Advanced Analytical Methods: Mass-Spring System, Newton's Law of Cooling, Toughness of Stainless Steel

UNIVERSITY OF DERBY
COLLEGE OF ENGINEERING AND TECHNOLOGY
MODULE TITLE: Advanced Applied Analytical Methods 2
MODULE CODE: 5ME526
Year: - 2018-2019
Applications of Advanced Analytical Methods
Pranav D. Mirani

Assignment brief:
You must complete the following 4 investigations and select suitable techniques
to analyse the data you have been given to answer the problems posed.
Your report should clearly demonstrate how you have selected and applied
suitable advanced analytical techniques to analyse the data and how your
solutions address the initial problems. You should clearly explain your
methods, why they are suitable and what the relevance is of your outcomes.
Investigation 1: Mass-Spring System (25%)
This investigation uses the TecQuipment TM16 Range Universal Vibration
Apparatus. The following diagrams are taken from the TQ User Guide with
permission (Figure 1 and Figure 2).
Figure 1. Force against Deflection
Number all the solutions.

Figure 2. Setup for Mass-Spring Systems
a) Determine the spring constant k
An experiment is conducted where masses are suspended from a spring and
resulting deflections recorded as shown in Table 1.
Mass
(kg)
Force
(N)
Depth gauge reading
(mm)
Mean reading
(mm)
Extension
(mm)
Stiffness
(N/mm)
Loading Unloading
0.0 0 60 60 60 0 -
0.4 3.924 65 65 65 5 784.8
0.8 7.848 70 69 69.5 9.5 826.1053
1.2 11.772 74 74 74 14 840.8571
1.6 15.696 78 78 78 18 872
2.0 19.62 83 83 83 23 853.0435
2.4 23.544 87 87 87 27 872
2.8 27.468 92 91 91.5 31.5 872
3.2 31.392 97 96 96.5 36.5 860.0548

3.6 35.316 101 101 101 41 861.3659
4.0 39.24 105 105 105 45 872
Coil diameter 44.6 mm
Wire diameter 3.1 mm
Number of
coils
18
Table 1. Experimentally Measured Spring Deflections for Applied Masses
Complete/fill in the blank cells in the Table 1 with your calculation.
Extension = (gauge reading - gauge reading with no mass on the platform)
Plot a graph of extension (mm) against load (N) and determine the spring stiffness,
k in N/m.
0 5 10 15 20 25 30 35 40 45 50
0
5
10
15
20
25
30
35
40
45
0
3.924
7.848
11.772
15.696
19.62
23.544
27.468
31.392
35.316
39.24
Force vs. Extention
Mass vs Extention. Linear (Mass vs Extention.)
Extention (mm)
Force (N)

Mean stiffness of the spring = 851.4226532 N/m
b)
Figure 3. Mass-Spring Ooscillation Measurement
An experiment (Figure 3) is performed and the data is recorded in Table 2 below:
Mass Time for
20
oscillation
s (s)
Period
t (s)
Mass
(kg) 3.d.p. 3.d.p. (N)
0.8 6.84 0.342 0.11696 7.848 85.31901 0.014904
1.2 7.4 0.37 0.1369 11.77 78.86244 0.011629

2
1.6 7.925 0.3963 0.15701 15.69
6
73.63811 0.010003
2 8.15 0.4075 0.16606 19.62 71.60516 0.008464
2.4 8.48 0.424 0.17978 23.54
4
68.81864 0.007636
2.8 8.905 0.4453 0.19825 27.46
8
65.5342 0.007217
3.2 9.65 0.4825 0.23281 31.39
2
60.47482 0.007416
3.6 9.98 0.499 0.249 35.31
6
58.47515 0.007051
4 10.22 0.511 0.26112 39.24 57.10196 0.006654
Coil diameter: 44.6 mm
Wire diameter: 3.1 mm
Number of coils: 18
Table 2. Measured Time for 20 Oscillations for Series of Added Masses
Fill in the blank cells in the Table 2 with your calculation, and complete the
following tasks: