Eigenvalue Formulas for the Uniform Timoshenko Beam: the Free-free Problem

This announcement presents asymptotic formulas for the eigenvalues of a free-free uniform Timoshenko beam. Suppose a structural beam is driven by a laterally oscillating sinusoidal force. As the frequency of this applied force is varied, the response varies. Experimental frequencies for which the response is maximized are called natural frequencies of the beam. Our goal is to address the question: if a beam’s natural frequencies are known, what can be inferred about its bending stiffnesses or its mass density? To answer this question we need to know asymptotic formulas for the frequencies. Here we establish these formulas for a uniform beam. One widely used mathematical model for describing the transverse vibration of beams was developed by Stephen Timoshenko in the 1920s (see [5], [6]). In this model, two coupled partial differential equations arise, (EIψx)x + kAG(wx − ψ)− ρIψtt = 0, (kAG(wx − ψ))x − ρAwtt = P (x, t). The dependent variable w = w(x, t) represents the lateral displacement at time t of a cross-section located x units from one end of the beam. ψ = ψ(x, t) is the cross-sectional rotation due to bending. E is Young’s modulus, i.e., the modulus of elasticity in tension and compression, and G is the modulus of elasticity in shear. The nonuniform distribution of shear stress over a cross-section depends on cross-sectional shape. The coefficient k is introduced to account for this geometry dependent distribution of shearing stress. I and A represent cross-sectional inertia and area, ρ is the mass density of the beam per unit length, and P (x, t) is an applied force. If we suppose the beam is anchored so that the so-called “free-free” Received by the editors January 5, 1998. 1991 Mathematics Subject Classification. Primary 34Lxx; Secondary 73Dxx.