Mechanical vibrations The following figure shows an industri

The following figure shows an industrial mixer and its driving system (motor-shafts-gears) used to rotate the two mixing blades. The shear modulus of elasticity for the steel in all shafts is G (x 10^6 psi = 10^6 pound/inch^2) Use Lagrange\'s theorem to find the equations of motion of the above-mentioned system. Deduce the mass matrix [M]. Deduce the stiffness matrix [K_t] Determine the natural frequencies and the principal modes of vibration of the system by using any specific software (include the name of the software and the printout of your program) Units of diameters and lengths = inches Units of Inertia = lb.s^2.in Units of Modulus of Elasticity = (10)^6 psi (pound/inch^2)

Solution

Lagrange theorem in the mathematics of group theory, states that for any finite group G the order of every sub group H of G divides the order of G . The theorem is named after Joseph Louis Lagrange.

This can be shown usig the concept of left cosets. of H in G . The left coests are the equiivalenc classes of a certain equivalence relation on G and thereform form a partition of G . specially x an Y in G are related if and only if there exists h in H such that x =yh . If we can show that all cosests of H have the same number of elements.