Let $\phi: \chi \to B$ be a proper holomophic submersion with smooth fiber $X$.


Then, we get local systems $R^i \phi\_\* \mathbb C$ and corresponding flat connection $(B, \bigtriangledown)$


In this case,there are tons of beautiful constructions even in such an elementary level: 


infinitesimal VHS, Mixed Hodge structure, Period mapping $P^{n,k} : B \to Grass(b^{n,k}, H^n(X, \mathbb C)), $ Picard-Lefschetz monosromy representation $\rho: \pi\_1(B, b\_0) \to GL(n, \mathbb C) $ and so on.


But my knowledge of these topic remains too abstract to digest it well. So i am collecting enlightening toy examples. For example, I've worked with the Legendre family of elliptic curves


{$y^2=x(x-1)(x-\lambda)$} $ \to $ {$\mathbb C -(0,1)$}


and interpreted everything into a concrete term.(and it was fantastic)


But i still wants more. Because in my examples, no mixed Hodge structure, no Hodge structure of weight $\ge$ 2. If you have any other good examples, please tell me. Good reference will be extremly helpful. I also appreciate any suggestion. 



motivating examples of family of Hodge structure
I guess the implied question is: what are good references containing explicit calculations of variations of Hodge structure etc.? I might suggest taking a look at Griffiths' early pioneering papers "On periods of certain rational integrals I, II" Annals 1969, and
"Periods of integrals on algebraic manifolds III" IHES 1970. These papers contain a large number of explicit calculations on VHS and intermediate Jacobians for things like hypersurfaces in projective space. Regarding 
(variations of) mixed Hodge structures, take a look at the books by Carlson-Müller Stach-Peters, Peters-Steenbrink, and Voisin.


