The three-mass chain of the opening

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Linear algebra and vector spaces

Conditions of Validity, Ch The four subspaces and the Fredholm alternative

Problem

The three-mass chain of the opening, with masses m=1 kgm=1\,\mathrm{kg} and four springs of k=100 N m−1k=100\,\mathrm{N\,m^{-1}} between two fixed walls, is driven exactly at its lowest normal-mode frequency, ω12=(2−2)k/m\omega_1^2=(2-\sqrt2)k/m, whose mode shape is (1,2,1)/2(1,\sqrt2,1)/2. The steady amplitudes solve (K−mω12I)x=b(\mathsf{K}-m\omega_1^2\mathsf{I})\mathbf{x}=\mathbf{b}. ((a)) The drive is b=(0.010, 0, −0.010) N\mathbf{b}=(0.010,\,0,\,-0.010)\,\mathrm N, equal and opposite on the two end masses. Use the Fredholm alternative to decide whether a steady amplitude exists. If it does, find the solution orthogonal to the mode shape and write the whole solution set. ((b)) The drive is instead b=(0, 0.010, 0) N\mathbf{b}=(0,\,0.010,\,0)\,\mathrm N on the middle mass. Decide whether a steady amplitude exists, giving the value of the quantity that decides it.

Browse the problem library