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The torsional composite oscillator enables the mechanical damping, the strain amplitude, and the shear modulus of a specimen to be determined quickly and accurately, making it a powerful technique for measuring transient phenomena. In this paper the equivalent circuit for the composite oscillator is derived, and the resulting formulas for the mechanical damping and strain amplitude are confirmed by various alternative methods of calibration. Experimental arrangements and procedures are described which indicate how this technique can be used for strain amplitudes ranging from 10-12 to 10-3.