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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) = [itex]ψ_{1}(x)[/itex][itex]q_{1}(t)[/itex] + [itex]ψ_{2}(x)[/itex][itex]q_{2}(t)[/itex]

[itex]ψ_{1}(x)[/itex] = [itex](x/L)^{2}[/itex]; [itex]ψ_{2}(x)[/itex] = [itex](x/L)^{3}[/itex]

In this case, the generalized coordinates [itex]q_{i}[/itex] have the units of length.

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

Thanks

yogesh

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# Assumed modes and their units

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