Structural dynamics · influence coefficient

Modelling Coupled Structures

Apply equilibrium and influence-based modelling to structural examples.

Approx. 32 min

Learning objectives

By the end of this lesson

Define influence coefficient in physical engineering terms.
Use linear restoring vector with consistent quantities and units.
Change model inputs and explain the resulting trend.
Separate an educational model result from a code-based safety decision.

Before you calculate

What this means in the real world

What the formula is for

The stiffness matrix maps a displacement pattern to the complete set of restoring forces.

fₛ = Ku

Where an engineer uses it

A frame with floor translation and torsion: This application turns influence coefficient into a real modelling decision.

Select coordinate directions and include eccentric stiffness paths that couple translation and rotation.

What a wrong value can do

A wrong unit, sign, scale, boundary condition, or property can produce a believable number with the wrong physical meaning. It may understate demand, exaggerate it, or lead to the wrong failure check.

Definition

influence coefficient

A response at one coordinate caused by a unit action applied at another coordinate.

Why it mattersIt provides the central modelling idea for modelling coupled structures and connects observed motion to a solvable engineering representation.

Build the physical picture

Apply equilibrium and influence-based modelling to structural examples. Start by identifying the system boundary, response coordinates, applied action, and the properties that store or dissipate energy. A useful model is the simplest one that preserves the behaviour needed for the engineering question.

Formula

Linear restoring vector

fₛ = Ku

The stiffness matrix maps a displacement pattern to the complete set of restoring forces.

u: response displacement or displacement vectorm
t: times
ω: angular frequency where usedrad/s

Force-balance lab

Explore influence coefficient

Change the available model inputs, watch the motion or force relationship, and describe the trend before reading the numerical result.

Displacement, u 20 mmVelocity, u̇ 0.12 m/sAcceleration, ü 0.60 m/s²Educational force limit 3.0 kN

Illustrative linear SDOF: m = 1000 kg, k = 20 kN/m, c = 5 kN·s/m

/\/\/\/

Fᵢ 0.60 kN ←→ Fₛ + Fᴅ

Stiffness forceFₛ = ku = 0.40 kN

Damping forceFᴅ = cu̇ = 0.60 kN

Inertia forceFᵢ = mü = 0.60 kN

Educational demand/limit ratio0.53

Calculated component-force sum is within the limit you entered.

Not a structural safety verdict. The entered limit is only a learning comparison. Real safety requires code-defined load combinations, strength and serviceability checks, geometry, materials, detailing, and professional review.

Real-world application

A frame with floor translation and torsion

This application turns influence coefficient into a real modelling decision.

Modelling move: Select coordinate directions and include eccentric stiffness paths that couple translation and rotation.
Engineering question: Which physical connection creates each coupling term?

Calculate, compare, interpret

Editable formula question

Relate deformation to restoring force: Modelling Coupled Structures

Change stiffness and displacement, then calculate the linear restoring-force magnitude.

Fₛ = ku

Stiffness k = 800 kN/mMin 10 · Textbook example 800 · Max 5000 kN/mDisplacement u = 20 mmMin 1 · Textbook example 20 · Max 100 mm

Substitute your current valuesFₛ = 800 × (20 ÷ 1000)Answer format: enter the numerical result only in kN. Example: 12, not “answer = 12”.

Your calculated answer

Round only at the final step. The accepted tolerance for this exercise is ±0.02.

The displayed minimum, maximum, and textbook values define this learning exercise only. They are not universal design limits.

Editable formula question

Compare a model demand with your entered criterion

Enter a demand and a comparison value chosen for this exercise, then interpret the ratio.

utilization indicator = model demand / entered criterion

Model demand D = 4 unitsMin 0.1 · Textbook example 4 · Max 20 unitsEntered criterion L = 5 unitsMin 0.1 · Textbook example 5 · Max 20 units

Substitute your current valuesratio = 4 ÷ 5Answer format: enter the numerical result only (no unit for this dimensionless result). Example: 12, not “answer = 12”.

Your calculated answer

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Round only at the final step. The accepted tolerance for this exercise is ±0.01.

The displayed minimum, maximum, and textbook values define this learning exercise only. They are not universal design limits.