Sm it sm sm jor

a set of mode shapes for any problem.

Completeness also implies that there can be no missing terms between the lowest and highest ones used in any series.

4. The set of functions must be linearly independent. This means that n ai(pi(x) = 0 =>• a, = 0 for all i. (2.286)

A set of functions that satisfies all these criteria is said to be "admissible."

By use of the series approximation, we have reduced a problem with an infinite number of degrees of freedom to one with n degrees of freedom. Instead of being governed by a partial differential equation, the behavior of this system is now defined by n second-order, ordinary differential equations in time. This reduction from a continuous system modeled by a partial differential equation with an infinite number of degrees of freedom to one described by a finite number of ordinary differential equations in time is sometimes called spatial discretization. The number n is usually increased until convergence is obtained. (It should be noted that if inertial forces are not considered, so that the kinetic energy is identically zero, then a system described by an ordinary differential equation in a single spatial variable is reduced by the Ritz method to one described by n algebraic equations.)

Now, let us illustrate how the approximating functions are actually used. Let </>,-, i — 1, 2,..., oo, be a complete set of p-times differentiable, linearly independent functions that satisfy the displacement and rotation boundary conditions. Thus, U can be written as

The contributions of any springs that restrain the structure, as well as conservative loads, must be added to obtain the full potential energy P. The kinetic energy of the beam is j n n n I

Contributions of any additional particles and rigid bodies must be added to obtain the complete kinetic energy K.

The virtual work term must account for distributed and concentrated forces resulting from all other sources, such as damping, aerodynamics, etc. This can be written as where x„ is a value of a at which a concentrated force is located. Here the first term accounts for a distributed force f(x. t) on the interior of the beam, and the second term accounts for a concentrated force on the interior (see Eq. 2.112). In aeroelasticity, the loads f(x, t) and Fc(xo, t) may depend on the displacement in some complicated manner.

The integrands in the above quantities all involve the basis functions and their derivatives over the length of the beam. It should be noted that these integrals involve only known quantities and often can be evaluated analytically. Sometimes they are too complicated to undertake analytically, however, and they must be evaluated numerically. Numerical evaluation is often facilitated by nondimensionalization. Symbolic computation tools such as Mathematica and Maple may be quite helpful in both situations.

With all such things considered, the equations of motion can be written in a form that is quite common, that is, where {£} is a column matrix of the generalized coordinates, {/*' [ i s a column matrix of the generalized force terms that do not depend on (') is the time derivative of ( ), [M] is the mass matrix, [C] is the gyroscopic/damping matrix, and [iT] is the stiffness matrix. The most important contribution to [M] is from the kinetic energy, and this contribution is symmetric. The most important contribution to [iT] is from the strain energy of the structure and potential energy of any springs that restrain the motion of the structure. There can be contributions to all terms in the equations of motion from kinetic energy and virtual work. For example, there are contributions from kinetic energy to [C] and [iT] when there is a rotating coordinate system. Damping makes contributions to [C] through the virtual work. Finally, because aerodynamic loads in general depend on the displacement and its time derivatives, aeroe-lastic analyses may contain terms in [M], [C], and [iT] that stem from aerodynamic loads.

An interesting special case of this method occurs when the system is conservatively loaded. The resulting method is usually referred to as the Rayleigh-Ritz method, and there are many theorems that can be proved about the convergence of approximations to the natural frequency. Indeed, one of the most powerful of such theorems states that the approximate natural frequencies are always upper bounds, and another states that adding more terms to a given series always lowers the approximate natural frequencies (i.e., making them closer to the exact values).

A further specialized case is the simplest approximation, in which only one term is used. Then, an approximate expression for the lowest natural frequency can be written as a ratio called the Rayleigh quotient. This simplest special case is of more than merely academic interest. It is not at all uncommon that a rough estimate of the lowest natural frequency is needed early in the design of flexible structures.

Example 12: The Ritz Method Using Clamped-Free Modes

In the first example we consider a uniform, clamped-free beam that we modify by adding a tip mass of mass /uti L The exact solution can be easily obtained

for this modified problem using the methodology spelled out earlier. However, it is desired here to illustrate the Ritz method, and we already have calculated the modes for a clamped-free beam (i.e., without a tip mass) in Section 2.3.4. These mode shapes are solutions of an eigenvalue problem, and so, provided we do not skip any modes between the lowest and highest mode number that we use, this set is automatically complete. The set is also orthogonal and therefore linearly independent. Of course, these modes automatically satisfy the boundary conditions on displacement and rotation for our modified problem (since they are the same as for the clamped-free beam), and they are infinitely differentiable. Hence, they are admissible functions for the modified problem. Moreover, they satisfy the zero moment boundary condition at the free end, which is a boundary condition for our modified problem. However, because of the presence of the tip mass in the modified problem, the shear force, which the reader will recall is proportional to the third derivative of the displacement, does not vanish as it does for the clamped-free mode shapes.

The strain energy becomes

Substituting the mode shapes of Eq. (2.251) into Eq. (2.291) and taking advantage of orthogonality, one can simplify it to

z i=i where or,- is the set of constants in Table 2.1. Similarly, accounting for the tip mass, the kinetic energy of which is

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