- 2.6.1: It is found experimentally that a 1 kg mass stretches a spring 49/3...
- 2.6.2: Lety(t)=Aerlr+Ber2r, with (AI+IBI#O. (a) Show that y(t) is zero at ...
- 2.6.3: Let y(t)=(A+ Bt)er', with JAI+IBJ#O. (a) Show that y(t) is zero at ...
- 2.6.4: A small object of mass 1 kg is attached to a spring with spring con...
- 2.6.5: A small object of mass 1 kg is attached to a spring with spring-con...
- 2.6.6: A small object of mass 4 kg is attached to an elastic spring with s...
- 2.6.7: The gun of a U.S. M60 tank is attached to a spring-mass-dashpot sys...
- 2.6.8: A spring-mass-dashpot system has the property that the spring const...
- 2.6.9: A spring-mass-dashpot system with m = 1, k = 2 and c = 2 (in their ...
- 2.6.10: A 1 kg mass is attached to a spring with spring constant k = 64 N/m...
- 2.6.11: A 1 kg mass is attached to a spring with spring constant k = 4 N/m,...
- 2.6.12: A small object of mass 1 kg is attached to a spring with spring con...
- 2.6.13: Determine a particular solution +(t) of my" + cy' + ky = F,,cosot, ...
Solutions for Chapter 2.6: Mechanical vibrations
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition
Tv = Av + Vo = linear transformation plus shift.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Column space C (A) =
space of all combinations of the columns of A.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.