Question 2.2.1: Examine the units for computing the forced response of a dam......
Examine the units for computing the forced response of a damped system. Often the equation-of-motion quantities (forces) are given in Newtons, whereas the initial displacement and velocity are given in mm. It is important to write the initial conditions in the correct units.
First, examine units in the mass-normalized equation of motion as given in equation (2.27) repeated here:
\ddot{x} + 2ζw_{n} \dot{x} + w²_{n}x = f_{0} \cos wtThe units for f_{0} are N/kg = m/s², the units of acceleration, agreeing with the first term in the equation. The damping ratio ζ has no units, so the units of the damping term are those of ω_{n} or rad/s, which when multiplied by the velocity yields m/s². Likewise, the units of the natural frequency squared are rad²/s² so the stiffness term also has units of m/s². Thus equation (2.27) is consistent in terms of units.
Next, consider the solution. Since the amplitude of the particular solution, X, has the units of m (the units of f_{0} are N/Kg or m/s²)
X = \frac{f_{0}}{\sqrt{(w²_{n} – w²)² + (2ζw_{n}w)²}} \left(\frac{m/s²}{rad/s²}\right) = \frac{f_{0}}{\sqrt{(w²_{n} – w²)² + (2ζw_{n}w)²}}m
the initial condition x_{0} must also be given in m because the value of amplitude A contains the numerator term x_{0} – X cos(θ). The same is true for the phase angle ϕ, which also contains the initial velocity added to Xζω_{n} which will have units of m/s. Thus in solving for the force response of a damped system it is important that the initial conditions are stated in terms of the same units that the equation of motion is expressed in.