
In this lecture we'll utilise the formulas we've developed so far to determine the dynamic characteristics of a SDoF system. Make sure to try and answer the question on your own before working through the solution below.

Consider the spring-mass-damper system below. The mass is displaced vertically by $10 \:mm$ and allowed to oscillate freely. The next oscillation peak is noted to be $7.3 \:mm$.

The system has the following parameters:

- Mass, $M = 150 \:kg$

- Spring stiffness, $k = 12\times 10^3 \: N/m$

<Image
	src="/images/courses/fundamentals-of-structural-dynamics-with-python/worked-example-1/img1.jpg"
	alt="Freely oscillating spring-mass-damper system. | EngineeringSkills.com"
	width="543"
	height="239"
	caption="Fig 1. Freely oscillating spring-mass-damper system."
/>

Determine the following dynamic parameters:

1. The natural frequency of the undamped system (in $rads/s$ and $Hz$)

2. The oscillation period of the undamped system

3. The logarithmic decrement

4. The damping ratio

5. The damping coefficient

6. The damped natural frequency (in $rads/s$)

We can start by establishing the constants provided in the question and then working down through the parameters one by one.

```python
# Constants
m = 150 # (kg) Mass
k = 12*10**3 # (N/m) Stiffness
u1 = 10 # (mm) Peak 1
u2 = 7.3 # (mm) Peak 2
```

### 1. Natural Frequency, $\omega_n$ & $f_n$

$$
\omega_n = \sqrt{\frac{k}{m}}
\tag{1}
$$

$$
f_n = \frac{\omega_n}{2\pi}
\tag{2}
$$

```python
# Natural frequency and period
omega_n = round(math.sqrt(k/m),3) # (rads/sec) Angular natural frequency
fn = round(omega_n/(2*math.pi),3) # (Hz) Natural frequency
print(f"1.0(a) The angular natural frequency is {omega_n} radians per second")
print(f"1.0(b) The natural frequency is {fn} Hz")
```

- `1.0(a) The angular natural frequency is 8.944 radians per second`

- `1.0(b) The natural frequency is 1.423 Hz`

### 2. The oscillation Period, $T$

$$
T=\frac{1}{f_n}
\tag{3}
$$

```python
T = round(1/fn,2)
print(f"2.0 The period is {T} seconds")
```

- `2.0 The period is 0.7 seconds`

### 3. Logarithmic decrement, $\delta$

$$
\delta = \ln{\frac{U_n}{U_{n+1}}}
\tag{4}
$$

```python
delta = round(np.log(u1/u2),3)
print(f"3.0 The logarithmic decrement is {delta}")
```

- `3.0 The logarithmic decrement is 0.315`

### 4. Damping ratio, $\xi$

$$
\xi \approx \frac{\delta}{2\pi}
\tag{5}
$$

```python
xi = round(delta/(2*math.pi),3)
print(f"4.0 The damping ratio is {xi} or {round(xi*100,2)} percent of critical damping")
```

- `4.0 The damping ratio is 0.05 or 5.0 percent of critical damping`

### 5. Damping coefficient, $c$

$$
c = 2\xi m \omega_n
\tag{6}
$$

```python
c = round(xi*2*m*omega_n,2)
print(f"5.0 The damping coefficient is {c} Ns/m")
```

- `5.0 The damping coefficient is 134.16 Ns/m`

### 6. Damped natural frequency, $\omega_d$

$$
\omega_d = \omega_n\sqrt{1-\xi^2}
\tag{7}
$$

```python
omega_d = round(omega_n*math.sqrt(1-(xi**2)),3)
print(f"6.0 The damp natural frequency is {omega_d} radians per second")
```

- `6.0 The damp natural frequency is 8.933 radians per second`

Although it wasn't asked for in the question, we can easily plot the response using the equation for damped free vibration that we derived back in [lecture 15](/course/fundamentals-of-structural-dynamics-with-python/under-damped-systems),

$$
u(t) = e^{-\xi\omega_n t}\left[\frac{v_0 + u_0\xi \omega_n}{\omega_d}\sin(\omega_d t) + u_0\cos(\omega_d t) \right]
\tag{8}
$$

```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

# Initial conditions
u0 = -0.01
v0 = 0

# Response
u = math.e**(-xi*omega_n*t)*(((v0+u0*xi*omega_n)/omega_d)*np.sin(omega_d*t) + u0*np.cos(omega_d*t))
```

```python
#Plotting
fig = plt.figure()
axes = fig.add_axes([0.1,0.1,2,2])
axes.plot(t,u,label='Under-damped response')
axes.set_xlim([0,5])
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/worked-example-1/img2.jpg"
	alt="Fig 2. Damped free vibration of the system from the specified initial conditions. | EngineeringSkills.com"
	width="1000"
	height="674"
	caption="Fig 2. Damped free vibration of the system from the specified initial conditions."
/>

We note that as expected, the second peak has an amplitude of $7.3\:mm$. In the next lecture we'll work through a slightly more challenging question.


Please log in or enroll to continue

Please log in or enroll to continue