This course is for anyone who wants to understand multi-degree of freedom (MDoF) structural dynamics. The course is built around the theme of seismic analysis and determining the response of structures to earthquake-induced ground motion. In reality, that’s just one form of excitation; when you complete the course, you’ll be able to handle ground motion as well as any directly applied dynamic loads such as wind loading or blast pressure waves, for example. Once you complete this course, you’ll have a toolbox full of dynamic analysis tools and the knowledge and confidence to apply them to your projects.

Introduction and course overview

Course prerequisites and do I need to know Python?

Getting your coding environment set up

Section 2 overview

El-Centro N-S ground acceleration record (.csv)

20 Largest Earthquakes in the World as of September 2021, USGS

Isoseismal map of the Tohoku Earthquake, Japan, USGD

Overview of earthquake ground motion

Capturing the influence of ground motion

Motion transmissibility and isolation

Simulating ground motion response

Realistic ground motion response

Integrating ground acceleration

Introduction to response spectra

Section 3 overview

Introduction to MDoF systems

MDoF Equations of Motion

Direct integration of the coupled EoM

Solving the coupled EoM by direct integration

Using SciPy’s ODEINT solver

Section 4 overview

What is modal analysis and how does it help us?

Finding natural frequencies and mode shapes

Matrix representation and the eigenproblem

Modal matrix and mass normalisation

Orthogonality and Decoupling Equations of Motion

Modal superposition - forced vibration

Comparing modal superposition with direct integration - forced response

Visualising the forced vibration response

Modal superposition - free vibration

Compare modal superposition with direct integration - free vibration

Section 5 overview

Rayleigh damping

General determination of the damping matrix

Calculating the damping matrix

Direct integration versus modal superposition for a damped system

Section 6 overview

Equations of motion N-storey shear building

Automating modal analysis for NDoF

Ground motion response using modal superposition

Base shear and overturning moments

Visualising the structural response for NDoF

Course wrap up and completion certificate

Build the knowledge and tools to decode the dynamic response of real-world structures to real-world loads.

Multi-Degree of Freedom Dynamics, Modal Analysis and Seismic Response Simulation in Python

This course has two simple objectives: (1) to help you build a solid understanding of the core concepts in structural dynamics and (2) to equip you with some practical tools you can deploy to analyse real-world dynamic structural behaviour. We’ll start by covering the fundamentals of dynamics, including free vibration and harmonic excitation. Following this we’ll explore ways to tackle general (non-harmonic) dynamic loading. To do this, we’ll introduce the Piecewise Exact Method - a numerical technique that gives us huge flexibility in the types of dynamic behaviour we can simulate. We’ll use Python throughout the course to help bring our analyses to life and build an intuition for dynamic behaviour.

Our approach to Python

Getting started with Jupyter Notebooks

Static versus dynamic analysis

Introducing inertia

Dynamic response and inertia 

Lumped mass analysis

The spring-mass-damper model

The equation of motion

Undamped free vibration

Natural frequency

Damped free vibration

Critically-damped and over-damped systems

Under-damped systems

Damped natural frequency

Logarithmic decrement

Worked Example #1

Worked Example #2

Harmonic forcing

Solving the equation of motion

Characterising the steady-state response

Dynamic magnification factor and resonance

Worked Example #3 - Part A

Worked Example #3 - Part B

Worked Example #3 - Part C

Pedestrian-induced vibrations of the Clifton Suspension Bridge, UK

General dynamic loading and feedback

The Piecewise-Exact Method - concept

The Piecewise-Exact Method - development

Implementing the Piecewise-Exact Method

Force segmentation

Time discretisation

Course wrap up

Leverage fundamental structural dynamics to build your own flexible numerical solutions in Python.

Fundamentals of Engineering Structural Dynamics with Python

Analysis techniques for dynamic analysis and excitation modelling of Civil Engineering structures

Dynamics

Part 1: Understanding human-induced floor vibration and assessing acceptability | EngineeringSkills.com

A practical guide to floor vibration serviceability assessment

Part 1: Understanding human-induced floor vibration and assessing acceptability

In this project you’ll learn about the Duhamel Integral & how it can be used to simulate crowd-induced vibrations in Python

The Duhamel Integral & Crowd-Induced Vibrations | EngineeringSkills.com

Learn about the Duhamel Integral & how it can be used to simulate crowd-induced vibrations in Python

Simulating crowd vibrations using the Duhamel Integral


By the end of this post you should clearly understand when and why a dynamic analysis is performed instead of a (usually more straightforward) static analysis. Clearly understanding structural dynamics and the difference between static and dynamic analysis really is essential before you can move on to understand dynamic analysis techniques.

When we study engineering (civil, structural, mechanical etc.), the great majority of our time is spent getting to grips with static analysis. As a result most engineers will be very comfortable when things are stationary, but once a problem ‘becomes dynamic’ and things start moving – many engineers start to get jumpy! This isn’t really surprising, most of the loads we deal with in general day-to-day civil/structural engineering can be considered static loads, e.g. typical permanent and variable loads. It’s worth nothing that mechanical engineers tend to have a higher comfort level with structural dynamics. This is hardly surprising given the nature of mechanical engineering.

> 🎓 If you want to learn more about Structural Dynamics, check out my course on the **[Fundamentals of Engineering Structural Dynamics with Python.](/course/fundamentals-of-structural-dynamics-with-python/)**

<EmbeddedCourseAd infoPageSlug="fundamentals-of-structural-dynamics-with-python" />

## 1.0 Introduction

Let’s start off by considering a static load or force. A static load has two important characteristics that distinguish it from a dynamic load:

1. The magnitude of a static load does not vary with time
2. For a load to be considered static, it must be applied ‘slowly’. We’ll discuss this in more detail below.

A quick aside here: The term ‘variable load’ above is used here in the Eurocode context, i.e. it describes loads that do _have the potential to vary with time_, e.g. loading due to occupancy in a building, but this is not considered a dynamic load in the context of this discussion as the rate of change of the load magnitude can be considered low. What we mean by this will become clearer as we proceed.

If a load applied to a structure qualifies and can be considered a static load, we can expect a static structural response, i.e. for a given load magnitude, we observe a single structural response (deflection). The deflection of the structure will be governed by the load magnitude, $p$ and the stiffness of the structure $K$ according to the standard relationship:

$$
p = K\times\delta
\tag{1}
$$

💡 A static force will induce a single static structural response, i.e. deflection. In other words for every load magnitude we have a corresponding single structural response.

We can illustrate the basic idea of static loading and response by considering the beam below subject to an external force $p(t)$.

<Image
	src="/images/posts/understanding-structural-dynamics-and-inertia/Structural-Dynamics-1.jpg"
	alt="Structural-Dynamics-1-EngineeringSkills.com"
	width="450"
	caption="Simple cantilever subject to a slowly applied static load. Every value of load magnitude corresponds to a single value of deflection, governed by the stiffness of the structure."
	height="566"
/>
Notice that although the external force is actually a function of time $t$, **the
duration of load application is highlighted as being long**, i.e. the load is slowly
increased in magnitude from zero to its maximum value. This is a classic static structural
response that most of us are comfortable with. 

## 2.0 Dynamic versus Static Structural Analysis

Although we said that we typically encounter static loading, dynamic loads occur with sufficient frequency that we need to understand how to assess their influence on a structure. Typical forms of dynamic loading can include loading due to earthquakes, wind, vehicle or pedestrian traffic or wave loading on coastal or offshore structures.

As you might assume from the preceding discussion, a dynamic load has a magnitude that varies as a function of time. It may also be a load that is applied rapidly to a structure. Consider for example the same beam as above, with the same applied load except this time the load is applied more rapidly $(t1>>t2)$, notice the different structural response.

<Image
	src="/images/posts/understanding-structural-dynamics-and-inertia/Structural-Dynamics-2.jpg"
	alt="Structural-Dynamics-2-EngineeringSkills.com"
	width="450"
	caption="Dynamic response of the cantilever when subject to a rapidly applied load. The response is characterised by a peak followed by a decaying oscillation about the static deflection point."
	height="463"
/>
We can see that when the same load is applied to the cantilever, but faster, we get
a very different structural response. We observe a peak dynamic response which is
greater than the static deflection. This is then followed by a decaying oscillation
or vibration about the static deflection point. This is a classic dynamic response.
**But the question remains, what is it about the speed of load application that causes
this totally different structural behaviour? The answer, in a word is, INERTIA!**

### 2.1 Inertia

In this case the load is applied ‘dynamically’ (quickly) and as a result we observe a dynamic structural response. This response is the result of inertia.

💡 Inertia is a bodies resistance to a change in its velocity. This resistance can be modelled with an inertia force.

<Image
	src="/images/posts/understanding-structural-dynamics-and-inertia/Structural-Dynamics-3.jpg"
	alt="Structural-Dynamics-3-EngineeringSkills.com"
	width="600"
	height="248"
/>
<Caption props="">
	Ball with mass $M$ experiencing a disturbing force $F(t)$. The ball’s
	resistance to a change in its velocity, i.e. its inertia, is modelled with an
	inertia force $F_I=M\times \ddot{u}$.
</Caption>

The role of inertia here can be captured as follows:

- If I want to change the velocity of the mass, $M$, this will give rise to an acceleration $\ddot{u}$ (rate of change of velocity). Therefore there will be an inertia force I must overcome if I want to change the velocity of the ball.
- Once the ball is moving, if I remove the external force, $F(t)$, the ball will stay moving with a constant velocity (Newton 1). Therefore the inertia force disappears because there is no change in velocity.
- When the velocity of the ball is constant, it is in a state of dynamic equilibrium. Newton 1 tells us that a mass always wants to maintain either static or dynamic equilibrium. This tendency of a mass to maintain equilibrium (i.e. resist changes in its velocity) can be modelled or by applying an inertia force.

<Callout type="info">
	⚠️ It’s important to realise that **the inertia force is fictitious**; it is
	not an externally applied force (like $F(t)$), but it models the behaviour of
	the mass, i.e. it models the masses resistance to a change in its velocity.
</Callout>

So in summary:

- all mass resists changes in its velocity (mass wants to maintain static or dynamic equilibrium)
- we can model this physical behaviour in our free body diagrams and equations by applying a fictitious inertia force (**known as D’Alembert’s Principle**)
- the inertia force is proportional to the mass and the rate of change of velocity (acceleration)

$$
F_I = m\times \dot{v} = m\times a
\tag{2}
$$

### 2.2 Dynamic Beam Response

Now returning to our beam example; the displacement of the beam depends on two factors:

- the time-varying external force $F(t)$
- the beam’s inertia modelled using a fictitious inertia forces which oppose the accelerations producing them

As the external load is applied quickly or dynamically, the rate of change of velocity of the beam will be high or non-negligible. We now know that this means the beam will exhibit a correspondingly high resistance to a change in its velocity. We model this by applying inertia forces.

<Image
	src="/images/posts/understanding-structural-dynamics-and-inertia/Structural-Dynamics-4.jpg"
	alt="Structural-Dynamics-4-EngineeringSkills.com"
	width="600"
	caption="Cantilever beam exhibiting a dynamic response due to the speed of load application and the resulting inertia forces developed along the length of the beam."
	height="518"
/>
Remember the beam deflection now depends on the external force **and the relatively
large inertia forces induced by the beam’s acceleration.** This means that the internal
moments and shears in the beam must resist not only the externally applied force
but also the inertia forces resulting from accelerations of the beam.

💡 It’s the presence of inertia forces that fundamentally differentiate a dynamic problem from a static problem.

If inertia forces represent a significant portion of the total loading resisted by a structure, a dynamic analysis should be performed as the dynamic characteristics of the problem cannot be ignored and the structural response will vary with time.

Conversely if all external loads are applied slowly such that they do not induce significant inertia forces, the problem can be considered static with a single response corresponding to the externally applied load(s).

Hopefully this short tutorial has given you a sense of why a dynamic analysis may be required rather than a static analysis. Of course, once you know that the dynamic characteristics of a problem can’t be ignored, you still need to carry out some dynamic structural modelling. This is where the ubiquitous single-degree-of-freedom system usually comes in. If you want to continue exploring structural dynamics, take a look at my course on this, [Fundamentals of Engineering Structural Dynamics with Python](/course/fundamentals-of-structural-dynamics-with-python/). In this course we’ll cover the fundamental theory but also leverage Python to develop structural dynamics solvers that you can deploy on your own projects (FYI you don’t need to know Python to take this course! 😉).

That’s all for now, see you in the next one.


By the end of this post you'll better understand structural dynamics and why dynamic analysis is performed instead of simpler static analysis.

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Validating our numerical solution code

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Dynamic analysis: Bridge + N pedestrian crowd

Animating bridge response & traffic flow

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Analysis techniques for dynamic analysis and excitation modelling of Civil Engineering structures

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