Structural dynamics · modal force

Coupled Response to Applied Forces

Transform force patterns into modal coordinates and recover physical response.

Approx. 32 min

Learning objectives

By the end of this lesson

Define modal force in physical engineering terms.
Use modal force 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

A load excites a mode strongly when its spatial pattern aligns with that mode.

pᵣ(t) = φᵣᵀp(t)

Where an engineer uses it

Synchronized crowd load on a grandstand: This application turns modal force into a real modelling decision.

Represent both the temporal frequency and spatial distribution of the applied action.

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

modal force

The generalized action obtained by projecting a physical force vector onto a mode shape.

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

Build the physical picture

Transform force patterns into modal coordinates and recover physical response. 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

Modal force

pᵣ(t) = φᵣᵀp(t)

A load excites a mode strongly when its spatial pattern aligns with that mode.

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

Force-balance lab

Explore modal force

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

Synchronized crowd load on a grandstand

This application turns modal force into a real modelling decision.

Modelling move: Represent both the temporal frequency and spatial distribution of the applied action.
Engineering question: Which modes are strongly excited by the load pattern?

Calculate, compare, interpret

Editable formula question

Assemble instantaneous dynamic demand: Coupled Response to Applied Forces

Edit the model state and calculate the force needed to balance inertia, damping, and stiffness.

p(t) = mü + cu̇ + ku

Mass m = 1000 kgMin 100 · Textbook example 1000 · Max 10000 kgAcceleration ü = 0.5 m/s²Min 0 · Textbook example 0.5 · Max 3 m/s²Damping c = 3000 N·s/mMin 0 · Textbook example 3000 · Max 20000 N·s/mVelocity u̇ = 0.1 m/sMin 0 · Textbook example 0.1 · Max 1 m/sStiffness k = 20000 N/mMin 1000 · Textbook example 20000 · Max 100000 N/mDisplacement u = 20 mmMin 0 · Textbook example 20 · Max 100 mm

Substitute your current valuesp = (1000×0.5) + (3000×0.1) + (20000×0.02)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 ±0.03.

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.