Numerical Analysis of Vibrations of Structures under Moving Inertial Load
Moving inertial loads are applied to structures in civil engineering, robotics, and mechanical engineering. Some fundamental books exist, as well as thousands of research papers. Well known is the book by L. Frýba, Vibrations of Solids and Structures Unde
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Analytical Solutions
A concentrated load acting on a continuous medium is usually described by a Dirac delta function. The point force or mass whose area of influence is limited, must be described in the entire spatial domain of the structure, for example 0 ≤ x ≤ l. Multiplication of the force by the Dirac delta function δ (x) leads to such an effect. Then we have the load terms δ (x − x0)P or δ (x − x0 )m d2 w/dt 2 described in the domain of the problem. Unfortunately, the mathematical treatment of the term of the first type is relatively simple. It does not contain the solution variable. The treatment of a term of the second type, which describes the inertial force induced by the material particle, is much more complex. It includes the acceleration of the selected point x0 as the second derivative of the solution of the differential equation w. What is more, the argument x0 = vt moves with velocity v and the inertial term is a function of both x and t: d2 w δ (x − vt)m 2 . dt Due to the presence of the Dirac delta function in this result, the solutions obtained to these partial differential equations are not solutions in the classical sense, but are called ‘weak’ or ‘distributional’ solutions. So we must extend the concept of solution, arranging that any limit of an almost uniformly convergent sequence of classical solutions will be regarded as a generalized solution in the sense of a distribution. Distributions are therefore defined as the limit of sequences of continuous functions. This is called the sequential theory of distributions [4], in contrast to the functional theory [125]. For each distribution in the sense of L. Schwartz (functional) there is exactly one distribution in the sense of Mikusi´nski–Sikorski (sequential), and vice versa, so there is a mutual uniqueness [152]. Distributions are thus a generalization of functions. The purpose of the concept of a distribution is to give the correct meaning qua mathematical concept to objects such as the Dirac delta δ (x), which is much used in mathematical physics. An important feature of a distribution is that it ensures the posibility of differentiation, which is not always allowed for an arbitrary set of functions. The starting point for the sequential theory of distributions is the set of functions which are continuous on some fixed interval A < x < B (−∞ ≤ A < B ≤ ∞). If a sequence fn (x) of such continuous functions converges C.I. Bajer and B. Dyniewicz: Numerical Analysis of Vibrations of Structures, LNACM 65, pp. 21–30. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com
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2 Analytical Solutions
almost uniformly to a function f (x), it is also convergent in the sense of distributions to f (x) [4]. Every convergent sequence of distributions can be differentiated term by term (analogous to a uniformly convergent series). Of course, every uniformly convergent sequence is convergent almost uniformly. This allows, in the distribution sense, the differentiation of any function, changing the order of differentiations, and passing to the limit,
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