1 3 J Braz. Soc. Mech. Sci. Eng. DOI 10.1007/s40430-015-0447-9 TECHNICAL PAPER Vibration analysis of moving anisotropic thin plate, wrapped in translation process Amirali Sadeqi 1 · Mojtaba Mahzoon 2 · Mohamad Hassan Kadivar 2 Received: 2 December 2014 / Accepted: 8 October 2015 © The Brazilian Society of Mechanical Sciences and Engineering 2015 Keywords Natural vibration · Anisotropic · Wrapped plate · Numerical solution · Moving plate · Mode shape List of symbols A ij Extensional stiffness elements D ij Bending stiffness elements E Elastic modulus G Shear modulus, gyroscopic inertia matrix h Thickness of plate K b Bending matrix K t Stretching matrix K r Curved region stiffness matrix K v Velocity matrix k Curvature L 1 , L 3 Lengths of flat segments L 2 Length of wrapped or curved segment N Resultant force M Resultant moment, mass matrix m Number of approximation function terms, number of nodes—longitudinal n Number of approximation function terms, number of nodes—lateral Q, Q Stiffness matrixes in principal and material directions Q(t ), ˙ Q(t ) Reduced order matrix of spatial vector q q ij (t) Temporal or time-dependent variable(vector) R(x, y) Radius of roller or wrapped region T Tension, transposing operator U Transformation matrix u Displacement in x direction V Axial translating speed v Displacement in y direction W ij (x, y) Spatial or time-independent variable (matrix) w Transverse displacement in z direction α Fiber angle Abstract Natural vibration analyses of moving aniso- tropic plates under tension in helically translation pro- cess, considering the effect of stiffness, caused by tension and curvature in both stationary and moving manners, are presented. The paper aims to investigate conflicts between fiber orientation and geometrical asymmetry due to helical wrapping and translation dynamics on modal properties of moving narrow thin plates. Owing to the low bending rigid- ity, the plate is continuously wrapped around the roller, and it is assumed that the structure is composed of three flat– curved–flat segments. The governing equations of motion are extracted by the virtual displacement method and based on the Kirchhoff–Love’s plate–shell theory. In order to achieve the eigenvalue set, the Rayleigh–Ritz method is used together with the orthogonal approximation func- tions. Convergence of the method is also discussed, which shows an acceptable accuracy for less computational costs, compared to the finite element solution. Effects of the fiber orientation and velocity on the natural frequencies are also studied. On account of discrepancy between fiber and helix angles, different modal results are reported for two sides of the plate which are wrapped around the roller. Moreover, mode shapes are illustrated, and contributions of longitudi- nal and lateral directions are highlighted. Technical Editor: Fernando Alves Rochinha. * Amirali Sadeqi a-sadeqi@phdstu.scu.ac.ir 1 Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran 2 Mechanical Engineering School, Shiraz University, Shiraz, Iran