 3.4.1: Determine the period and frequency of the simple harmonic motion of...
 3.4.2: Determine the period and frequency of the simple harmonic motion of...
 3.4.3: A mass of 3 kg is attached to the end of a spring that is stretched...
 3.4.4: A body with mass 250 g is attached to the end of a spring that is s...
 3.4.5: Two pendulums are of lengths L1 and L2 andwhen located at the respe...
 3.4.6: A certain pendulum keeps perfect time in Paris, where the radius of...
 3.4.7: A pendulum of length 100:10 in., located at a point at sea level wh...
 3.4.8: A pendulum of length 100:10 in., located at a point at sea level wh...
 3.4.9: Derive Eq. (5) describing the motion of a mass attached to the bott...
 3.4.10: Consider a floating cylindrical buoy with radius r, height h, and u...
 3.4.11: A cylindrical buoy weighing 100 lb (thus of mass m D 3:125 slugs in...
 3.4.12: . Assume that the earth is a solid sphere of uniform density, with ...
 3.4.13: Suppose that the mass in a massspringdashpot system with m D 10, c ...
 3.4.14: Suppose that the mass in a massspringdashpot system with m D 10, c ...
 3.4.15: The remaining problems in this section deal with free damped motion...
 3.4.16: The remaining problems in this section deal with free damped motion...
 3.4.17: The remaining problems in this section deal with free damped motion...
 3.4.18: The remaining problems in this section deal with free damped motion...
 3.4.19: The remaining problems in this section deal with free damped motion...
 3.4.20: The remaining problems in this section deal with free damped motion...
 3.4.21: The remaining problems in this section deal with free damped motion...
 3.4.22: A 12lb weight (mass m D 0:375 slugs in fps units) is attached both...
 3.4.23: This problem deals with a highly simplified model of a car of weigh...
 3.4.24: (Critically damped) Show in this case that x.t / D .x0 C v0t C px0t...
 3.4.25: (Critically damped) Deduce from that the mass passes through x D 0 ...
 3.4.26: (Critically damped) Deduce from that x.t / has a local maximum or m...
 3.4.27: (Overdamped) Show in this case that x.t / D 1 2 .v0 r2x0/er1t .v0 r...
 3.4.28: (Overdamped) If x0 D 0, deduce from that x.t / D v0 ept sinh t:
 3.4.29: (Overdamped) Prove that in this case the mass can pass through its ...
 3.4.30: (Underdamped) Show that in this case x.t / D ept x0 cos !1t C v0 C ...
 3.4.31: (Underdamped) If the damping constant c is small in comparison with...
 3.4.32: (Underdamped) Show that the local maxima and minima of x.t / D Cept...
 3.4.33: (Underdamped) Show that the local maxima and minima of x.t / D Cept...
 3.4.34: (Underdamped) A body weighing 100 lb (mass m D 3:125 slugs in fps u...
 3.4.35: Suppose that m D 1, c D 2, and k D 1 in Eq. (26). Show that the sol...
 3.4.36: Suppose that m D 1 and c D 2 but k D 1 102n. Show that the solution...
 3.4.37: Suppose that m D 1 and c D 2 but that k D 1 C 102n. Show that the s...
 3.4.38: Whereas the graphs of x1.t / and x2.t / resemble those shown in Fig...
Solutions for Chapter 3.4: Mechanical Vibrations
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 3.4: Mechanical Vibrations
Get Full SolutionsChapter 3.4: Mechanical Vibrations includes 38 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. Since 38 problems in chapter 3.4: Mechanical Vibrations have been answered, more than 15182 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pascal matrix
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).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.