1 For the dynamic system a Determine the equilibrium equatio

1. For the dynamic system,

(a) Determine the equilibrium equations in the generalized coordinate system (degrees of freedom) shown in the figure. m=mass, J=rotary inertia, k=stiffness (spring or torsional stiffness in shafts), etc.

(b) Compute the natural frequencies of the system.

(d) Compute the mass normalized modes and verify that the modes are mass orthogonal.

(3) 2k X2 no S

a) As two mass are connected together by linear springs. Then independent coordinates are position mass of left side & right side so their will two degrees of freedom.

Looking at mass on left side, change in length of spring k2depends on both mass positions (see attached F.B.D)

-k₁x₁ + k₂(x₂-x₁) + f₁(x) = m₁x₁

Above equation suggest there will be two natural frequencies with two possible solutions

= sqrt (k/m)

= sqrt (3k/m)

solving Eigen vector equation

2k -(k/m)m -k X11 0

0 -k 2k-(k/m)m X21 0

Natural frequency is

X1 = X11 {1

b)Modal shape is

ø = 1 1

1 -1

c) Solution for each mass is

x(t) =x_i e^jit

1. For the dynamic system, (a) Determine the equilibrium equations in the generalized coordinate system (degrees of freedom) shown in the figure. m=mass, J=rota

1 For the dynamic system a Determine the equilibrium equatio

Solution

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