
The focus of this lecture is to get a better understanding of the relationship between support motion and the motion transmitted into the structure; the so-called transmissibility of the system. Recall from the last lecture that for a SDoF system subject to a harmonic support motion, the (absolute) steady-state response of the SDoF mass was given by,

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

where,

$$
D = U_0 k \sqrt{1+(2\xi\beta)^2}
\tag{2}
$$

We know from our earlier study of harmonic excitation that the steady-state amplitude is given by,

$$
U = \frac{D}{k}\left[\frac{1}{\sqrt{(1-\beta^2)^2+(2\xi\beta)^2}} \right]
\tag{3}
$$

After we substitute in $D$ we get,

$$
U = U_0\left[\frac{\sqrt{1+(2\xi\beta)^2}}{\sqrt{(1-\beta^2)^2+(2\xi\beta)^2}} \right]
\tag{4}
$$

We can now divide across by the amplitude of support motion, $U_0$ to obtain the transmissibility, $T$,

$$
\boxed{ T = \frac{U}{U_0} = \frac{\sqrt{1+(2\xi\beta)^2}}{\sqrt{(1-\beta^2)^2+(2\xi\beta)^2}} }
\tag{5}
$$

So, the transmissibility is simply the ratio of the amplitude of SDoF motion to the amplitude of support motion. We can see that transmissibility is analogous to the **Dynamic Magnification Factor** for a SDoF system subject to a harmonic force. This is not surprising since, from the point of view of the SDoF system, harmonic support motion is _experienced_ as a harmonic inertia force.

The dynamic magnification factor is a measure of the effectiveness of a harmonic force in exciting a structure beyond its equivalent static deflection. Transmissibility is a measure of how effectively base motion is transmitted into the structure. As usual, we will derive more understanding from the equation if we plot it.

### Plotting transmissibility against frequency ratio

```python
#Generate a vector of frequency ratios
max_B = 2 #Max value of Beta
delta_B = 0.01 #Step size in range of Beta values
nPoints = max_B/delta_B #Number of datapoints in the range
B = np.linspace(0,max_B, int(nPoints)) #Vector of Beta values
Xi = [0.05, 0.1, 0.2, 0.33, 0.5] #List of damping ratios

#Set up the figure
fig = plt.figure()
axes = fig.add_axes([0.1,0.1,2,1.5])
axes.plot([math.sqrt(2), math.sqrt(2)],[0,3],'--')
axes.set_xlim([0,max_B])
axes.set_ylim([0,6])
axes.set_xlabel('Frequency ratio $\\beta$')
axes.set_ylabel('Transmissibility, T')
axes.set_title('Transmissibility, T')
axes.grid()

#For each damping ratio in the list, add a plot to the figure
for xi in Xi:
    damping = xi*100 #percentage damping (for label)

    T = np.sqrt(1+(2*xi*B)**2) / np.sqrt(((1-B**2)**2+(2*xi*B)**2))
    axes.plot(B,T, label = "{A}% damping".format(A=damping))
    axes.legend()
```

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/motion-transmissibility-and-isolation/img1.jpg"
	alt="Plot of transmissibility versus frequency ratio for a range of damping ratios. | EngineeringSkills.com"
	width="1000"
	height="541"
	caption="Fig 1. Plot of transmissibility versus frequency ratio for a range of damping ratios."
/>

At this point we can make some observations about transmissibility. As expected, for a frequency ratio $\beta\approx 1$, we observe the maximum transmissibility. Also, a special boundary exists at $\beta= \sqrt{2}$, indicated by the vertical dashed line in the plot above.

- When $\beta < \sqrt{2}$, damping increases the isolation effect. In other words, increasing damping reduces transmission of support motion into the structure
- When $\beta>\sqrt{2}$, increasing damping has the counter-intuitive effect of increasing transmissibility.

The study of support motion and transmissibility for SDoF systems sets us up well for MDoF system analysis to come. We'll continue our study of SDoF systems in the next lecture and consider relative base motion. We'll see how this helps us simulate structural response when we can't rely on closed-form solutions.


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