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

The velocity component in the above expression could be rewritten as change in displacement
divided by time;

u (t+∆t) − u t u − u t−∆t
∆t − t ∆t u − 2u − u t−∆t
Acceleration u = = (t+∆t) t 2
t + ∆t − t − ∆t 2 ∆t

Since we have computed velocity, acceleration and displacement, their respective expression
could be substituted in the equation of motion and simplified to get the value of u t+∆t , using

the displacement data of the previous step. The accuracy of this method depends on square of
the time step ∆t chosen.
2
Newmark (Implicit time step)

This method, the displacement computation for average acceleration and average velocity, say,
from step 1 (s) to step 2 (s+1), is given by:

1
2
2
Displacement = u s+1 = u + u ∆t + ∆t − β u + ∆t β u s+1
s
s
2 s

Where, β is a constant of value (1/4) or 0.25. Substituting the value of β in the above
expression, we get:

1
1
1
Displacement = u s+1 = u + u ∆t + ∆t − u + ∆t u s+1
2
2
s
s
2 4 s 4

∆t 2 u ∆t 2 u s+1
s
Displacement = u s+1 = u + u ∆t + +
s
s
2 2 2 2

∆t 2 u + u s+1
s
Displacement = u s+1 = u + u ∆t + 2 2

s
s

The above expression is comparable to:
1
Initial displacement u + ut + at
2
0
2
It can be observed that, the Newmark’s equation contains average acceleration, given by:

s
u + u s+1
Average acceleration =
2

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