## L

3.5.3 Frequency Comparison for Lumped-Mass, Dynamic-Stiffness, and Consistent-Mass Models

In general, the computation efforts associated with response analysis for the lumped-mass, consistent-mass, and dynamic-stiffness models are reduced in the order of the individual models for a given problem, and the solution accuracy, however, may also be reduced, respectively. The accuracy sensitivity is illustrated by frequency comparison of a rigid frame shown in Figure 3.29a using these mathematical models: (A) the lumped-mass method, (B) the consistent-mass method, and (C) the dynamic-stiffness method (based on Bernoulli-Euler theory). For ''A'' and ''B'' the structural members are divided into three, six, and nine elements, respectively, and the masses of''A'' are lumped at the center of the divided segments. Dynamic-stiffness, lumped-mass, and consistent-mass models are shown in Figure 3.29b,c, and d, respectively. Assume that all members are identical with m = 0.04837 kgs2/m2, I = 0.00286 cm , E = 20684.27 kN/cm2, and L = 0.2413 m.

Eigenvalues of the first three modes are shown in Table 3.3 for comparison with the accurate solution by the dynamic stiffness method. Observation of the solutions reveals that the lumped-mass method needs six elements for the first two modes and nine for the third mode, while the consistent-mass method needs three elements for the first mode and six for the second and third modes. The lumped-mass model can give eigenvalues higher or lower than the dynamic stiffness's solution depending on how much mass is lumped at each node of the structure. The consistent-mass model always gives frequencies higher than the dynamic stiffness.

The consistent-mass model does not yield accurate solutions for higher modes because the shape functions, which are based on four generalized coordinates (see Equation 3.87) in deriving the mass and stiffness coefficients of a typical member, cannot be flexible enough to represent the deformed shape of

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