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101) The first term is the exact value for a simply supported beam without prestress, and the second term is the correction due to the axial loading. One sees that a compressive load reduces ω1 and, when P reaches Pcr , ω1 = 0; conversely, a traction load tends to increase ω1 . 98) is the exact mode shape of the problem. Any other assumption (satisfying the geometric boundary conditions) would have led to a larger value of the ω1 , because the Rayleigh-Ritz method tends to overestimate the natural frequencies.
These induction effects are present in all conductors, but they become very important if the conductor consists of closely packed coils with many turns. In air, the magnetic behavior of a coil is linear (the flux linkage λ is proportional to the current i), but a ferromagnetic core is often added to the coil to enhance the magnetic flux density by several orders of magnitude; the behavior of such coils is nonlinear and often hysteretic. Faraday’s law states that the voltage developed across an inductor is equal to the rate of change of flux linkage λ 44 2 Dynamics of electrical networks (a) (b) i i (õ; i) ã e Wm(i) Wm (õ) õ Fig.
19) where M is the number of circuit elements. 20) k=1 We is the electrical energy function of the system, and Ek represents the generalized voltage corresponding to the nonconservative elements and conjugate to the independent generalized charge variable qk . ne is the number of independent generalized charge coordinates. The negative sign on δWe is due to the fact that when the voltage across a conservative circuit element does work on the circuit, the electrical energy in the element decreases.