9 Ritz method

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Aerospace Structure – Exercises 10 1 Exercise #10 Ritz Method Aerospace Structure – Exercises 10 2 Ritz Method - EX. 1 Find the first two natural axial frequencies of the following system. Apply a Ritz approximation by us ing as shape functions two polynomials of the first and th e second order, which satisfy essential Boundary Conditions. Consider: E = 70000 MPa A = 900 mm 2 L = 600 mm K = 12000 N/mm ρ = 2800 kg/m 3 Solution a) Formulation of PVW Conceptually the spring at the end of the rod shoul d be considered a part of the system in this type o f free vibration problem. Actually, the same results could be obtained by considering the spring an external element that apply forces to the rod. The problem will be solved by applying PVW. The vir tual work of the spring will be expressed in terms of the displacement at the end z = L. The essential boundary conditions in the problem is simply: ( ) 00 =zw The separate contributions to the complete expression of PVW are hereby listed: 1) Deformation work in the rod: ∫ ′′=− L zzrod d dz wEA wW 0 δδ 2) Deformation work in the spring: ( ) ( )LKw L wW spring d δδ =− The expression of the virtual work can be seen as t he variation of the elastic energy stored in the spring: ( ) 2 2 1 LKw spring e =Π − ⇒ ( ) ( )LKw L wW sping ispring e δδδ ==Π −− If the spring is considered an external element, th e external applied force is -Kw(L), because it is opposite to w(L). Hence the external work (that wil l be positive or negative according to the sign of δw(L)) reads: ( ) ( )LKw L wW spring e δδ − =− Hence, the spring contribution change sign dependin g on the how spring is considered (internal or external). However, such choice also lead to insert the spring contribution on different side of the P VW statement and the final result will…