A free-ended chain of equal masses joined in a line by springs of stiffness has the…

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

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

Problem

A free-ended chain of n≥2n\ge2 equal masses joined in a line by n−1n-1 springs of stiffness kk has the n×nn\times n stiffness matrix K\mathsf K with kk in the two end places of the diagonal, 2k2k in the other diagonal places, −k-k in the places just above and just below the diagonal, and zeros elsewhere. The chain from the text is the case n=3n=3. ((a)) Show that xTKx=k∑i=1n−1(xi+1−xi)2\mathbf x^{\mathrm T}\mathsf K\mathbf x=k\sum_{i=1}^{n-1}(x_{i+1}-x_i)^2 for every x∈Rn\mathbf x\in\mathbb R^n, and use it to prove that N(K)\mathcal N(\mathsf K) is spanned by (1,1,…,1)(1,1,\ldots,1) for every nn. ((b)) Prove that the static problem Kx=F\mathsf K\mathbf x=\mathbf F has a solution exactly when ∑iFi=0\sum_iF_i=0, and describe the whole solution set when it does. ((c)) Name the result about a continuous body in the previous chapter that part (b) is the discrete form of, and match each piece of part (b) to its continuum counterpart.

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