# Assumed modes and their units

1. May 28, 2013

### boeing_737

Hi,

I am trying to understand the assumed-modes approach to solving vibration problems. For example, the transverse deformation of a cantilever beam is using two assumed modes is given as

ζ(x,t) = $ψ_{1}(x)$$q_{1}(t)$ + $ψ_{2}(x)$$q_{2}(t)$

$ψ_{1}(x)$ = $(x/L)^{2}$; $ψ_{2}(x)$ = $(x/L)^{3}$

In this case, the generalized coordinates $q_{i}$ have the units of length.

My question is, is it necessary for the assumed modes to be dimensionless? For instance, can we have : $ψ_{1}(x)$ = $x^{2}$; $ψ_{2}(x)$ = $x^{3}$, in which case $q_{1}$ has the unit [1/length] and $q_{2}$ has [1/length$^{2}$]

Thanks
yogesh

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