
For under-damped systems,

$$
c^2-4mk<0
\tag{1}
$$

As such, the roots of the characteristic equation are complex and with some rearranging, they can be expressed as,

$$
\lambda_1 = \frac{-c}{2m} +i\sqrt{\frac{k}{m}-\left(\frac{c}{2m}\right)^2}
\tag{2}
$$

$$
\lambda_2 = \frac{-c}{2m} -i\sqrt{\frac{k}{m}-\left(\frac{c}{2m}\right)^2}
\tag{3}
$$

Remember, these roots appear in our initial assumed solution, $u(t) = e^{\lambda t}$. We've also seen in previous lectures that the general solution can be expressed as,

$$
u(t) = A e^{\lambda_1 t} + B e^{\lambda_2 t}
\tag{4}
$$

So, substituting our expressions for $\lambda_1$ and $\lambda_2$ into the general solution yields,

$$
u(t) = \exp\left(\frac{-ct}{2m}\right) \left[ A \exp\left\{ i\:t \sqrt{\frac{k}{m} - \left(\frac{c}{2m}\right)^2} \right\} + B\exp\left\{-i\:t \sqrt{\frac{k}{m} - \left(\frac{c}{2m}\right)^2}\right\} \right]
\tag{5}
$$

Again using Euler's formula just as we did for the free vibration case in [lecture 11](/course/fundamentals-of-structural-dynamics-with-python/undamped-free-vibration), this reduces to,

$$
\boxed{u(t) = e^{-\xi \omega_n t}\left[C\sin\left(\omega_d t \right) + D\cos\left(\omega_d t \right) \right]}
\tag{6}
$$

where, $\omega_d$ is defined as the damped angular natural frequency and is given by,

$$
\boxed{\omega_d = \sqrt{\frac{k}{m} - \left(\frac{c}{2m}\right)^2}}
\tag{7}
$$

Our general solution has two unknown constants of integration, $C$ and $D$. We can obtain numerical values for these in the usual way, by using knowledge of the system initial conditions to determine two equations that can be used to identify numerical values for the two unknown constants.

Another approach is to assign the initial position and velocity of the system to variables, $u_0$ and $v_0$ respectively. Using this approach we determine expressions for $C$ and $D$ rather than numerical values. This way we retain the variables and can substitute actual values in later for $u_0$ and $v_0$. We can demonstrate this approach now.

### Initial condition #1

Replacing the left side of equation 6 with $u_0$ and substituting $t=0$ into the right side yields,

$$
D = u_0
\tag{8}
$$

### Initial condition #2

To make use of the second initial condition, we need to first differentiate equation 6 to get the corresponding expression for velocity. First let's rewrite equation 6,

$$
u(t) = C\underbrace{\left[e^{-\xi\omega_nt}\:\sin(\omega_d t) \right]}_{f_1(t)} + D\underbrace{\left[e^{-\xi\omega_nt}\:\cos(\omega_d t) \right]}_{f_2(t)}
\tag{9}
$$

We've divided the equation into two functions so we can process the differentiation in two parts,

$$
\frac{\mathrm{d}}{\mathrm{d}t}f_1(t) = e^{-\xi\omega_nt}\omega_d\cos(\omega_dt) - \sin(\omega_dt)\: \xi\omega_n \: e^{-\xi\omega_nt}
\tag{10}
$$

$$
\frac{\mathrm{d}}{\mathrm{d}t}f_2(t) = -e^{-\xi\omega_nt}\omega_d\sin(\omega_dt) - \cos(\omega_dt)\:\xi\omega_n\:e^{-\xi\omega_nt}
\tag{11}
$$

Recombining these two expressions we obtain our expression for the velocity, $\dot{u}(t)$,

$$
\begin{split}
\dot{u}(t) =  & \:C\left[e^{-\xi\omega_nt}\omega_d\cos(\omega_dt) - \sin(\omega_dt)\: \xi\omega_n \: e^{-\xi\omega_nt} \right] + \\  & \:D\left[ -e^{-\xi\omega_nt}\omega_d\sin(\omega_dt) - \cos(\omega_dt)\:\xi\omega_n\:e^{-\xi\omega_nt}\right] \\
\end{split}
\tag{12}
$$

Now we can replace the left side of the equation with $v_0$ and substitute $t=0$ into the right side. When we also substitute in the expression we've already found for $D$ and rearrange for C$$ we finally get,

$$
C = \frac{v_0 + u_0\xi\omega_n}{\omega_d}
\tag{13}
$$

Now we can substitute these expressions for $C$ and $D$ back into equation 6 to reveal the final expression for the damped free oscillation,

$$
u(t) = e^{-\xi\omega_n t}\left[\frac{v_0 + u_0\xi \omega_n}{\omega_d}\sin(\omega_d t) + u_0\cos(\omega_d t) \right]
\tag{14}
$$

Using this expression, we can model the damped free vibration response as a function of the initial conditions and level of damping, $\xi$. We can start by defining an arbitrary initial position and velocity.

```python
#Define initial conditions
u0 = 1 #Initial position
v0 = 0 #Initial velocity
```

Before progressing any further, we need to import some libraries (we'll omit this step from the text in future lectures).

```python
import math #Basic math functionality
import numpy as np #Numpy for working with arrays
import matplotlib.pyplot as plt #Plotting functionality
```

Next we define some system parameters.

```python
#Define some parameters
m = 2000 #(kg) Mass
fn = 0.4 #(Hz) Natural Frequency
wn = 2*math.pi*fn #(rads/sec) Angular natural frequency
k = m*wn**2 #(N/m) Stiffness
```

We'll define a time vector that runs from $0\:s$ to $100\:s$ in steps of $0.01\:s$.

```python
#Define a time vector
tmax = 100 #(sec) Max time
delta_t = 0.01 #(sec) Time-step
nPoints = tmax/delta_t #Number of equally spaced data points
t = np.linspace(0,tmax, int(nPoints)) # Generate time vector
```

Now we can evaluate equation 14 assuming $5\%$ damping.

```python
# Response for 5% of critical damping (Under-damped system)
xi_1 = 0.05 #Damping ratio
wd_1 = wn*math.sqrt(1-xi_1**2) #(rads/sec) Damped angular natural frequency
u_1 = math.e**(-xi_1*wn*t)*(((v0+u0*xi_1*wn)/wd_1)*np.sin(wd_1*t) + u0*np.cos(wd_1*t))
```

To visualise the response of a critically-damped system (almost), we can construct a second response and set $\xi = 0.999$.

```python
# Response for 99.9% of critical damping (Critically damped system)
xi_2 = 0.999
wd_2 = wn*math.sqrt(1-xi_2**2)
u_2 = math.e**(-xi_2*wn*t)*(((v0+u0*xi_2*wn)/wd_2)*np.sin(wd_2*t) + u0*np.cos(wd_2*t))
```

Now it's just a matter of plotting both responses.

```python
#Plotting
fig = plt.figure()
axes = fig.add_axes([0.1,0.1,2,2])
axes.plot(t,u_1,label='Under-damped response')
axes.plot(t,u_2,label='Critically damped response')
axes.set_xlim([0,40])
axes.set_xlabel('Time (sec)')
axes.set_ylabel('Displacement (m)')
axes.set_title('Displacement-time history')
axes.grid()
axes.legend(loc='lower right')
plt.savefig('plot1.png',dpi=100, bbox_inches='tight', facecolor='w')
plt.show()
```

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/under-damped-systems/img1.jpg"
	alt="Free vibration response of under-damped and critically-damped systems. | EngineeringSkills.com"
	width="1000"
	height="681"
	caption="Fig 1. Free vibration response of under-damped and critically-damped systems."
/>

Notice the behaviour of the critically-damped system (orange). The system does not oscillate but decays exponentially to zero. The blue line exhibits a classic under-damped response, oscillating about the equilibrium position with the amplitude of oscillation decaying over time due to the sub-critical level of damping.

To get a better understanding of how the under-damped behaviour varies as a function of the level of damping, we can plot a range of different oscillation responses for different levels of damping. Calling our plot inside a simple `for` loop will help us do this efficiently.

```python
#Plot multiple damping levels
Xi = [0.05, 0.1, 0.2, 0.5, 0.75 ,0.99]

fig = plt.figure()
axes = fig.add_axes([0.1,0.1,2,2])
axes.set_xlim([0,40])
axes.set_xlabel('Time (sec)')
axes.set_ylabel('Displacement (m)')
axes.set_title('Displacement-time history')
axes.grid()

for xi in Xi:
    wd = wn*math.sqrt(1-xi**2)
    u = math.e**(-xi*wn*t)*(((v0+u0*xi*wn)/wd)*np.sin(wd*t) + u0*np.cos(wd*t))
    axes.plot(t,u,label=f"{xi*100} % damping")

axes.legend(loc='lower right')
plt.savefig('plot2.png',dpi=100, bbox_inches='tight', facecolor='w')
plt.show()
```

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/under-damped-systems/img2.jpg"
	alt="System response for a range of sub-critical damping ratios. | EngineeringSkills.com"
	width="1000"
	height="681"
	caption="Fig 2. System response for a range of sub-critical damping ratios."
/>

We can clearly see that as the damping ratio increases, the rate at which energy is dissipated from the system increases and the amplitude reduces more rapidly. Over the next two lectures, we will spend a little more time exploring damping as it has a big impact on the behaviour of the system.


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