Question

2. The equations of motion of this system are; 12Y"+7Y'+247-62-122=0 62"' +62 +12Z- 6Y-12Y=f(t) 7W"+7W'+14W-7Y-14Y=ft) Where Y,Z and W are deformations of the masses and springs. Put these equations into state variable form and express the model as matrix vector equation if output of the system is Y. Energy Storage Element State Variable mi X = Y' m2 X2 = Z' m3 X3 =W' ki X4 = Y k2 Xς k3 X6 = W Faz Seats Body Suspension Wheel Road Datum level

Answer 1

State Variable form: Given equation of motions are; 12Y" + 7Y' + 24Y – 6Z' – 12Z = 0 6Z" + 6Z' + 12Z - 6Y' – 12Y = f(t) 7W" + 7W' + 14W – 7Y' – 14Y = f(t) State variables are; X1 = Y' X2 = Z' X3 = W' X4 = Y X5 = Z X6 = W Rewrite the given equations of motions: 1 XL = = Y" = (-7Y' – 248 + 62° + 122) = (-12x; – 2x. +3x2+x3) 12

1 X2 = Z" = 6 (-6Z' – 12Z + 6Y' + 12Y + f(t)) f(t) 6 = (-X2 – 2x3 + x2 + 2X. + 109) 1 X3 = W" = (-7W' – 14W+7Y' + 14Y + f(t)) = (-x3 – 2% +X, + 2X. + 19) *4 = Y' = X X = Z' = X2 Xo = W' = X3

0 1 0 IrX1 1/6 X2 X2 X3 | 1/7 [f(t)] = ܢܢ + X4 The state space model is represented as, • Å = AX + Bu [X 7 1 0 -2 X2 12 2 1 -1 0 2 0 -1 2 1 0 0 0 0 0 0 0 0 0 7 1 0 -2 12 2 1 -1 0 2 [X] = 1 0 -1 2 1 0 0 0 0 0 0 0 0 1 0 -2 0 0 0 0 0 -2 0 0 0 X. X5 LX Lie 1 0 0 0 0 176 1 0 11/17 [u] -2 0 0 0 0 0 -2 [X] + 0 0 0 1 0 0 0

• Y = CX + Du Y2 Xi X2 X3 Y3 = = [1 1 1 0 0 0] YA + [0]u Y LY LX -