Wave propagation in a cantilever beam

Responses of the free end of a cantilever beam made of two different materials under the effect of an impact force. The beam was excited by a transverse triangular force impulse modulated by a harmonic motion. Three methods are used for investigating the responses. Mass, damping and stiffness matrices are obtained by using finite element method and then by using these matrices classical dynamic stiffness matrix is obtained. In the second method, motion equations consisting of mass matrix, damping matrix, stiffness matrix, load vector and unknown vector which are obtained by the finite element method mentioned before are solved in the time domain by using Newmark direct integration scheme. In the third method which is known as spectral element method, governing homogeneous differential equation for a beam part is solved exactly and then by using this solution dynamic stiffness matrix for a spectral element is obtained. The load is transformed into frequency domain numerically by using the discrete Fourier transform technique. Then the solution obtained in the frequency domain is transformed in the time domain numerically by using inverse discrete Fourier transform technique. In the first and third methods, the only difference is in obtaining the stiffness matrices and apart from that the solution algorithm in the two methods is the same: Namely the solution is obtained in the frequency domain and then it is transformed into the time domain by using inverse discrete Fourier transform technique. Solution algorithms for the three methods mentioned above are made and they are programmed by using Matlab packet programme. The results of the three methods for the special case, a beam which is made of only one material, are obtained and compared with each other. Only the second method is used for obtaining the numerical solutions of a cantilever beam made of two different materials. It is seen in the numerical investigations that because of two different materials, additional secondary waves appear between the primary waves. The relations between additional waves and location and physical properties of the additional secondary waves are found. It is proved that the location and length of the different material can be obtained by investigating the additional secondary waves. Keywords: Wave propagation in a cantilever beam, Additional wave, Fourier transform, Spectral element method, Direct time integration, Newmark method.