Mathcad in Teaching Rotor and Structural Dynamics - download pdf or read online

By G Broman; S Oestholm

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7 Depression of Equation 45 ✼ ✽ 1) 2 ✶ d2 ✽ d ✷ 2 ✾ y(x) + ✸ dx dx y(x) ✹✻ f(x) substitute, ✺ ✿❁ d ❀ dx d dx ✼ 2) ✷ ✽ ✷ ✾ d2 2 ✸ y( x) + dx d dx y( x) + y(x) z( x) + z(x) substitute, ✶ ✽ z( x) → y(x) ❀ 2 ✺ 0 ✹ ✻ ❁✿ d dx y(x) ❀ ✺ 2 f (x) p ( y(x)) → simplify ❀ t ← y( x) ⋅ t ← x 1 2 + p ( y(x)) + y( x) 2 0 w d y( t) dt d p (t) dt ✶ ✷ d ✸ p ⋅p + ✹✻ dy ✺ 2 p+ y 0 Fig. 3. Depression of differential equation For the second case we duplicate the result (the first-order equation) in a more precise form by hand (see final string).

11) will take the form: ∂ϕ =γ. 2) The divergence operator is omitted, as spatial distribution of dependent variable is uniform. We deal with the ordinary differential equation containing the unique independent variable – time . 2) with Eq. 1) we present its left part as rate of increase y′ of some physical quantity y in a control volume due to a source (Eq. 2), term on the right), ✑ ✒ ✓ γ ≡ f ( τ) ⋅ y + g ( τ) , having two components. 8). The second component g is the internal production or inflow from an environment, independent of y.

7a). The localization of parameters area, in which there is an ambiguity, is especially well determined in Fig. 7b – this projection extends the bifurcation diagram in Fig. 6. The title “catastrophe” is used because the indicated shape predetermines a jump, a disastrous development of the system, as shown in the next section. 7 Catastrophic Jumps at Smooth Variation of Parameters The changes of a system state can be viewed under two aspects: • As time evolution from some initial state at given constant values of control parameters • As transferring from one equilibrium state to another at change of control parameters.

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