
Now we'll confirm the validity of our Duhamel function by comparing the calculated response with the closed-form (analytical) solution for harmonic loading of a SDoF system. We could plot the complete closed-form response including the transient and steady-state components but for our validation exercise it will be sufficient to only plot the steady-state response, $u_s(t)$ given by,

$$
u_s(t) = \frac{P}{k}\left[ \frac{1}{(1-\beta^2)^2 + (2\xi\beta)^2} \right] \left[ (1-\beta^2)\sin(\omega t)-2\xi\beta\cos(\omega t) \right]
\tag{1}
$$

If you've studied fundamental structural dynamics, you will recognise this as the dimensionless form of the steady-state response equation - if this is completely unfamiliar and you don't like equations dropping out of thin air, [go here](/course/fundamentals-of-structural-dynamics-with-python).

```python
#Steady-state response
beta = wf/wd #Frequency ratio
k=m*wd**2 #(N/m) Stiffness

#Break equation into two for convenience
O = (P/k)*(1/((1-beta**2)**2 + (2*xi*beta)**2))
response_cf = O*((1-beta**2)*np.sin(wf*time) - 2*xi*beta*np.cos(wf*time))
```

```python
fig = plt.figure()
axes = fig.add_axes([0.1,0.1,2,1])
axes.plot(time,response, '-', label='Duhamel')
axes.plot(time,response_cf,'-',label='Closed-form')

#Housekeeping
axes.set_xlabel('time (s)')
axes.set_ylabel('Disp (m)')
axes.set_title('SDoF system response to harmonic loading')
axes.legend(loc='lower right')
axes.set_xlim([0,tMax])
plt.grid()
plt.show()
```

<Image
	src="/images/courses/simulating-crowd-induced-vibrations-using-the-duhamel-integral/validating-our-numerical-solution-code/img1.jpg"
	alt="Comparison of closed-form and numerical solution for harmonic excitation. | EngineeringSkills.com"
	width="1000"
	height="369"
	caption="Fig 1. Comparison of closed-form and numerical solution for harmonic excitation."
/>

We can see that both solutions agree once the transient response has decayed away. This gives us confidence in the Duhamel integral code we've written. This is quite a powerful tool we have at our disposal now. Although we only used it to calculate the response due to harmonic loading, we can feed it any forcing function and it will return the SDoF system response.

So, in the next phase of this project, we'll use our Duhamel integral function to do something a little more interesting. We're going to estimate the vertical vibration response of a footbridge to pedestrian traffic. Although, we'll focus on a simply-supported footbridge, the same principles can be applied to crowd-induced vibration of any structure, although there are some specific caveats we'll discuss along the way.


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