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

Similarly the velocity is given by:

2
Velocity = u s+1 = u + u ∆t 1 − γ + ∆t γ u s+1
s
s

Where, γ is a constant of value (1/2) or 0.5. Substituting the value of γ in the above
expression, we get:

Velocity = u s+1 = u + u ∆t 1 − 0.5 + ∆t 0.5 u s+1
s
s

1
Velocity = u s+1 = u + u ∆t + ∆t u s+1 = u + (average acceleration ∗ ∆t)
s 2 s s

The above expression is comparable to [v = U + at]

Since, this scheme is for an implicit analysis, the choice of a larger time step does not affect the
accuracy of the solution. This scheme is unconditionally stable.

For an explicit analysis, a larger time step adversely affects the accuracy of the solution. Further,

the choice of time step also determines the damping introduced numerically. Therefore, if the
analyst were to assume a physical damping, then, it must be well justified, as, the combination of
physical and numerical damping can overdamp the system, leading to inaccurate computation of
response.

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