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Document Title
Fundamentals of Stress and Vibration Chapter Title
[A Practical guide for aspiring Designers / Analysts] 2. Engineering Mechanics

ʹǤͷǤͶ ‘Ž‡ ‘ˆ “—ƒ–‹‘• ‘ˆ ‘–‹‘ ‹ ”ƒ•‹‡– ƒ† ‘-Ž‹‡ƒ” ƒŽ›•‡•

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

Š‡ ‡“—ƒ–‹‘ ‘ˆ ‘–‹‘ ‹• ‹–‡‰”ƒ–‡† •–‡’ „› •–‡’ ™‹–Š‘—– –”ƒ•ˆ‘”‹‰ –Š‡ ‡“—ƒ–‹‘•Ǥ Š‡ ‹†‡ƒ ‹•
–Šƒ–ǡ ƒ– ‡ƒ Š –‹‡ •–‡’ǡ –Š‡ ‡“—‹Ž‹„”‹— ‹• •ƒ–‹•ϐ‹‡† —†‡” –Š‡ ƒ –‹‘ ‘ˆ ‹‡”–‹ƒǡ †ƒ’‹‰ǡ •–‹ˆˆ‡••
ƒ† ‡š–‡”ƒŽ ˆ‘” ‡•Ǥ

Šƒ– ‹• š’Ž‹ ‹– ƒƒŽ›•‹•ǫ
Explicit methods do not involve the solution of a set of linear equations at each step. Basically, these
methods use the differential equation at time “t” to predict a solution at time “t+∆t”.

The equation of motion is given by, Newton’s II law.

F = mu + cu + ku

The equation of motion is solved in several time steps till the solution at time “t+∆t” is got. These
time steps have to be carefully determined, as they in turn determine the accuracy of the solution.

There are many direct integration methods available to solve the equation of motion. A few popular
ones are listed below.

Ø Central difference scheme
Ø The Newmark family of methods
Ø The Hilber, Hughes and Taylor method.
Ø The Houbolt method.
Ø The Wilson method.

By average slope/central difference method, the velocity and acceleration in the equation of motion
could be expressed in terms of displacement as follows:
Change in displacement
Velocity =
time interval ∆t
If we consider the time intervals to be t + ∆t and t − ∆t , we get:

u t+∆t − u t−∆t u t+∆t − u t−∆t
Velocity u = =
t + ∆t − t − ∆t 2∆t

Let us now find acceleration by considering the rate of change of velocity as follows:

u t+∆t − u t−∆t
Acceleration u =
t + ∆t − t − ∆t

Page 34 QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,