On the application of integral and variational principles of mechanics to problems of vibrations

Analytical Mechanics

“…-(WI sma -W2 cos a) sm(3 = --sma --cos a cos (3, are values of higher orders of smallness than those in (22). To prove this we construct the ratio of terms in eq.…”

“…A more general case is the definition of quasivelocities by means of linear forms with free terms a s ,n+1 Ws = a s 1tI1 + ... + asntIn + a s ,n+1 (8 = 1, ... , n) , (1.5.22) where the factors ask may explicitly depend on time. There is no need to repeat the above, it suffices to make use of notation (3.7) and add (22) with an additional line W n +1 = tIn+1 = 1 (an +1,z = bn +1,Z) .…”

“…The consequence of equalities (22) is that the linear terms in the expression for the potential energy vanish. Now using eqs.…”

“…It is reasonable to take those two angles as the generalised coordinates which remain small in the near vicinity of the When a and 13 are small we arrive at the system of two differential equations (7.18.26) The solution of the system can be found, for example, in [56], see also Subsection 7.9.3 of the present book. The integrals (22) and (24) …”

“…(23). Then the second equation in (22) gains a term with q. Stability of the stationary rotation can be achieved when power mg<,Dg > 0 is supplied to shaft 2 whereas shaft 1 acts as its receiver, i.e.…”