bridge, it can, and the filter corner should be checked before a record is used near it.
The synthetic records in this course are generated with an explicit high-pass corner at 0.15 Hz, stated in each record's notes, precisely so that this is visible rather than hidden.
Try it
A ground motion, and what it does to one structure
The ground moves; the structure is dragged along by its supports. What the structure feels depends entirely on its own period.
Record
0.24 g
Show the numbers behind this plot
- Relative displacement u (mm)
- Absolute acceleration (m/s²)
Show the numbers behind this plot
- Peak ground acceleration
- 2.4 m/s² (0.24 g)
- Peak ground velocity
- 0.237 m/s
- Peak ground displacement
- 109 mm
- 5–95% significant duration
- 9.01 s
- Total Arias intensity
- 0.808 m/s
- Peak RELATIVE displacement
- 42.8 mm
- Peak ABSOLUTE acceleration
- 2.65 m/s²
- Peak inertia force per tonne
- 2.65 kN/tonne
- Amplification over the ground
- 1.11×
Full record 20 s
This is what deforms the structure and generates member forces.
This is what the people and the equipment on the floor feel.
At T = 0.80 s the structure is amplifying the ground motion by 1.1×. Both quantities matter and they are different questions.
A note on these records
- Synthetic record generated for teaching. It is not a recorded earthquake and must not be used for design.
- Kanai–Tajimi filter with a ground period of 0.30 s and 60% ground damping.
- High-pass filtered at 0.15 Hz, then integrated to velocity and displacement with a linear trend removed from each.
- Linearly scaled by 1.00 to a peak ground acceleration of 2.40 m/s².
What this shows: An earthquake applies no force to a structure. It moves the ground, and the resulting inertia force −m·üg is what the structure has to resist — which is why mass is a liability in an earthquake and an asset under wind.
Worked example
Harmonic base motion: relative against absolute
Given
- A machine foundation subject to ground-borne vibration from nearby piling
- Ground motion idealised as harmonic: 30 mm amplitude at 1.5 Hz
- The supported structure: natural frequency 3.33 Hz, damping 5%
Find
The relative displacement, the absolute displacement and the absolute acceleration.
Assumptions
- Steady-state harmonic ground motion, so the closed forms of Module 5 apply
- Uniform base motion
Predict first
Two identical buildings differ only in mass: one is twice as heavy. Both are subjected to the same earthquake. How do the effective earthquake forces compare?
Practice
A structure of mass 40 tonnes experiences a peak ground acceleration of 3.2 m/s². What is the peak effective earthquake force, in kN?
Practice
A structure has a relative acceleration of −2.1 m/s² at an instant when the ground acceleration is 1.4 m/s². What is its absolute acceleration, in m/s²?
Practice
Ground motion is harmonic at 20 mm amplitude and 2 Hz. A structure with a natural frequency of 8 Hz and 5% damping sits on it. What is the relative displacement amplitude, in mm? Use u = β²Rd Ug.
Check yourself
Why does the base-excitation equation contain relative displacement on the left and absolute acceleration on the right?
Practice
A record has a peak ground acceleration of 0.28g. What is that in m/s²?
Practice
A very stiff structure of 900 tonnes sits on ground with a PGA of 0.28g. What horizontal force does it attract, in MN?
Check yourself
Which displacement strains the structure during an earthquake?
Worked example
A rigid-structure sanity check before any analysis
Given
- A four-storey building, 350 tonnes per floor
- Site peak ground acceleration 0.22g
- A response-spectrum analysis is about to be run
Find
The force a rigid structure would attract, as a lower bound on the answer.
Assumptions
- All the mass moves with the ground — the zero-period limit
Worked example
Relative and absolute, on the same building
Given
- A structure whose roof reaches a peak displacement of 74 mm relative to the ground
- Peak absolute roof acceleration 5.9 m/s²
- Peak ground acceleration 2.16 m/s²
- A rigid raft foundation that itself translated 180 mm during the event
Find
Which quantity governs each of four different questions.
Assumptions
- Rigid-body site translation; linear elastic response
Worked example
Why the equation is written in relative displacement
Given
- A structure on a base moving with ground acceleration ü_g(t)
- Two candidate unknowns: total displacement, and displacement relative to the base
Find
Which choice makes the problem tractable, and why
Summary
- m ü + c u̇ + k u = −m üg, with u relative to the ground
- Inertia uses ABSOLUTE acceleration; spring and damper use RELATIVE motion
- The effective force is proportional to MASS — unlike wind, where it is not
- For MDOF, peff = −Mι üg, and getting ι wrong is a silent error
- Relative displacement deforms the structure; absolute acceleration shakes its contents
- A rigid structure rides with the ground; a very flexible one is left behind
- Every published record has been filtered, and the corner is a processing choice
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint