PROGRAMA DE DOUTORAMENTO EM ENGENHARIA MECÂNICA Non-linear Vibrations of Tow Placed Variable Stiffness Composite Laminates Hamed Akhavan Doctoral Program in Mechanical Engineering Supervisor: Pedro Manuel Leal Ribeiro October 2015 c Hamed Akhavan, 2015 Abstract Vibrations of structures and mechanical systems have plenty of advantages, but also disadvantages. Notably, vibrations can lead to failures or damage of equipment. Therefore, knowing more about the vibratory behaviour helps engineers to control damage and to take preventive measures in order to avoid damages or malfunctions. For a long time, optimising material properties was a goal in material design. An important aspect in the optimisation of fibrous composite materials is the fibre angles in different layers of the laminate. In traditional composite laminates, the fibre orientation is constant in a ply, but some researchers paid attention to the possibility of changing the fibre orientation angle inside a ply. A laminate with plies which have curvilinear fibre paths is considered as a variable stiffness composite laminate (VSCL). Due to the variation of in-plane stiffness in a VSCL, these laminates find special applications in naval, aerospace, or other structures (e.g. in solar array wings, aircraft wings, fuselages and tail sections). Vibrations of VSCL plates are analysed in this thesis. To define a curvilinear fibre path, it is assumed that the fibre angle varies linearly along one of the Cartesian axes, say x, from a value T0 at the centre of the panel to the value T1 at the panel edges, x = ±a=2, where the panel has length a. The fibre path defined in this fashion is called the reference fibre path; other fibre paths in the ply can be produced by shifting - or copying - this path with a constant distance through direction y. During the manufacture process, there is a limitation on the fibre curvature radius; if the fibre path curvature is less than a determined amount, there will be kinking or buckling of the tows in the ply. In this thesis, a p-version finite element based on a third-order shear deformation theory is used to model the displacements. Non-linear terms based on von Kármán strain–displacement relations are included. Using the principle of virtual work, the equations of motion are obtained in matrix form. The order of the equations of motion (regarded as the order of the full model) can be reduced using static condensation and modal coordinates (reduced-order models). The transient responses of the equations of motion of VSCL models are obtained using Newmark method; the same method is employed to obtain solutions in static problems, using time as an auxiliary parameter. The program is implemented in an in-house Visual Fortran code. To find the periodic oscillations of VSCL plates in free and forced vibrations, the shooting method is used, together with a Runge-Kutta-Fehlberg method for integration, again in an in-house Visual Fortran computational code. In a traditional composite laminate the stiffness does not change - at the macro scale - in the laminate, while, in a VSCL laminate stiffnesses change spatially with respect to at least one coordinate. It is found that this change in local stiffness changes the natural frequencies and mode shapes of vibration of the laminate, both in the linear and the non-linear regimes. Using VSCL, the stress field and the strengths in the laminate may change in a different way from what would occur in a laminate with rectilinear fibres. Consequently, the load magnitudes at which damage onset appears, as well as the position of damage onset, can be changed. i ii The effect of geometrical imperfections (out-of-planarity) is included in the p-version finite element too. The free and forced, non-linear, periodic vibrations of VSCL plates are analysed using displacement time-histories, phase plane plots, frequency spectra and shapes assumed by the plates. Bifurcations from main branches of the backbone of frequency response curves are detected and analysed. The effect of imperfection of the plate on the hardening and softening of the free vibration backbone curves are detailed. Resumo Materiais compósitos laminados são frequentemente utilizados em elementos estruturais sujeitos a vibrações. Um aspeto importante na otimização de materiais compósitos reforçados com fibras é a orientação destas últimas. Em laminados compósitos tradicionais, a orientação das fibras dis- postas na mesma lâmina é constante, mas alguns investigadores têm explorado a possibilidade de alterar a orientação das fibras numa lâmina, utilizando fibras curvilíneas. Um laminado com fibras curvilíneas é um laminado compósito com rigidez variável (VSCL do inglês Variable Stiffness Composite Laminates). Vibrações e a deformação em regime estático de placas VSCL são analisadas nesta tese. Considera-se, em particular, o caso de fibras cujo ângulo é uma função bilinear de um dos eixos cartesianos. Limitações impostas à curvatura das fibras pelo processo de fabrico e pela ocorrência de encurvadura são tomadas em consideração. Um aspeto fundamental desta tese, é o desenvolvimento de um elemento finito da versão p para placas VSCL, baseado numa teoria de deformação de corte de terceira ordem. Os termos não-lineares das relações entre deformações e deslocamentos de von Kármán são considerados, para se conseguir estudar grandes deslocamentos (regime geometricamente não-linear). Usando o princípio dos trabalhos virtuais, as equações de movimento são obtidas na forma matricial. O número de equações de movimento do modelo completo foi reduzido utilizando condensação estática e coordenadas modais (modelos de ordem reduzida). No que concerne à resolução das equações do movimento, foram essencialmente utilizados dois métodos. Para obter soluções em regime estático e respostas dinâmicas em regime transiente, as equações do movimento foram resolvidas pelo método de Newmark, no primeiro caso utilizando o tempo como um parâmetro artificial auxiliar. Para encontrar as oscilações periódicas de placas, tanto em vibrações livres como forçadas, foi utilizado o método do disparo (método shooting), em conjunto com um método Runge-Kutta-Fehlberg para a integração no tempo das equações do movimento. Em todos os casos, recorreu-se a códigos computacionais escritos em Fortran 90. Enquanto numa placa laminada compósita com fibras retilíneas a rigidez não muda - à escala macro - de um ponto para o outro do plano médio, numa placa VSCL a rigidez muda espacial- mente. Verificou-se que esta alteração na rigidez local provoca alterações nas frequências naturais e nos modos de vibração do laminado, tanto em regime linear como no regime geometricamente não linear. Usando VSCL, o campo de tensões e a resistência do laminado muda de uma forma diferente do que ocorreria num laminado com fibras retilíneas. Por conseguinte, as magnitudes de carga para a qual dano se inicia, bem como o local em que se dá esse início, são alterados. O efeito de imperfeições geométricas (alteração da planaridade) foi incluído no elemento finito da versão p desenvolvido. Vibrações periódicas nos regimes livre e forçado, não lineares, de pla- cas VSCL foram analisadas utilizando gráficos dos deslocamentos em função do tempo, planos de fase, espectros de frequência e as formas assumidas pelas placas. Bifurcações de ramos prin- cipais da curvas dorsais (backbone curves) e de curvas de resposta de frequência foram detetadas e analisadas. O efeito de imperfeições geométricas da placa sobre a inclinação das curvas dorsais iii iv foi descrito. Acknowledgments I am grateful to Almighty Allah, whose blessings have always been enormous, who gave me the ability to complete this task. I express my heartiest thanks to my parents and family members who encouraged me at every stage of life and saw in me a successful person. It is a great honor and privilege for me to record this deep sense of gratitude and indebtedness to my supervisor Professor Pedro Ribeiro1 who guided me during all the course of this work. He was always there to help me find my way out of any problems. His critical observations helped me improve the analytical skills required for the research process as well as the writing skills required for disseminating the results. I am also thankful to Professor Marcelo Moura1 and to Professor José Rodrigues1 who ex- tended every help to me to have a better understanding of the subject. Here, I am going to appreci- ate the support from Professor Bogdan Epureanu2 who received me very well during a short visit at the University of Michigan, Ann Arbor. I do not forget all the supports from Professor Antonio Ferreira1 during my study in Porto. I appreciate his great helps. Joining the program in University of Porto proved to be a door to make friends from around the world, know new cultures and understand new perspectives of life. I would also like to mention my colleagues and friends whose silent support led me to achieve the goal of completing this work. They are Dr. Julija Vasiljevska, Dr. Rosa Crespo, Dr. Stanislav Stoykov, Dr. Hadi Fanaee, Dr. Marzieh Moeenfard, Dr. Arif Ur Rahman, Dr. Gloria Lema, Dr. Jakov Krstulovic, Dr. David Miranda, Dr. Mariana Seabra, Dr. Mohsen Mirkhalaf, Eng. Tiago Lima, Eng. Romina Fernandes, Eng. Raul Moreira, Eng. Gonçalo Carneiro, and Eng. João Reis. Also, during my stay in Ann Arbor, I had very good time with Dr. Hossein Rokni and his wife. It was a wonderful experience with all of them. During part of this work, I benefited from a research grant of project "Vibrations of variable stiffness composite laminated panels", reference PTDC / EME-PME / 098967 / 2008, a project funded by the Portuguese Science and Technology Foundation (Fundação para a Ciência e a Tec- nologia - FCT3), with national funds (PIDDAC). Also, I would like to mention the support by another grant from FCT with reference number SFRH / BD / 81707 / 2011. These supports are gratefully acknowledged. Finally, but most importantly, I wish to thank my wife Dr.