- #1

lalala1plus1

I am trying to derive the (Euler-Bernoulli) beam equation of a composite beam to determine the vibration and natural frequencies (up to 3 modes) of the beam.

The composite beam can be modeled as in the picture below:

In order to apply the general equation:

(EI)

_{eff}w'''' = m'

_{eff}(d

^{2}w/dt

^{2})

in which

(EI)

_{eff}: Effective Flexural Rigidity of the composite beam

m'

_{eff}: Effective mass per unit length of the composite beam

I know that for pure layered composite beam (with n layers), we can simply add up the flexural rigidity and mass per unit length of each layer: (EI)

_{eff,layered}= Σ E

_{n}I

_{n}and m'

_{eff,layered}= Σ ρ

_{n}b

_{n}t

_{n}

However, for my case, I have 3 different materials laminated in parallel in the middle layer. Is there any way to get the effective EI and m' of this layer, so that I can add them with those from the top and bottom layer to find the total effective flexural rigidity and mass per unit length of the whole beam?

I am urgently looking forward for explaination, and if better, with literature suggestion, which I can read it in detail.

Thank you very much for the help!