FVSDOF-SYS Example 2 - Natural period of spring max system

Consider a spring mass system with natural period of 0.45 [s]. Calculate the new period if a spring constant is icreased by 10, 50, 100, 200, 500\%. $$\begin{array}{rcl}{\tau }_1& =& 0.21[s]=2\pi \sqrt{\frac{m}{k}}\\ {\tau }_{10}& =& \frac{2\pi \sqrt{m}}{\sqrt{1.1k}}=\frac{2\pi \left(\frac{0.21\sqrt{k}}{2\pi }\right)}{\sqrt{1.1k}}=\frac{0.21\sqrt{k}}{\sqrt{1.1k}}=0.20022714374157435[s]\end{array}$$ $$\begin{array}{rcl}{\tau }_{20}& =& \frac{2\pi \sqrt{m}}{\sqrt{1.5k}}=\frac{2\pi \left(\frac{0.21\sqrt{k}}{2\pi }\right)}{\sqrt{1.5k}}=\frac{0.21\sqrt{k}}{\sqrt{1.5k}}=0.17146428199482247[s]\end{array}$$ $$\begin{array}{rcl}{\tau }_{30}& =& \frac{2\pi \sqrt{m}}{\sqrt{2k}}=\frac{2\pi \left(\frac{0.21\sqrt{k}}{2\pi }\right)}{\sqrt{2k}}=\frac{0.21\sqrt{k}}{\sqrt{2k}}=0.14849242404917495[s]\end{array}$$ $$\begin{array}{rcl}{\tau }_{40}& =& \frac{2\pi \sqrt{m}}{\sqrt{3k}}=\frac{2\pi \left(\frac{0.21\sqrt{k}}{2\pi }\right)}{\sqrt{3k}}=\frac{0.21\sqrt{k}}{\sqrt{3k}}=0.12124355652982141[s]\end{array}$$ $$\begin{array}{rcl}{\tau }_{50}& =& \frac{2\pi \sqrt{m}}{\sqrt{5k}}=\frac{2\pi \left(\frac{0.21\sqrt{k}}{2\pi }\right)}{\sqrt{5k}}=\frac{0.21\sqrt{k}}{\sqrt{5k}}=0.09391485505499116[s]\end{array}$$

Solution in Pyhton

For this example we will also need two libraries matplotlib and numpy.

import numpy as np
import matplotlib.pyplot as plt

We have a constant value (natural period) that we will define as:

tau_0 = 0.21

Then we will define the increase of the spring constant or spring stiffness as list:

Spring_Const = [1.1, 1.5, 2, 3, 5]

Then we will define the empty list named tau_res.

tau_res = []

Now we will create a for loop that will calculate new natural period and store its value into the taur_res list.

for i in range(len(Spring_Const)): 
    res = tau_0/np.sqrt(Spring_Const[i])
    tau_res.append(res)

The np.sqrt is the square root function from the numpy library. Now we can plot results in which we will show how natural period changes versus spring constant inscrease:

plt.rcParams["font.family"] = "Times New Roman"
SMALL_SIZE = 20
MEDIUM_SIZE = 22
BIGGER_SIZE = 24
plt.rc('font', size=SMALL_SIZE)          # controls default text sizes
plt.rc('axes', titlesize=SMALL_SIZE)     # fontsize of the axes title
plt.rc('axes', labelsize=MEDIUM_SIZE)    # fontsize of the x and y labels
plt.rc('xtick', labelsize=SMALL_SIZE)    # fontsize of the tick labels
plt.rc('ytick', labelsize=SMALL_SIZE)    # fontsize of the tick labels
plt.rc('legend', fontsize=SMALL_SIZE)    # legend fontsize
plt.rc('figure', titlesize=BIGGER_SIZE)  # fontsize of the figure title
SpringPER = [10,50,100,200,500]
plt.figure(figsize=(12,8))
plt.plot(SpringPER, tau_res)
plt.xlabel("Spring Constant Increase [%]")
plt.ylabel("Natural period [s]")
plt.grid(True)
plt.xlim(0,500)
plt.ylim(0,0.21)
plt.show()

Figure 1 - The natural period versus spring constant increase. As seen from Figure 1 as the spring constant or spring stiffness increases the natural period of the system decreases from initial 0.2002 up to 0.09 [s] when spring constant increased from 10 up to 500%.