Page 153 - C:\Users\trainee\AppData\Local\Temp\msoEAA3.tmp

Fundamentals of Stress and Vibration
[A Practical guide for aspiring Designers / Analysts] 1. Mathematics for Structural mechanics
ͳǤͳͲ ‹‰‡ ƒŽ—‡• ƒ† ‹‰‡ ‡ –‘”•

˜‡”› ”‡ƒŽ ƒ–”‹š ȋ™Š‘•‡ ‡Ž‡‡–• ƒ† †‡–‡”‹ƒ–• ƒ”‡ ”‡ƒŽȌ Šƒ• ”‡ƒŽ ”‘‘–•ǡ ™Š‹ Š †‡’‡†• ‘
–Š‡ ‘”†‡” ‘ˆ –Š‡ ƒ–”‹šǤ

šƒ’Ž‡ǣ ƒ ȋ͵ š ͵Ȍ ƒ–”‹š Šƒ• ͵ ”‡ƒŽ ”‘‘–•Ǥ

Š‡•‡ ”‘‘–• ƒ”‡ ƒŽŽ‡† –Š‡ ‹‰‡ ˜ƒŽ—‡• ‘” Šƒ”ƒ –‡”‹•–‹ ”‘‘–•Ǥ Š‡•‡ ƒ „‡ ’”ƒ –‹ ƒŽŽ›
‘””‘„‘”ƒ–‡† –‘ ’”‹ ‹’ƒŽ ˜ƒŽ—‡•ǡ •— Š ƒ•ǡ ƒ–—”ƒŽ ˆ”‡“—‡ ‹‡• ‘ˆ ƒ •›•–‡ǡ ’”‹ ‹’ƒŽ ‹‡”–‹ƒ• ‘ˆ ƒ
•›•–‡ǡ ‡– Ǥ

Mathematically, for a matrix, say, ‘A’, assuming ‘λ’ to be the root/Eigen value, we have:

Eigen value equation = det A − λI = 0 , where I is the identity matrix

Example 1: find the Eigen value of the matrix ‘A’ in equation (1.65).

50 10
A = - - - - (1.65)
10 25

50 10 λ 0 1 0
The Eigen value equation is given as: det − - - - - (1.66)
10 25 0 λ 0 1
50 − λ 10
Simplifying the equation . , we get: det - - - - (1.67)
10 25 − λ
The determinant of equation (1.67) is given as:

2
50 − λ 25 − λ − 100 = 0 = [1250 − 50λ − 25λ + λ − 100 = 0]
1150 − 75 + = 0 - - - - (1.68)
2
Rearranging the quadratic equation . we get: λ − 75λ + 1150
2
The roots of the quadratic equation are got as follows:

2
−b ± b − 4ac
, where, a = 1 , b = −75 and (c = 1150)
2a
2
2
−b + b − 4ac 75 + 75 − 4 1150
First root = = = λ = 53.51 - - - - (1.69)
1
2a 2
2
2
−b − b − 4ac 75 − 75 − 4 1150
Second Root = = = λ = 21.49 - - - - (1.70)
2a 2 2

QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,
Page 56