Modal Superposition Method

In this method, the equations of motions are transformed from a set of n simultaneous differential equations to a set of n independent equations by the use of normal coordinates. The equations are solved for the response of each mode, and the total response of the system is obtained by superposing individual solutions. Two concepts are necessary for the understanding of the modal superposition method: (1) the normal coordinates; and (2) the property of orthogonality. Normal Coordinates

In a static analysis, it is common to represent structural displacements by a Cartesian system of coordinates. For example, in a planar system, coordinates x and y and rotation 0 are used to describe the position of a displaced structure with respect to its static position. If the structure is restrained to move only in the horizontal direction and if rotations are of no consequence, only one coordinate x is sufficient to describe the displacement. The displacements can also be identified by using any other independent system of coordinates. The only stipulation is that a sufficient number of coordinates are included to capture the deflected shape of the structure. These coordinates are commonly referred to as generalized coordinates and their number equal the number of degrees-of-freedom of the system. In dynamic analysis, however, it is advantageous to use free-vibration mode shapes known as normal modes to represent the displacements. While a mathematical description of normal modes and their properties may be intriguing, there is nothing complicated about their concept. Let us indulge in some analogies to bring home the idea. For example, normal modes may be considered as being similar to the primary colors red, blue, and yellow. None of these primary colors can be obtained as a combination of the others, but any secondary color such as green, pink, or orange can be created by combining the primary colors, each with a distinct proportion of the primary colors. The proportions of the primary colors can be looked upon as scale factors, while the primary colors themselves can be considered similar to normal modes. To further reinforce the concept of generalized coordinates, recall beam bending problems in which the deflection curve of a beam is represented in the form of trigonometric series. Considering the case of a simply supported beam subjected to vertical loads, as shown in Fig. 2.69, the deflection y, at any point can be represented by the following series:

Geometrically, this means that the deflection curve can be obtained by superposing simple sinusoidal shown in Fig. 2.69.

The first term in Eq. (2.70) represents the full-sine curve, the second term, the half-sine, etc. The coefficients a1, a2, a3,..., represent the maximum ordinates of the curves, while the numbers 1, 2, and 3, the number of waves or mode shapes. By determining the coefficients a1, a2, a3,..., the series can represent the deflection curve to any desired degree of accuracy, depending on the number of terms considered in the series. Orthogonality

This force-displacement relationship is rarely used in static problems, but is of great significance in structural dynamics. This is best explained with an example shown in Fig. 2.70.

Consider a two-story, lumped-mass system subjected to free vibrations. The system's two modes of vibrations can be considered as elastic displacements due to two different loading conditions, as shown in Fig. 2.70b and c. We will use a theorem known as Betti's reciprocal theorem to demonstrate the derivation of orthogonality conditions. This theorem states that the work done by one set of loads on the deflections due to a second set of

Figure 2.69. Generalized displacement of a simply supported beam (a) loading; (b) full-sine curve; (c) half-sine curve; (d) one-third sine curve; (e) one-fourth sine curve.

loads is equal to the work done by the second set of loads acting on the deflections due to the first. Using this theorem with reference to Fig. 2.70, we get

This can be written in matrix form

(W2 - W2{x,}^[M]{xa} = 0 If the two frequencies are not the same, i.e., wx ^ w2, we get {xhy[M]{xa} = 0

This condition is called the orthogonality condition, and the vibrating shapes, {xa} and {xb}, are said to be orthogonal with respect to the mass matrix, [M]. By using a similar procedure, it can be shown that

The vibrating shapes are therefore orthogonal with respect to the stiffness matrix as they are with respect to the mass matrix. In the general case of the structures with damping, it is necessary to make a further assumption in the modal analysis that the orthogonality

Figure 2.70. Two-story lumped-mass system illustrating Betti's reciprocal theorem: (a) lumped model; (b) forces acting during first mode of vibration; (c) forces acting during second mode of vibration.

Figure 2.70. Two-story lumped-mass system illustrating Betti's reciprocal theorem: (a) lumped model; (b) forces acting during first mode of vibration; (c) forces acting during second mode of vibration.

condition also applies for the damping matrix. This is for mathematical convenience only and has no theoretical basis. Therefore, in addition to the two orthogonality conditions mentioned previously, a third orthogonality condition of the form xT \c{xb} = 0 (2.75)

a is used in the modal analysis.

To bring out the essentials of the normal mode method, it is convenient to consider the dynamic analysis of a two-degree-of-freedom system. We will first analyze the system by a direct method and then show how the analysis can be simplified by the modal superposition method.

Consider a 2-story dynamic model of a shear building shown in Fig. 2.71a, b, and c, subject to free vibrations. The masses m1 and m2 at levels 1 and 2 can be considered connected to each other and to the ground by two springs having stiffnesses k1 and k2. The stiffness coefficients are mathematically equivalent to the forces required at levels 1 and 2 to produce unit horizontal displacements relative to each level.

It is assumed that the floors, and therefore the masses m1 and m2, are restrained to move in the direction x and that there is no damping in the system. Using Newton's second law of motion, the equations of dynamic equilibrium for masses m1 and m2 are given by m1jc1 = -k1 x + k2(x2 - x1) (2.76)

m2X2 = -k2(x2 - x1) (2.77) Rearranging terms in these equations gives m1xx1 + (k1 + k2)x1 - k2x2 = 0 (2.78)

m2xx2 - k2x1 + k2x2 = 0 (2.79) The solutions for the displacements x1 and x2 can be assumed to be of the form x1 = A sin (wt + a) (2.80)

Figure 2.71. Two-story shear building; free vibrations: (a) building with d masses; (b) mathematical model; (c) free-body diagram with d masses.

where w represents the angular frequency and a represents the phase angle of the harmonic motion of the two masses. A and B represent the maximum amplitudes of the vibratory motion. Substitution of Eqs. (2.80) and (2.81) into Eqs. (2.78) and (2.79) gives the following equations:

To obtain the solution for the nontrivial case of A and B ^ 0, the determinant of the coefficients of A and B must be equal to zero, thus

Expansion of the determinant gives the relation

or m,m2w4- m,k2 + m2(k, + k2)w2 + k,k2 = 0 (2.86) Solution of this quadratic equation yields two values for w2 of the form

2a where a = m1m2

As mentioned previously, the two frequencies w1 and w2, which can be considered intrinsic properties of the system, are uniquely determined.

The magnitudes of the amplitudes A and B cannot be determined uniquely, but can be obtained in terms of ratios r1 = A1/B1 and r2 = A2/B2 corresponding to w2 and w 2, respectively. Thus

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