In this work, according to the first-order torsional mode shape function computed by ANSYS software and the optimization principle, the relationship between the shear modulus of materials and the first-order torsional frequency of the free plate was established in terms of the energy method. Mode shape coefficients of orthotropic materials (lumbers of Sitka spruce, Scots pine, Beech, et al.) and isotropic materials (steel, aluminum, and glass) were calculated. The correlation between the mode shape coefficients of lumbers in tangential, radial, and cross sections and length-width and width-thickness ratios of the free plate was obtained. Also, the correlation between the mode shape coefficients of isotropic materials and the length-width ratio of the free plate were achieved. The correctness of these correlations was verified by mathematical simulation, dynamic testing, and static testing for shear modulus. The transient excitation method can be used to measure the spectrum of free plates. With the aid of the cross-power spectrum based on the frequency spectrum, the first-order torsional frequency of the free plate can be directly identified. The free-plate torsional mode shape method proposed in this work is applicable not only to testing the shear modulus of isotropic materials such as steel but also to testing shear modulus in the three main sections (GTL, GLR, and GRT) of orthotropic materials such as lumber.
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