By Khanh Chau Le (auth.)

The above examples may still clarify the need of knowing the mechanism of vibrations and waves on the way to keep watch over them in an optimum method. although vibrations and waves are ruled by means of differential equations which require, in most cases, quite complex mathematical equipment for his or her research. the purpose of this textbook is to assist scholars collect either a great clutch of the 1st ideas from which the governing equations may be derived, and the enough mathematical equipment for his or her fixing. Its specific beneficial properties, as visible from the identify, lie within the systematic and in depth use of Hamilton's variational precept and its generalizations for deriving the governing equations of conservative and dissipative mechanical structures, and in addition in delivering the direct variational-asymptotic research, each time on hand, of the power and dissipation for the answer of those equations. will probably be confirmed that many famous tools in dynamics like these of Lindstedt-Poincare, Bogoliubov-Mitropolsky, Kolmogorov-Arnold-Moser (KAM), and Whitham are derivable from this variational-asymptotic analysis.

This e-book grew up from the lectures given through the writer within the final decade on the Ruhr college Bochum, Germany. considering vibrations and waves are parts of varied disciplines (physics, mechanics, electric engineering etc.) and can't be dealt with in one textbook, i've got constrained myself in most cases to vibrations and waves of mechanical nature. the cloth of this publication could be prompt for a 365 days direction in greater dynamics for graduate scholars of mechanical and civil engineering. For this circle of readers, the emphasis is made at the *constructive* equipment of answer and never at the rigorous mathematical proofs of convergence. As reimbursement, numerous numerical simulations of the precise and approximate options are supplied which show vividly the validity of the used equipment. to assist scholars turn into more adept, each one bankruptcy ends with workouts, of which a few could be solved successfully through the use of *Mathematica*.

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**Example text**

3. Thus, for small damping the eigenfrequencies remain the same as those of the undamped coupled oscillators which are given by 1 2 (ω − ωϕ2 )2 + α 4 . 4 x 1 2 ω1,2 = (ωx2 + ωϕ2 ) ∓ 2 Fig. 8 shows the plots of dimensionless frequencies (ω1,2 /ωx )2 versus the ratio of frequencies (ωϕ /ωx )2 at different coupling ratios (α /ωx )2 . It can be seen that for the zero coupling α = 0 the eigenfrequencies coincide with those of uncoupled oscillators ωϕ and ωx . The larger the coupling factor, the farther the eigenfrequencies lie apart.

Alternatively, we can present the solution in the form ϕ = q1 a1 cos(ω1t − φ1 ) + q2 a2 cos(ω2 t − φ2 ). 7) Recalling the addition theorem for cos(ω t − φ ) we find aj = A2j + B2j , tan φ j = Bj , Aj j = 1, 2. 4). Normal modes and coordinates. 7) the solution is the superposition of two harmonic cosine functions with different frequencies. If the frequency ratio is not a rational number, the motion is no longer periodic in general2. 5). 1 Conservative Oscillators 41 the motion is purely harmonic with the frequency ω1 or ω2 .

N. 26) For any conservative mechanical system the Lagrange function equals ˙ = K(q, q) ˙ − U(q), L(q, q) ˙ is the kinetic energy and U(q) the potential energy. The kinetic enwhere K(q, q) ˙ is a positive definite quadratic form with respect to q˙ ergy K(q, q) ˙ = K(q, q) Thus, n 1 2 n ∑ m jk (q)q˙ j q˙k . j,k=1 ∂K ˙ ∑ ∂ q˙ j q˙ j = 2K(q, q). j=1 Any function possessing this property is called homogeneous function of order two ˙ We want to show now that the conservation of energy follows with respect to q.