
The dynamic magnification factor (DMF) is the ratio of the amplitude of steady-state dynamic response to the deflection due to a static load of the same magnitude. This is a very helpful metric because it allows us to easily identify the degree to which the dynamic nature of the force amplifies the structural response. We obtained the amplitude of the steady-state response in the previous lecture as,

$$
|u_p(t)|=\frac{P_0}{k}\frac{1}{\sqrt{(1-\beta^2)^2 + (2\xi\beta)^2}}
\tag{1}
$$

Therefore we can write the dynamic magnification factor as,

$$
DMF = \frac{|u(t)|}{\frac{P_0}{k}} = \frac{1}{\sqrt{(1-\beta^2)^2 + (2\xi\beta)^2}}
\tag{2}
$$

Recall also from the previous lecture that we obtained the phase lag between force and response as,

$$
\phi = \tan^{-1}\frac{2\xi\beta}{1-\beta^2}
\tag{3}
$$

The DMF and phase of the response $\phi$ are functions of the forcing frequency, the system's natural frequency and the level of damping in the system. Taken together the DMF and $\phi$ allow us to fully determine the steady-state response of the system provided we know the frequency and amplitude of the harmonic exciting force.

We can get a better sense for the variation by graphing these quantities. First, we define a range of frequency ratios $(\omega/\omega_n)$ and damping values.

```python
beta = np.linspace(0,2,500) #Range of frequency ratios
Xi = np.array([0.05, 0.1, 0.2, 0.5]) #Range of damping ratios
```

Next we plot the DMF within a `for` loop so that each iteration through the loop plots the DMF for a different damping ratio.

```python
#Plot DMF
fig = plt.figure()
axes = fig.add_axes([0.1,0.1,1.5,1])

#Loop through each damping ratio and plot the DMF as a function of beta
for xi in Xi:
    DMF = 1/np.sqrt((1-beta**2)**2 + (2*xi*beta)**2)
    Label = str(xi*100) + '% damping'
    axes.plot(beta,DMF,'-',label=Label)

axes.set_xlim([0,2])
axes.set_xlabel('Frequency Ratio, $\omega/\omega_n$')
axes.set_ylabel('DMF')
axes.set_title('Dynamic Magnification Factor')
axes.grid()
axes.legend(loc='upper right')
plt.show()
```

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/dynamic-magnification-factor-and-resonance/img1.jpg"
	alt=" Dynamic magnification factor. | EngineeringSkills.com"
	width="982"
	height="489"
	caption="Fig 1. Dynamic magnification factor."
/>

This graph gives us a very easy to interpret picture of the DMF's variability with damping and frequency ratio. There are two headlines here; first, we recognise the phenomenon of resonance. When the forcing frequency is close to the natural frequency of the SDoF system, the amplification of the response is at a maximum. This peak amplification actually occurs when,

$$
\beta = \sqrt{1-2\xi^2}
\tag{4}
$$

The second observation is that increased damping dulls or inhibits the magnifying influence of the dynamic load. However, since most civil engineering structures have damping levels less than about $5\:\%$, the magnifying influence of a dynamic load can be quite high with the resonance case presenting a particular challenge.

Next we can visualise the behaviour of $\phi$ by repeating the same code structure from above.

```python
#Plot Phase Angle
fig = plt.figure()
axes = fig.add_axes([0.1,0.1,1.5,1])

#Loop through each damping ratio and plot the phase angle as a function of beta
for xi in Xi:
    angle = np.arctan2((2*xi*beta),(1-beta**2)) #Angle in radians
    angleDeg = angle*180/math.pi #Angle in degrees
    Label = str(xi*100) + '% damping'
    axes.plot(beta,angleDeg,'-',label=Label)

axes.set_xlim([0,2])
axes.set_ylim([0,180])
axes.set_xlabel('Frequency Ratio, $\omega/\omega_n$')
axes.set_ylabel('Phase Angle in Degrees')
axes.set_title('Phase angle between force and response')
axes.grid()
axes.legend(loc='lower right')
plt.show()
```

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/dynamic-magnification-factor-and-resonance/img2.jpg"
	alt="Phase angle between force and response. | EngineeringSkills.com"
	width="991"
	height="489"
	caption="Fig 2. Phase angle between force and response."
/>

We note that for values of $\beta < 1$, the phase lag between force and response is less than $90^{\circ}$ but as $\beta$ passes through $1$, the component of the response in phase with the force is eliminated - refer to the phase diagram introduced in the previous lecture and repeated below.

This means that the only component of the response remaining is that component that is always $90^{\circ}$ behind the applied force. As $\beta$ increases beyond $1$, the phase lag grows beyond $90^{\circ}$.

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/dynamic-magnification-factor-and-resonance/img3.jpg"
	alt="Two components of the steady-state response represented as rotating vectors on the complex plane. | EngineeringSkills.com"
	width="400"
	height="397"
	caption="Fig 3. Two components of the steady-state response represented as rotating vectors on the complex plane."
/>

Referring to figure 2, we can also see that influence of increased damping is to smooth the transition in phase lag, particularly around the resonance case when $\beta\approx 1$.

### Resonance

As we've seen above from the DMF, when the excitation frequency approaches the natural frequency of the system, the steady-state amplitude reaches a maximum. This condition is known as resonance and can cause a dynamic response far in excess of the equivalent static force response.

In civil engineering structures a resonance condition must be avoided. We can see the influence of resonance on bridges that have natural frequencies that coincide with the typical walking or pacing frequency of pedestrians.

In these relatively rare situations, large bridge oscillations can develop, leading to feedback between the occupying crowd and bridge response. The classic example of this type of positive feedback loop is the London Millennium Footbridge which experienced large amplitude lateral oscillations on its opening day in 2000.

Over the next several lectures we will work our way through an example question that allows you to implement much of what you've learned so far in this section. As always, try to work your way through the question independently before working through the next few solution lectures. The question will be presented at the start of the next lecture.


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