Saskia Baltrusch

131 Chapter 5 In order to calculate the displacement V(X) as a function of along a cantilever beam (Figure 3A) the following second order differential equation has to be solved: (1) with the displacement V(X) , the moment M , the Young’s modulus E , and the second moment of area I . The Young’s modulus E is a material property, related to stiffness, while the second moment of area I is related to the geometric properties. For a round cross section beam, like the one used here, the second moment of area amounts to: (2) with the radius R of the beam. For a cantilever beam, where a force F is applied at the top of the beam, the moment along the beam has the following form: (3) with the force F , the length of the beam L and the coordinate . This results in the following equation for the displacement V : (4) Integrating this equation twice, with the boundary conditions, that (a) the displacement at the base is zero, i.e., V(X=0)=0 and (b) that the deflection at the base is zero, i.e., yields: (5) This equation can be used to explicitly calculate the displacement V(X) as a function of the force F , the Young’s modulus E , the second moment of area I the length L and the position along the beam. In a similar fashion can the angle Ө( ) which is defined as the difference of the displacements 5