# ## Advanced Dynamics and Vibrations: Numerical methods for vibrating systems

[Jason’s part goes here] where we can have a different mass as the equation is different. For SHM If we choose then  Therefore: Note: If  Now using the result above Consider  Note: We can apply this idea to various dynamical systems which are more complicated than the simple spring/mass. Consider an approximation to longitudinal vibration of a bar

Consider a beam of total mass and cross section modulus which we approximate as if , go to if    Therefore: Therefore: N = 4 To calculate the mode shapes one can start at the free end e.g.: Assume  Now we use the recursion relationship Therefore: For the 4 storey building for the lowest natural frequency is Therefore:    Check       Therefore: if ( is low) then: for the lower frequencies  where is density

For other boundary conditions, the expression for is determined then the natural frequencies are known. For fixed ends.  Consider the fixed boundaries:       Therefore:   Therefore: ##### Example        Therefore:  