Viscoelastic damping material has been widely used in engineering machinery to absorb vibration and noise. Dynamic behavior of the viscoelastic damping material is mainly affected by temperature and frequency. Classical dynamic behavior equations of the viscoelastic damping material have complex structures with multiple and ambiguous parameters. So a novel variable-order fractional constitutive model (VOFC) is established based on the variable-order fractional operator. Then the viscoelastic dynamic equations are derived by Laplace transform of the VOFC model. The dynamic thermomechanical analysis tests by the three-point bending mode are carried out at variable temperatures and frequencies, and the frequency spectrum of the dynamic behavior, i.e. the loss modulus, storage modulus and loss factor are obtained. Against the test data, the VOFC model is compared with classical models such as the integer-order Maxwell model, constant fractional-order Kelvin-Voigt model, constant fractional-order Maxwell model and constant fractional-order standard linear solid model. The results show that the VOFC model can describe dynamic behavior of the viscoelastic damping material at different temperatures and frequencies more accurately. And the VOFC model has simpler structure and only two parameters with clearly physical meaning. Finally, a calculation software is designed and developed, which simplifies the process of predicting viscoelastic dynamic behavior in a wide frequency range.