Assembling mode shapes by least squares

Assembling (or 'gluing') mode shapes identified from multiple setups is a problem frequently encountered in full-scale modal tests that cover a large number of locations with a limited number of sensors. Mode shapes identified in individual setups can have different sense and scaling. Depending on the number of reference degrees-of-freedom (dofs) and the quality of identified mode shapes, implementation issues can arise when determining the optimal mode shape that compromise among different setups. This paper presents a theory with an automated procedure for determining the optimal mode shape that fits the mode shapes identified from multiple setups in a least square sense. The measure-of-fit function is defined as the squared difference between the theoretical and identified mode shapes suitably oriented and scaled to the same norm. Due to the nonlinear nature of the objective function, the optimal mode shape cannot be determined analytically as in conventional least square problems. A fast iterative procedure is proposed, making use of partially optimal solutions that can be derived analytically. The proposed method can be implemented in an automated manner without the need to select the reference dof or setup for scaling purpose. It is applied to assembling mode shapes identified from ambient vibration tests of two full-scale structures.