
The logarithmic decrement can be used to estimate the level of damping in a structure based on its observed oscillation behaviour. This is a helpful tool that can be used to confirm the in-service level of structural damping. This is a particularly valuable since there is no easy way to _analytically_ determine a structure's damping ratio.

The logarithmic decrement, $\delta$ is expressed as the natural log of the ratio of two successive oscillation peaks with amplitudes $U_i$ and $U_{i+1}$,

$$
\delta=\ln\left(\frac{U_i}{U_{i+1}}\right)
\tag{1}
$$

So, we can calculate $\delta$ directly by evaluating two successive oscillation peaks. The task now is to relate this quantity to the damping ratio, $\xi$.

If we assume the two peaks $U_i$ and $U_{i+1}$ are separated by the damped oscillation period $T_D$, they can be represented as,

$$
U_i=C\:e^{-\xi\omega_nt_i}
\tag{2}
$$

$$
U_{i+1} = C\:e^{-\xi\omega_n(t_i+T_D)}
\tag{3}
$$

Substituting these expressions into our expression for $\delta$,

$$
\delta=\ln(e^{-\xi\omega_nt_i})-\ln(e^{-\xi\omega_n(t_i+T_D)})
\tag{4}
$$

$$
\delta=\xi\omega_nT_D
\tag{5}
$$

Remember, $T_D$ is the time between two successive peaks and is therefore the period of the damped oscillation. Therefore it can be expressed as,

$$
T_D=\frac{2\pi}{\omega_d} = \frac{2\pi}{\omega_n\sqrt{1-\xi^2}}
\tag{6}
$$

The relationship between the observed logarithmic decrement $\delta$ and the damping ratio $\xi$ is therefore given by,

$$
\boxed{
\delta=\frac{2\pi\xi}{\sqrt{1-\xi^2}}
}
\tag{7}
$$

However, when the damping ratio is small and $\omega_n\approx\omega_d$, the relationship simplifies to,

$$
\boxed{
\delta\approx2\pi\xi
}
\tag{8}
$$

We can test this by identifying the damping ratio for one of our under-damped systems from [lecture 15](/course/fundamentals-of-structural-dynamics-with-python/under-damped-systems). We can generate a damped oscillation with $\xi=5\%$ and then confirm this value directly from the oscillation time-history.

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

#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

#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

# 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))
```

Now plot the response and identify two oscillation peaks at $t=2.5\:s$ and $t=5.0\:s$.

```python
ind = int(2.5/delta_t) #Index of displacement value at 2.5 sec
U1 = u_1[ind] #Displacement at 2.5 sec

ind = int(5/delta_t) #Index of displacement value at 5 sec
U2 = u_1[ind] #Displacement at 5 sec

#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(2.5,U1,'ro', markersize=5)
axes.plot(5,U2,'ro', markersize=5)
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.show()
```

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/logarithmic-decrement/img1.jpg"
	alt="Damped free vibration with two successive oscillation peaks identified. | EngineeringSkills.com"
	width="1000"
	height="681"
	caption="Fig 1. Damped free vibration with two successive oscillation peaks identified."
/>
First we need to calculate $\delta$.

```python
delta = math.log(U1/U2)
```

Now we can determine the damping ratio using equation 8.

```python
Xi = delta/(2*math.pi)
print(f'The damping ratio is {round(Xi*100,2)} %')
```

The damping ratio is $5.01 \%$

We can see that the damping ratio obtained directly from the oscillation time-history is in agreement with what we know the level of damping to be.

In the next two lectures we'll work through example questions that test what we've learned so far.


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