THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

90 5. Vibrating BarsFigure 5.1. A bar undergoing longitudinal strain in the x-direction.Figure 5.2. An element of the bar undergoing compression.and is fairly independent of lateral coordinates y and z. Thus,ξ = ξ(x, t)The x-coordinate of the bar is established by placing the left end of the bar at x =0,with the right end terminating at x = L. Consider an incremental element formedby dx of the unstrained bar positioned between x and x + dx as shown in Figure 5.2.The application of a force in the positive x-direction causes a displacement of theplane at x by a distance ξ to the right and the plane at x + dx by a distance ξ + dξalso to the right. A force acting in the opposite direction will likewise causecorresponding negatively valued displacements to the left. Because the element dxis small, we can represent the displacement at x + dx by the first two terms of aTaylor series expansion of ξ about x:( ) ∂ξξ + dξ = ξ + dx∂xThe left end of the element dx has been displaced a distance ξ and the right end adistance ξ + dξ, thus yielding a net increase dξ in the length of the element given by( ) ∂ξ(ξ + dξ) − ξ = dξ = dx∂xIn solid mechanics the strain ε of an element is defined as the ratio of the changeof its length to the original length, i.e.,( ) ∂ξε =∂xdxdx = ∂ξ∂x(5.1)