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The solution for x will be the real part of the solution for z. The equation becomes: m¨ z + bz˙ + kz = a0 + Fn ei(nωt−αn ) The solution is the sum of a particular solution plus the solution to the homogeneous equation (right-hand side set to 0, see above). The solution to the homogeneous equation is called the transient solution. The transient solution will contain two constants that are determined by initial conditions. We proceed to obtain the particular solution by substituting a trial solution of the form, z = B0 + Bn ei(nωt−γn ) Solution: x = (a0 /k) + where, Bn = Bn cos(nωt − γn ) Fn m2 (ω02 − n2 ω 2)2 + ω 2 n2 b2 and, tanβn = bnω − n2 ω 2 ) m(ω02 and, γn = αn + βn and ω02 = k/m.
Computer Project 3 (see Appendix) 18. Computer Project 4 (See Appendix) 19. Computer Project 5 (See Appendix) 20. Computer Project 6 (See Appendix) 21. 1 Definitions and Theorems Consider a particle which moves in a two or three-dimensional region of space under the influence of a force F. Consider two points, a and b in the region of space. 1 Definition The net force is the vector sum of all forces acting on the particle. 2 b a F · ds. Theorem Let a and b refer to any two points on the trajectory of a particle.
You will be relieved to know that for our purposes this is a formal result essential to the derivation of Lagrange’s equations that follow, but not one that we will use as a practical tool. 2 Application to Acceleration The mathematics we have introduced can now be extended in several directions. It was used by Albert Einstein as the original language of his General Theory of Relativity and a simple modification of it is also the basis of the most elegant formalism of the Special Theory of Relativity (see Chapter 10).