Quadratic Mode Shape Components From Ground Vibration Testing

Points on a vibrating structure generally move along curved paths rather than straight lines. For example, the tip of a cantilever beam vibrating in a bending mode experiences axial displacement as well as transverse displacement. The axial displacement is governed by the inextensibility of the neutral axis of the beam and is proportional to the square of the transverse displacement; hence the name “quadratic mode shape component.” Quadratic mode shape components are largely ignored in modal analysis, but there are some applications in the field of modal-basis structural analysis where the curved path of motion cannot be ignored. Examples include vibrations of rotating structures and buckling. Methods employing finite element analysis have been developed to calculate quadratic mode shape components. Ground vibration testing typically only yields the linear mode shape components. This paper explores the possibility of measuring the quadratic mode shape components in a sine-dwell ground vibration test. This is purely an additional measurement and does not affect the measured linear mode shape components or the modal parameters, i.e., modal mass, frequency, and damping ratio. The accelerometer output was modeled in detail taking into account its linear acceleration, its rotation, and gravitational acceleration. The response was correlated with the Fourier series representation of the output signal. The result was a simple expression for the quadratic mode shape component. The method was tested on a simple test piece and satisfactory results were obtained. The method requires that the accelerometers measure down to steady state and that up to the second Fourier coefficients of the output signals are calculated. The proposed method for measuring quadratic mode shape components in a sine-dwell ground vibration test seems feasible. One drawback of the method is that it is based on the measurement and processing of second harmonics in the acceleration signals and is therefore sensitive to any form of structural nonlinearity that may also cause higher harmonics in the acceleration signals. Another drawback is that only the quadratic components of individual modes can be measured, whereas coupled quadratic terms are generally also required to fully describe the motion of a point on a vibrating structure.