The time-dependent portion of the solution for these rigid-body type motions is seen to be aperiodic. This means that natural frequencies for both rigid-body modes are zero. The two arbitrary constants contained in Y(t) can be evaluated from the initial rigid-body displacement and velocity associated with the translation and rotation. Thus, the complete solution for the free-free beam bending problem can now be written in terms of all of its modes as

This example provides a convenient vehicle for further discussion of symmetry. It has already been noted in the case of a vibrating string that systems exhibiting geometric symmetry have two distinct types of mode shapes, namely, those that are symmetric about the midpoint and those that are antisymmetric about the midpoint. As can be seen in the results, this is indeed true for the modes of the free-free beam. In particular, the rigid-body translation mode and the first and third elastic modes are clearly symmetric about the mid-point of the beam, whereas the rigid-body rotation mode and the second elastic mode are antisymmetric about the midpoint (see Fig. 2.40).

This observation suggests that the symmetric mode shapes could be obtained by calculating the mode shapes of a beam that is half the length of the original beam and that has the sliding condition at one end and is free at the other. Similarly, the antisymmetric modes could be obtained by calculating the mode shapes of a beam with half the length of the original beam and that has one end pinned and the other free. It should also be evident that a symmetric aircraft with high-aspect-ratio wings, modeled as beams and attached to a rigid-body fuselage, could be represented in terms of the symmetric and antisymmetric modes of the combined body and wing system in a similar way. That is, one may model the whole system by only considering one wing attached to a rigid body with half the mass and half the rotational inertia with appropriate boundary conditions.

2.4 Approximate Solution Techniques

There are several popular methods that make use of a set of modes or other functions to approximate the dynamic behavior of systems. In this section, without going into details on the theories associated with this subject, we will illustrate within the framework already established how one can use a truncated set of modes or other set of functions to obtain an approximate solution. Details of the theories behind modal approximation methods may be found in texts that treat structural dynamics at the graduate level. The two main approaches are Galerkin's method, applied to ordinary or partial differential equations, and the Ritz method, applied to the principle of virtual work. These two methods yield identical results in certain situations. Thus, if time is limited it would only be necessary to discuss one of the two methods to give the student an introduction to the method and an appreciation of results that can be obtained this way. The Ritz method is to be preferred in the present context oc

;=-i because of the ease with which it can be presented within the framework of Lagrange's equations. Nevertheless, both of these methods will be presented at a level suitable for undergraduate students.

2.4.1 The Ritz Method

Building on the earlier treatment, we start with Lagrange's equations, given by d (dL\ dL

where the Lagrangean, L — K — P, the total kinetic energy is K, the total potential energy is P, n is the number of generalized coordinates retained, the generalized coordinates are and E, is the generalized force. Although it can be helpful, as we shall see below, it is not necessary to make use of potential energy, which can only account for conservative forces. The generalized force, however, can be used to include the effects of any loads. So as not to count the same physical effects more than once, the generalized force should only include those forces that are not accounted for in the potential energy. Generalized forces stem from virtual work, which can be written as n

;=i where <5£, is an arbitrary increment in the ith generalized coordinate.

Let us consider a beam in bending as an example. The total kinetic energy must include that of the beam as well as of any attached particles or rigid bodies. The contribution of the beam is

where m is the mass per unit length of the beam. The total potential energy P — U + V comprises the internal strain energy of the beam, denoted by U, plus any additional potential energy, V, attributed to gravity, springs attached to the beam, or applied static loads. All other loads, such as aerodynamic loads, damping, follower forces, etc., must be accounted for in E,.

The strain energy for a beam in bending is given by i re (d2v\2

The expression for V varies depending on the problem being addressed, as does the virtual work of all forces other than those accounted for in V. The virtual work can be written as

J o where Sv is an increment of v in which time is held fixed and f(x. t ) is the distributed force per unit length and is positive in the direction of positive v.

To apply the Ritz method, one needs to express P, K, and 8W in terms of a series of functions with one or more terms. For a beam in bending, this means that n v(*,i) = to- (2.285)

There are several characteristics that these "basis functions" </); must possess:

1. Each function at least must satisfy all boundary conditions on displacement and rotation (often called the "geometric" boundary conditions). It is not necessary that they satisfy the force and moment boundary conditions, but satisfaction of them may improve accuracy. However, it is not easy in general to find functions that satisfy all boundary conditions.

2. Each function must be continuous and p times differentiable, where p is the order of the highest spatial derivative in the Lagrangean. The p\h derivative of at least one function must be nonzero. Here, from Eq. (2.283), p — 2.

3. If more than one function is used, they must be chosen from a set of functions that is complete. This means that any function on the interval 0 < x < i and having the same boundary conditions as the problem under consideration can be expressed to any degree of accuracy whatsoever as a linear combination of the functions in the set. Examples of complete sets of functions on the interval 0 < x < i include

0 0

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