In this paper, the full development and analysis of four models for the transversely vibrating uniform beam are presented. The four theories are the Euler}Bernoulli, Rayleigh, shear and Timoshenko. First, a brief history of the development of each beam model is presented. Second, the equation of motion for each model, and the expressions for boundary conditions are obtained using Hamilton's variational principle. Third, the frequency equations are obtained for four sets of end conditions: free}free, clamped}clamped, hinged}hinged and clamped}free. The roots of the frequency equations are presented in terms of normalized wave numbers. The normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes. Fourth, the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented. Finally, a numerical example is shown for a non-slender beam to signify the di!erences among the four beam models.
1999 Academic Press *v *x "0, v"0 for hinged end; *v *x "0, v"0 for clamped end;
These conditions are shown in Figure 1 where D, S, M, and Q represent displacement, slope, moment and shear respectively. The equation of motion, boundary conditions, and initial conditions form an initial-boundary-value problem which can be solved using the methods of separation of variables and eigenfunction expansion. First, we consider a homogeneous problem by setting f (x, t)"0 in order to obtain the natural frequencies and eigenfunctions. By separating v(x, t) into two functions such that v(x, t)"=(x)ΒΉ(t), the equation of motion (11) can be separated into two ordinary 940 S. M. HAN EΒΉ AΒΈ.
with the boundary conditions given by
where v is the dimensionless displacement, *v/*x the dimensionless slope, *v/*x the dimensionless moment and *v/*x! I(*v/*x*t) the dimensionless shear. Four possible end conditions are
The expression for shear might seem odd. Its validity can be veri"ed by summing the forces and moments on an incremental beam element, as shown in Figure 2. The sum of the forces on a beam element in the transverse direction isR
where can be approximated as *v/*x or *v*/*x*, and dQ* and d represent (*Q*/*x*) dx* and (* /*x*) dx* respectively. Expanding cos( #d ) about using a Taylor series expansion and using the small angle assumption,S we obtain ! *Q* *x* " *A* *v* *t* !f *(x*, t*). (25) Similarly, taking the sum of the moments about the center of the beam element, we obtain *M* *x* !Q*" *I* *v* *t**x* . (26) R Symbols with superscript * are dimensional quantities. S The small angle assumption means that 1.