Natural frequencies of a cantilever beam and block system with clearance while block staying on given position

In this paper, free vibration modes are experimentally and numerically investigated for a cantilever beam and block system with clearance while the block is assumed to be on several positions (the left end of the cantilever; 1 m from the left end; the middle of the cantilever; and the right end) without movement along the cantilever beam in the longitudinal direction. Mathematical expression of clearance stiffness between the beam guide and block is formulated, and the condition when the simplified stiffness formula hold is retrieved based on numerical computation. Stiffness and mass matrices for a beam with varying cross sections are derived. Frequencies and eigenvectors are obtained by Cholesky decomposition and the Jacobi method. The experiment setup is constructed, and the structure and tools for tunable clearance are considered. Modal properties of the system with clearance are measured and analyzed. Finally, experimental results are used to validate the computed frequencies, and there is good agreement between the measured and predicted natural frequencies.