11_Linear_Algebra - Linear Algebra Dr.B Santhosh Department...

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Linear Algebra Dr.B Santhosh Department of Mechanical Engineering Dr.B SanthoshDepartment of Mechanical Engineering Linear Algebra
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Matrices Square matrix Symmetric matrix Diagonal matrix Identity matrix Upper triangular matrix Lower triangular matrix Banded matrix Permutation matrix Augmented matrix Dr.B SanthoshDepartment of Mechanical Engineering Linear Algebra
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Linear Systems Any system whose response is proportional to the input is deemed to be linear Linear systems include structures, elastic solids, heat flow, electromagnetic fields, electric circuits Linear system can be classified into discrete or continuous Discrete system analysis lead directly to linear algebraic equations Dr.B SanthoshDepartment of Mechanical Engineering Linear Algebra
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Truss- a discrete system Equilibrium equations at node 1 V 1 + F 13 sin α = 0 H 1 + F 12 + F 13 cos α = 0 For node2 V 2 + F 23 sin β = 0 F 12 F 23 cos β = 0 For node 3 F 13 sin α F 23 sin β = W 3 F 13 cos α + F 23 cos β = 0 6 equations and 6 unknowns ( H 1 , V 1 , V 2 , F 12 , F 13 , F 23 ) Dr.B SanthoshDepartment of Mechanical Engineering Linear Algebra
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Truss-discrete system In matrix form 0 1 0 0 s α 0 1 0 1 0 c α 0 0 0 0 1 s β 0 0 0 1 0 0 c β 0 0 0 0 s α s β 0 0 0 0 c α c β H 1 V 1 F 12 V 2 F 13 F 23 = 0 0 0 0 W 3 0 Generally [ A ] { x } = { b } Dr.B SanthoshDepartment of Mechanical Engineering Linear Algebra
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Continuous System The behavior of continuous system is described by differential equations (ODE or PDE) Numerical methods can handle only discrete systems, the differential equations should be approximated by a system of algebraic equations The finite difference, finite element and boundary element methods can be used (discretization) Results in a system of algebraic equations which need to be solved Dr.B SanthoshDepartment of Mechanical Engineering Linear Algebra
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Finite Difference Methods for BVP of ODE Finite differences are substituted for the derivatives in the original equation The linear differential equation is thus transformed into a set of simultaneous algebraic equations Heated rod model d 2 T dx 2 + h ( T T ) = 0 Solution domain is divided into a series of nodes.At each node FD approximations can be written for the derivatives Dr.B SanthoshDepartment of Mechanical Engineering Linear Algebra
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For node i d 2 T dx 2 = T i 1 2 T i + T i + 1 Δ x 2 Substitute in the equation T i 1 2 T i + T i + 1 Δ x 2 + h ( T T i ) = 0 T i 1 + ( 2 + h Δ x 2 ) T i T i + 1 = h Δ x 2 T This is an algebraic equation Similar equations can be obtained for all interior nodes (n-1 in number) First and last node values are obtained from BC’s (n-1) simultaneous algebraic equations solve for (n-1) unknowns Dr.B SanthoshDepartment of Mechanical Engineering Linear Algebra
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Advantages
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