ASEN5022 Dynamics of Aerospace Structures Lecture 5 The Method of Lagrange s Multipliers This lecture covers Begin with the Lagrangian L T V Identify the constraints in the system Augment the constraints to the Lagrangain by multiplying with an unknown for each constraint Apply the formalism of the Euler Lagrange equations of motion by treating the constraint functional as part of the system Lagrangian Demonstrate the method via example problems 1 Fig 1 Two DOF Example Partitioned into Two Systems The Euler Lagrange equations of motion are obtained from the system energy that consists of the kinetic energy potential energy and virtual work due to nonconservative forces In other words while enabling the derivation process simpler the Euler Lagrange formalism eliminate an important information the reaction forces or moments on joints and kinematic constraints such as on the boundaries Often both the designers and analysts need to know joint force levels so that necessary 2 joint articulations can be carried without jeopardizing the system safety A case in point is the human joints which repeatedly over loaded can cause excessive bone deformations and arthritis The question is How do we re introduce the constraint forces reaction forces within the EulerLagrange formalism This was originally developed by Lagrange and described in his book Me canique Analytique published in 1788 This is accomplished as follows Let s consider holonomic cases viz the constraint conditions that can be explicitly stated in terms of position vectors and consider a two dof spring mass system shown above Suppose we would like to know the reaction force between mass m1 and spring k2 and the reaction force between spring k1 and the attached boundary We now proceed with the procedure that leads to the determination of the reaction forces Note that other partitionings are possible each of which may lead to different constraints 3 Step by step procedure for the method of Lagrange s multipliers Step 1 Partition the system into completely free subsystems Step 2 Identify the conditions of constraints between the completely free systems Step 3 Construct the energy of each of the completely free subsystems Step 4 Obtain the total energy by summing the energy of each of the completely free subsystems Step 5 Append the conditions of constraints by multiplying each with an unknown coefficient multiplier to the total system energy kinematic and potential 4 The system Lagrangian of the assembled system is given by L T V 1 1 T m1x 21 m2x 22 2 2 1 1 V k1x21 k2 x2 x1 2 2 2 W f1 t x1 f1 t x2 5 1 Hence the equations of motion for the assembled system are given via the Euler Lagrange formalism as m1 0 x 1 k1 k2 k2 x1 f1 t 5 2 x 0 m k k x f t 2 2 2 5 2 2 2 Derivation of equations of motion for partitioned system via the method of Lagrange s multipliers Step 1 Partition the system done in the preceding figure Step 2 Identify the conditions of constraints Between k1 and the fixed end 0g x0 xg 0 Between m1 and k2 13 x1 x3 0 5 3 Step 3 Energy of two completely free subsystems For subsystem 1 1 T1 m1x 21 2 1 V1 k1 x1 x0 2 2 W 1 f1 t x1 5 4 1 V2 k2 x2 x3 2 2 W 2 f2 t x2 5 5 For subsystem 2 1 T2 m2x 22 2 6 Step 4 Sum the total system energy T T1 T2 V V1 V2 W W 1 W 2 5 6 Step 5 Append the constraint functional 0g 0g 13 13 x0 xg 0g x1 x3 13 5 7 The total Lagrangian of the partitioned system with the appended constraint functional is thus given by L x 1 x 2 xg x0 x1 x3 0g 13 T1 T2 V1 V2 xg x0 x1 x3 0g 13 7 5 8 Partitioned equations of motion are obtained by utilizing the Euler Lagragne equations x1 term x2 term x3 term x0 term 0g term 13 term m1x 1 k1x1 k1x0 13 f1 t m2x 2 k2x2 k2x3 f2 t k2x2 k2x3 13 0 k1x1 k1x0 0g 0 x0 xg 0 x1 x3 0 8 5 9 Matrix form of the partitioned equations of motion m1D2 k1 0 0 k1 0 1 x1 f1 2 0 m D k k 0 0 0 x f 2 2 2 2 2 0 k k 0 0 1 x 0 2 2 3 k1 0 0 k1 1 0 x0 0 0 0 0 1 0 0 x og g 1 0 1 0 0 0 0 d2 D 2 dt 2 5 10 13 If the above equation is applicable when the ground is moving such as the earthquake motion or when the spring mass system is attached to a lager structures such as the main structures of a ship airplane or vehicle Symbolically the above equation can be expressed as A C x f x T C 0 b 9 5 11 It is noted that solution of 5 11 provides both the displacement x and the reaction forces Question Can one obtain the equations of motion for the assembled system 5 2 The answer is yes To this end we observe the following relation between the partitioned degrees of freedom and the assembled ones as x 1 0 0 1 x 1 x2 0 1 0 x Lxg 5 12 x2 x3 1 0 0 x 0 x0 001 Substituting the above assembling operator L into 5 11 we obtain T T T L f L A L L C x g T xb C L 0 10 5 13 Simple matrix multiplications show 0 0 T L C 0 0 10 2 k2 k1 m1D k1 k2 2 K LT AL k2 m2D k2 0 k1 0 k1 Therefore 5 13 reduces to 2 f1 x1 k2 k1 0 m1D k1 k2 2 f k m D k 0 0 x 2 2 2 2 2 k1 0 k1 1 x0 0 0 0 1 0 og xg 11 5 14 5 15 Finally invoking the boundary condition x0 xg 0 5 16 x1 f1 m1D k1 k2 k2 f 2 k2 m2D k2 x2 2 5 17 we arrive at 2 Note that solution of 5 15 provides the reaction force og at the left end support whereas 5 17 does not 12 Second Application of the method of Lagrange s multipliers Kinematics of Subsystem 1 rA0 xi yj vA0 x i y j 5 18 rC 0 xC 0 i yC 0 j L rA00 rC 0 sin i cos j 2 L rB 0 rC 0 sin i cos j 2 5 19 Kinematics of Subsystem 2 Step 1 Partitioning is done in the preceding figure 13 Fig 2 Partitioned sliding mass and pendulum system 14 Step 2 Identify the conditions of constraints 1 Between mass m and the ground y yg j 0 2 Joint at A rA0 rA00 0 L x xC 0 sin i 2 y yC 0 5 20 L cos j 0 …