
In this lecture we'll briefly consider harmonic loading and why we choose to analyse the influence of this form of loading first. A harmonic force or harmonic excitation is a force with a sinusoidally varying magnitude,

$$
F(t) = P_0\sin(\omega t)
\tag{1}
$$

where $P_0$ is the force amplitude and $\omega$ is the forcing frequency. To visualise the force we simply plot the sine wave.

```python
tmax = 10 #(sec) The max time
delta_t = 0.01 #(sec) The time-step
nPoints = tmax/delta_t #Number of equally spaced data points
t = np.linspace(0,tmax, int(nPoints)) # Time vector
```

```python
P0 = 1000 #Force amplitude
w = 2*math.pi #Angular forcing frequency
F = P0*np.sin(w*t) #Force magnitude values
```

```python
#Plotting
fig = plt.figure()
axes = fig.add_axes([0.1,0.1,2,1.5])
axes.plot(t,F,label='Harmonic force')
axes.set_xlim([0,tmax])
axes.set_xlabel('Time (sec)')
axes.set_ylabel('Force (N)')
axes.set_title('Force-time history')
axes.grid()
axes.legend(loc='lower right')
plt.show()
```

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/harmonic-forcing/img1.jpg"
	alt="An example of a harmonic force (sinusoidally oscillating magnitude). | EngineeringSkills.com"
	width="543"
	height="239"
	caption="Fig 1. An example of a harmonic force (sinusoidally oscillating magnitude)."
/>

A sinusoidally oscillating force may at first appear like an odd choice for our first forced vibration analysis. But there is a very good reason to focus on harmonic excitation. A great many dynamic loads are in fact periodic, meaning the force magnitude pattern repeats at a fixed time interval. A harmonic force (sine wave) is a simple mathematical representation of a periodic force. As such we can (and will) develop closed-form analytical solutions for the behaviour of a system subject to a harmonic force.

Another related reason to focus on harmonic excitation is that many real-world forces that do not necessarily appear periodic (or indeed harmonic) can be represented by the summation of a series of sinusoidal forces, also known as harmonics. The frequencies and magnitudes of these harmonic forces can be obtained using a mathematical process known as Fourier decomposition.

While Fourier decomposition is beyond the scope of this course, it's important to appreciate that once you know how to simulate the influence of harmonic excitation, with the aid of Fourier decomposition, you can simulate the dynamic influence of much more complex force-time histories. So, on this basis, understanding how structures respond to harmonic loading is far more valuable than it might initially appear. In the next lecture we'll set about solving the equation of motion.


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