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ABSTRACT
A reflected wave superposition method is proposed for an axially traveling string with classical and nonclassical boundaries, based on the reflection of the propagating wave on both sides of the string, combining its initial conditions and the continuity conditions in order to obtain the expressions for the reflected wave. The reflection process, in three phases, is deduced and a determinate expression for the transverse vibration is obtained. The correctness and superiority of the proposed method is verified by comparison with the Newmark-β method for an axially moving string with a fixed and a spring-dashpot boundary.
Nomenclature
| = | Initial amplitude of the middle point of the string | |
| = | Constant of integration | |
| = | Wave propagation speed | |
| = | Kinetic energy | |
| = | Potential energy | |
| F(x) | = | The function of right traveling wave |
| Fi (i = 1, 2, 3) | = | The function of right traveling wave after i − 1 reflections |
| G(x) | = | The function of left traveling wave |
| Gi (i = 1, 2, 3) | = | The function of left traveling wave after i − 1 reflections |
| H(x) | = | Heaviside function |
| = | Stiffness of spring at right end of string | |
| L | = | Function space |
| = | Length of the string | |
| R | = | One-dimensional Euclidean space |
| x | = | Axial position of the point in the string |
| = | Tension of the string | |
| = | Vibration cycle | |
| t | = | Time |
| ta | = | The time periods for the waves F to be reflected at the right boundary totally |
| tb | = | The time periods for the waves G to be reflected at the left boundary totally |
| u | = | Transverse displacement of the string |
| utt | = | Second-order partial derivative of u with respect to t |
| uxx | = | Second-order partial derivative of u with respect to x |
| uxt | = | First-order partial derivative of u with respect to x and t. |
| = | Axial velocity of the string | |
| vl | = | Velocity of the left traveling wave |
| vr | = | Velocity of the right traveling wave |
| α | = | Intermediate variable |
| β | = | Intermediate variable |
| ρ | = | Mass per unit length of string |
| = | Variation of a function | |
| = | Damping factor of a dashpot | |
| φ(x) | = | Initial transverse displacement of the string |
| ψ(x) | = | Initial transverse velocity of the string |
Disclosure statement
No potential conflict of interest was reported by the authors.