Aj Cj Fi A A

Close inspection of this total string displacement indicates that at any given instant the transverse deflection is represented by summation over a denumerably infinite set of shapes. Each shape is of indeterminate amplitude and is associated with a particular eigenfunction; these shapes are also called "mode shapes" in the field of structural dynamics. They will be represented here by </>, U ). Thus, for transverse deflection of a string the mode shapes may be written as

or any constant times <pi(x). It can be observed from this function (see Fig. 2.3) that the higher the mode number i, the more crossings of the zero axis on the interval 0 < x < t. These crossings are sometimes referred to as "nodes." The trend of increasing numbers of nodes with an increase in the mode number is generally true in structural dynamics.

In the above solution for the total displacement it may be noted that each mode shape is multiplied by a function of time. This multiplier is called the "generalized coordinate" and will be represented here by ยง, (/). For this specific problem the generalized coordinates are

and are thus seen to be simple harmonic functions of time. Since there were no external loads applied to the string, the preceding result is called the homogeneous solution. If there had been an external loading, the resulting time dependency of the generalized coordinates would reflect such a loading.

Thus, the total string displacement can be written as a sum of "modal" contributions of the form

This expression can be interpreted as a weighted sum of the mode shapes, each of which has a modal amplitude (i.e., the generalized coordinate) that is a function of time. For the homogeneous solution obtained above, this time dependency is simple harmonic at a frequency that is unique for each mode or eigenvalue. These are called the "natural frequencies" of the modes or "modal frequencies" and will be represented by &>,. For the string they are

<t>2 (-V)
(A)

with the lowest frequencies given by the lowest mode numbers. Indeed, just as the increase in the number of nodes with the mode number is generally true, so it is with the natural frequency. When the physical and geometric parameters of the problem are expressed in any consistent1 set of units, the units of the natural frequency will be rad/s. Division by 2tz converts the units of frequency into "cycles per second" or Hertz. The inverse of the natural frequency in Hertz is the period of the oscillatory motion.

1 For example, with SI units one has the units of T as N, m as kg/m, and f as m. With English units, one has the units of T as lb, m as, say, lb-s2/in.2, and f as in.

To summarize what has been accomplished in solving the wave equation, it may be said that the string displacement as a function of both x and t can be represented as a sum of modal contributions. Each mode in this representation is a structural dynamic property of the given system (string) and can be completely described by its mode shape and modal frequency. Such "modes of vibration" can be formulated for any conservatively loaded, linearly elastic structure. This statement includes two restrictions that must be observed for a modal representation. One restriction is linearity, which is satisfied here by the linear wave equation. The other restriction is that the system must be conservative, which means that there can be no addition or dissipation of energy during the dynamic response. A typical violation of this restriction would be the existence of damping such as structural or aerodynamic damping. When such damping is present, it can be adequately treated as an external loading.

2.1.3 Orthogonality of Mode Shapes

A most significant property of the mode shapes previously discussed is that they form a set of orthogonal mathematical functions. If the system is nonuniform then the mode shapes are orthogonal with respect to some inertial weighting function, such as the mass distribution for the string. This condition of functional orthogonality can be described analytically as

0 0

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