Some people say that it is better to use the "of contruction" for the possessive of animate nouns. For example.

sweater of Miss Grammes

bike of the child

And for inanimate objects, an " ' " apostrophee can be used. For example.

School's principles

Bike's radiator.

What is your opinion on this.

I think you may have your first 'rule' a little confused . I've never heard that you should not use a possessive apostrophe in this situation. Are there really people who avoid saying Miss Grammes's sweater or the child's bike? That instead of saying 'please pass Jane's plate' they would say 'please pass the plate of Jane'? It sounds as though Jane has been chopped up and put on a plate....Emotion: ick!

Are you sure you don't mean the other way around - that some people don't like the possessive apostrophe when the 'possessor' is inanimate, not the object being possessed? I've heard people say that but don't see any problem with using a possessive apostophe with inanimate objects. The alternative is rather long-winded and sounds oddly old-fashioned somehow. (i.e. 'please paint the radiator of the bike'. There nothing wrong with saying 'please paint the bike's radiator'. It reminds me of those old fashioned French language books - the pen of my aunt is on the table etc. I think all this went out in the 50s!)
Students: We have free audio pronunciation exercises.
Well nona, i think you'r right. I was wrong. I don't know what the hell was I thinking when I red the following passage.

"Many writers consider it bad form to use apostrophe -s possessives with pieces of furniture and buildings or inanimate objects in general. Instead of "the desk's edge" (according to many authorities), we should write "the edge of the desk" and instead of "the hotel's windows" we should write "the windows of the hotel." In fact, we would probably avoid the possessive altogether and use the noun as an attributive: "the hotel windows." This rule (if, in fact, it is one) is no longer universally endorsed. We would not say "the radio of that car" instead of "that car's radio" (or the "car radio") and we would not write "the desire of my heart" instead of "my heart's desire." Writing "the edge of the ski" would probably be an improvement over "the ski's edge," however.

For expressions of time and measurement, the possessive is shown with an apostrophe -s: "one dollar's worth," "two dollars' worth," "a hard day's night," "two years' experience," "an evening's entertainment," and "two weeks' notice" (the title of the Hollywood movie nothwithstanding).

Seems like you're right in sying that this rule had been went out in the 50s. Thank You for correcting me.

sweater of Miss Grammes as opposed to Miss Grammes' sweater ... this is not the inanimate usage as Miss Grammes is live! Of would not be preferred here