
In this lecture we'll generate an animation of the structural response in much the same way that we animated the mode shapes previously. This is completely optional but it's nice to spend a little time bringing the system to life once you've done the hard work of analysing it. We start by importing the required animation functionality.

```python
from matplotlib.animation import FuncAnimation
import matplotlib.patches as patch #For plotting polygons
%matplotlib notebook
```

```python
h1 = 6.5 #(m) Story height 0-1
h2 = 4.5 #(m) Story height 1-2
h3 = 4 #(m) Story height 2-3

#Scale displacements to make them visible
scale = 50
u1S = scale*u1
u2S = scale*u2
u3S = scale*u3
```

```python
#Animation parameters
delta_t = 0.005
animLength = 10 #(sec)
frameRate = 10 #(5,10,20) frames per second (too high and animation slows down)
plotInterval = 1/frameRate #(sec) time between frame plots
dataInterval = plotInterval/delta_t #Plot moving elements every 'dataInterval-th' point

th = 0.5 #Floor thickness
wth = 6 #Floor width

#Set up plots
fig, (ax1) = plt.subplots(nrows=1,ncols=1, figsize=(10, 6))
fig.gca().set_aspect('equal', adjustable='box')
ax1.set_xlabel('Disp (m)')
ax1.set_ylabel('Height (m)')
ax1.set_title('Total response\n ($\omega_F$ =  {one} rads/sec)'.format(one=round(w,1)))
ax1.set_xlim([-5,5])
ax1.grid()

#Plot initial state
colL, = ax1.plot([-wth/2,-wth/2,-wth/2,-wth/2],[0, h1, h1+h2, h1+h2+h3],'k',lw=2)
colR, = ax1.plot([wth/2,wth/2,wth/2,wth/2],[0, h1, h1+h2, h1+h2+h3],'k',lw=2)

#Floor plate m1
pt1 = np.array([-wth/2,h1-th/2]) #Bottom left
pt2 = np.array([-wth/2, h1+th/2]) #Top left
pt3 = np.array([wth/2, h1+th/2]) #Top right
pt4 = np.array([wth/2, h1-th/2]) #Bottom right
xm1 = np.array([pt1[0],pt2[0],pt3[0],pt4[0]]) #x-coords
ym1 = np.array([pt1[1],pt2[1],pt3[1],pt4[1]]) #y-coords

#Draw the patch representing floor mass 1
patchM1 = patch.Polygon(xy=list(zip(xm1,ym1)),alpha=1, facecolor='#33cc99', edgecolor='black')
ax1.add_patch(patchM1)

#Floor plate m2
pt1 = np.array([-wth/2,h1+h2-th/2]) #Bottom left
pt2 = np.array([-wth/2,h1+h2+th/2]) #Top left
pt3 = np.array([wth/2, h1+h2+th/2]) #Top right
pt4 = np.array([wth/2, h1+h2-th/2]) #Bottom right
xm2 = np.array([pt1[0],pt2[0],pt3[0],pt4[0]]) #x-coords
ym2 = np.array([pt1[1],pt2[1],pt3[1],pt4[1]]) #y-coords

#Draw the patch representing floor mass 1
patchM2 = patch.Polygon(xy=list(zip(xm2,ym2)),alpha=1, facecolor='#33cc99', edgecolor='black')
ax1.add_patch(patchM2)

#Floor plate m3
pt1 = np.array([-wth/2,h1+h2+h3-th/2]) #Bottom left
pt2 = np.array([-wth/2,h1+h2+h3+th/2]) #Top left
pt3 = np.array([wth/2, h1+h2+h3+th/2]) #Top right
pt4 = np.array([wth/2, h1+h2+h3-th/2]) #Bottom right
xm3 = np.array([pt1[0],pt2[0],pt3[0],pt4[0]]) #x-coords
ym3 = np.array([pt1[1],pt2[1],pt3[1],pt4[1]]) #y-coords

#Draw the patch representing floor mass 1
patchM3 = patch.Polygon(xy=list(zip(xm3,ym3)),alpha=1, facecolor='#33cc99', edgecolor='black')
ax1.add_patch(patchM3)

#Function to animate plot objects
def animate(i):
    frm = int(i*dataInterval)

    #Structure
    xm1 = u1S[frm] + np.array([pt1[0],pt2[0],pt3[0],pt4[0]]) #x-coords
    patchM1.set_xy(list(zip(xm1,ym1)))
    xm2 = u2S[frm] + np.array([pt1[0],pt2[0],pt3[0],pt4[0]]) #x-coords
    patchM2.set_xy(list(zip(xm2,ym2)))
    xm3 = u3S[frm] + np.array([pt1[0],pt2[0],pt3[0],pt4[0]]) #x-coords
    patchM3.set_xy(list(zip(xm3,ym3)))


    colL.set_data([-wth/2,-wth/2+u1S[frm],-wth/2+u2S[frm],-wth/2+u3S[frm]],[0, h1, h1+h2, h1+h2+h3])
    colR.set_data([wth/2,wth/2+u1S[frm],wth/2+u2S[frm],wth/2+u3S[frm]],[0, h1, h1+h2, h1+h2+h3])

    return

#Function to generate the animation
myAnimation = FuncAnimation(fig,
                            animate,
                            frames=int(1 + (animLength/plotInterval) ),
                            interval=plotInterval*1000,  #milliseconds
                            blit=True,
                            repeat=True)

plt.show()
myAnimation.save('Total driven responses.gif')
```

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/visualising-the-forced-vibration-response/img1.gif"
	alt="Animated mode shapes for mode 1 (left) and mode 2 (right). | EngineeringSkills.com"
	width="1000"
	height="600"
/>
<Caption>
	Fig 1. Animation of structural response when a harmonic force with frequency
	$2\pi\:rads/s$ is applied to mass (floor) 3.
</Caption>

### Exploring different forcing frequencies

Now that we have completed the analysis and animation of our structure, it's worth playing with the forcing frequency to highlight its influence on the response. The original forcing frequency was $2\pi$ which was close to resonance with the first or fundamental mode of vibration. As a result, the first mode dominated the response of the structure.

However, if we change the forcing frequency to say $15\:rads/s$, we can see in Fig 2 below that we're close to resonance with the second mode and as such, it dominates the response. We can see this in the completely different displacement pattern of the structure, Fig. 2.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/visualising-the-forced-vibration-response/img2.gif"
	alt="Animation of structural response when a harmonic force with frequency 15\:rads/s is applied to mass (floor) 3 | EngineeringSkills.com"
	width="1000"
	height="600"
/>
<Caption>
	Fig 2. Animation of structural response when a harmonic force with frequency
	$15\:rads/s$ is applied to mass (floor) 3.
</Caption>

Let's remind ourselves of the three mode shapes, Fig. 3. We can clearly see that the response is being governed by mode 2.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/visualising-the-forced-vibration-response/img3.gif"
	alt="Mode shapes for mode 1, 2 and 3 (left to right). | EngineeringSkills.com"
	width="1000"
	height="500"
	caption="Fig 3. Mode shapes for mode 1, 2 and 3 (left to right)."
/>

Obviously with a forcing frequency of $15\:rads/s$ the first mode of vibration alone is no-longer sufficient to approximate the behaviour of the structure; a single mode representation would clearly need to use the second mode.

We could repeat the test and demonstrate excitation of the third mode too, simply by incresing the excitation frequency closer to the third modal frequency. All of this should highlight how useful it is to be able to decompose a structure's dynamic response into its fundamentals modes. It gives us the information we need to judge how likely an anticipated forcing frequency is to excite our structure in service.

Remember that the modes also give us important information about _where_ on a structure is susceptible to excitation. Consider mode 2 $(\omega_2 = 16\:rads/s)$ and the influence of applying a harmonic force to mass 2 (second floor) versus applying it to mass 3 (third floor) as we did above. The mode shape tells us that mode 2 will be excited much more by the force applied to mass 3 (or 1). We can easily show this by applying a force of $15\:rads/s$ to mass 2 and observing the reduced amplitude response.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/visualising-the-forced-vibration-response/img4.gif"
	alt="Fig 4. Animation of structural response when a harmonic force with frequency 15\:rads/s is applied to mass (floor) 2. | EngineeringSkills.com"
	width="1000"
	height="600"
/>

<Caption>
	Fig 4. Animation of structural response when a harmonic force with frequency
	$15\:rads/s$ is applied to mass (floor) 2.
</Caption>


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