By studying the real and imaginary parts of complex natural frequencies varying with the axially velocity, we can determine the natural frequencies and critical speed beyond which the system loses its stability. In Fig. 2, the first two complex natural frequencies are plotted for different axially traveling velocities for the ratio of the elastic and shear modulus Y = 0 and Y = 0.01, respectively. For a homogeneous material, i.e., Y = 0, it is well known that the system presents a sequence of phenomena such as stable-divergence-stable-flutter with the increase of axial velocity. Starting from the velocity c = 0, the real parts of all orders remain zero while the imaginary parts decrease. At c = π, which is called the critical speed, the imaginary part of the first order natural frequency vanishes and the real part begins positive, where divergence occurs. In a small range beyond c = 2π, the real part becomes zero again and the system regains stability. We can find both the positive real part and positive imaginary parts when the natural frequencies of the first and second orders coincide. Hence, the flutter phenomena occur and the system loses stability for the second time in this case. For the axially moving sandwich beam, the natural frequencies are lowered because of the introduction of a soft core. The critical axial velocity is also reduced accordingly. It appears that it is less stable for the axially traveling sandwich material than the homogeneous material. In this investigation, the thin face layers of the sandwich beam are assumed to bear axial loading while the core layer bears only shear deformation. With the increase of the thickness of the core layer, the overall bending stiffness will increase because of the distance between the two layers. Hence, on one hand, the natural frequencies are lowered if the core layer is softer and, on the other hand, the natural frequencies will be greatly increased because of the introduction of the soft layer, which leads to the far distance of the face layers.