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Stiffness matrices and displacements in spring systems

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For the spring system shown in the attached diagram, find:
a) The stiffness matrix for each element
b) The unconstrained global stiffness matrix
c) The constrained (reduced) stiffness matrix
d) The displacements of node 2 and node 3
e) The reaction forces at node 1 and node 4

For the given spring system in the second attachment, find:
a) The stiffness matrix for each element (symbolically)
b) The unconstrained global stiffness matrix (symbolically)
c) The constrained (reduced) stiffness matrix (symbolically)

Take k1 = 100N/mm, k2 = 200N/mm, k3 = 300N/mm, k4 = 400 N/mm, P1 = 1000N, P3 = 3000N
d) The displacements of nodes 1, 2, 3 (numerically)
e) The reaction forces at node 4 and node 5 (numerically)

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Problem 1

a.) k1 -k1 100 -100
= (stiffness matrix for element 1)
- k1 k1 -100 100

k2 -k2 200 -200
= (stiffness matrix for element 2)
- k2 k2 -200 200

k3 -k3 100 -100
= (stiffness matrix for element 3)
- k3 k3 -100 100

b) k1 -k1 u1 f1ele1
X = (1)
- k1 k1 u2 f2ele1

k2 -k2 u2 f2ele2
X = (2)
- k2 k2 u3 f3ele2

k3 -k3 u3 f3ele3
X = (3)
- k3 k3 u4 f4ele3

P1= f1ele1 = k 1u 1-k1u 2 (4)
P2= f2ele1 + f2ele2 = - k1u1 +u2(k1 +k2) -k2u3 (5)

P3= f3ele2 + f3ele3 = - k2u2 +u3(k2 +k3) -k3u4 (6)

P4 = f4ele3 = k3u4 -k3u3 (7)

The above equations can be represented as

k1 -k1 0 0 u1 P1

- k1 k2 +k1 -k2 0 u2 P2
x = (8)
0 -k2 k2+k3 -k3 u3 P3

0 0 -k3 ...

Solution Summary

The solution calculates stiffness matrices for different elements of springs systems as well as displacements and reaction forces.

See Also This Related BrainMass Solution

Mechanical Vibration and Springs

The machine shown in the figure is rigidly anchored on a rigid concrete block and the mass of both is 3000 kg. The block has height h = 1 m and square horizontal section with a = 1.2 m and is supported at the corners by 12 springs of stiffness k = 2000 N/m.

i) Derive the equations of motion of the system using both Newton's law of motion and Lagrange's equations methods.

ii) Find the natural frequencies and mode shapes of the system using both hand calculations and Matlab.

iii) A 200-kg mass falls on the middle of one edge of the block and stayed in contact with it. Neglecting the change in the mass, determine the resulting motion.

iv) If the block is isolated by layers of elastic materials in a rigid concrete base. Design the isolation if the machine has parts that rotate at 250, 500, and 1500 rpm.

v) If there is unbalanced mass of 0.1 kg on the machine at a distance of 0.25 m from the axis of rotation and angular velocity of 1750 rpm. Find the resulting vibration.

The problem has 3 degrees of freedom x, y, and theta.
The moment of inertia for the system can be calculated by using standard formula, however, you can use a value of 640 kgm^2 for the block moment of inertia.

See attachment for diagram.

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