Foundations of structural motion
When Structures Begin to Move
Discover why loading speed and structural properties can turn an ordinary force into a vibration problem.
Learning objectives
By the end of this lesson
- Distinguish static and dynamic loading through the role of time.
- Explain why load rise time and structural flexibility influence dynamic effects.
- Recognise common dynamic actions including earthquake, wind, impact, and blast loading.
Before you calculate
What this means in the real world
What the formula is for
The natural period is the time required for one complete free-vibration cycle in an idealized undamped linear single-degree-of-freedom system. More mass increases the period; more stiffness reduces it.
Tₙ = 2π √(m / k)Where an engineer uses it
A building under lateral motion: Floors and roofs carry mass while frames, walls, and cores resist sideways deformation. Earthquake ground motion and wind can activate several modes of movement.
Begin with an effective lateral mass and stiffness for the motion being studied, then increase the number of coordinates when one displacement is not enough.
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
Structural dynamics
The study of structural response when loading, displacement, velocity, acceleration, and internal forces vary with time.
Loads have a history
In ordinary static analysis, a load is often introduced only by its final magnitude. Structural dynamics asks an additional question: how did that load develop with time? A gradually applied action can allow a structure to settle toward equilibrium, while a rapidly applied action can initiate vibration.
Formula
Natural period of an undamped SDOF system
The natural period is the time required for one complete free-vibration cycle in an idealized undamped linear single-degree-of-freedom system. More mass increases the period; more stiffness reduces it.
- Tₙ
- natural periods
- m
- lumped masskg
- k
- lateral or translational stiffnessN/m
Formula experiment
Experiment with mass and stiffness
Edit m and k, calculate the period, and replay one motion cycle at the corresponding timescale.
One full left-right cycle represents the calculated natural period.
Idealized undamped linear SDOF result. Changing mass or stiffness updates the model, not the safety of a real structure.
Interactive concept
Explore dynamic tendency
Change the way a load is applied and the flexibility of the structure. This is a qualitative concept explorer, not a numerical simulation.
A gradual action on a relatively rigid structure is more likely to remain close to static behaviour.
Where this idea appears in practice
The same mass-stiffness relationship appears in very different engineering systems. What changes is the physical meaning assigned to the mass, the deformation that supplies stiffness, and the action that may repeatedly excite the system.
Real-world application
A building under lateral motion
Floors and roofs carry mass while frames, walls, and cores resist sideways deformation. Earthquake ground motion and wind can activate several modes of movement.
- Modelling move
- Begin with an effective lateral mass and stiffness for the motion being studied, then increase the number of coordinates when one displacement is not enough.
- Engineering question
- Which natural periods and mode shapes contribute most to the expected loading?
Real-world application
An elevated water tank
A heavy container and its contents sit above a comparatively flexible supporting system. Structural motion may also interact with liquid movement.
- Modelling move
- Use the participating container and liquid mass with the lateral stiffness of the staging as a first model, then add fluid-structure effects when required.
- Engineering question
- Does the simplified support model capture the motion that controls demand?
Real-world application
A pedestrian structure
Walking, running, or coordinated movement can apply repeated forces. Comfort may become important even when member strength is adequate.
- Modelling move
- Estimate the participating deck mass and stiffness in the relevant direction, then compare structural and activity frequencies.
- Engineering question
- Could ordinary human activity produce noticeable vibration or discomfort?
Real-world application
Equipment mounted on a floor
Rotating machinery can apply a regular force at its operating frequency. Flexible mounts change how vibration reaches the supporting structure.
- Modelling move
- Represent the machine and its support as a mass-stiffness-damping system and examine both equipment motion and transmitted force.
- Engineering question
- Should the mount be adjusted to separate operating and natural frequencies while controlling movement?
Application studio
Compare four engineering applications
Choose an application, change its illustrative model properties, and compare the repeated action frequency with the calculated natural frequency.
The selected repeated action is not close to this model's natural frequency. Other modes and load effects may still govern.
Preset values are intentionally illustrative. The motion graphic shows frequency proximity, not predicted displacement or structural safety.
A repeatable engineering workflow
Start by naming the response that matters. Decide what mass participates, identify the deformation that creates stiffness, estimate the characteristic timescale, and describe the loading in time. Compare frequencies only as a screening step; damping, multiple modes, nonlinear behaviour, load combinations, and design-code checks determine the analysis that follows.
Editable formula question
Calculate a natural period
Choose the model mass and stiffness within the exercise limits, calculate Tₙ, and enter your answer.
The displayed minimum, maximum, and textbook values define this learning exercise only. They are not universal design limits.
Editable formula question
Convert period to frequency
Choose a natural period and calculate the corresponding natural frequency.
The displayed minimum, maximum, and textbook values define this learning exercise only. They are not universal design limits.
Dynamic loading in practice
Earthquakes, wind, impact, blast, and tsunami actions can vary significantly with time. Whether their effects are important depends on the duration and frequency content of the action as well as the dynamic properties of the structure.
The structural-dynamics problem
The engineering task begins with three ingredients: describe the load history, create a mathematical model of the structure, and select a response method. The desired response may be displacement, velocity, acceleration, internal force, or stress. Later lessons build each ingredient carefully.