This is Mechanical Engineering. Please answer with clear work and answers. Thank you **Problem Statement:**1. For the spring assemblage shown below, determine the nodal displacements, the forces in each element, and the reaction force.**Diagram Description:**The system consists of a series of springs arranged in a linear fashion. - **Node 1 to Node 2:** - A spring with a stiffness of 4000 N/m connects node 1 and node 2. An external force of 100 N is applied at node 2 in the direction towards node 3.- **Node 2 to Node 3:** - Another spring with a stiffness of 4000 N/m connects node 2 to node 3. An external force of 200 N is applied at node 3 in the direction away from node 2.Each node is represented by a small circle, and the springs are represented by zigzag lines indicating the connection between the nodes.

The stiffness matrix for each element is:here, k is the stiffness matrixFor element k1 is:For…

Answered: 1- For the spring assemblage shown…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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This is Mechanical Engineering. Please answer with clear work and answers. Thank you

**Problem Statement:**

1. For the spring assemblage shown below, determine the nodal displacements, the forces in each element, and the reaction force.

**Diagram Description:**

The system consists of a series of springs arranged in a linear fashion.

- **Node 1 to Node 2:**
- A spring with a stiffness of 4000 N/m connects node 1 and node 2. An external force of 100 N is applied at node 2 in the direction towards node 3.

- **Node 2 to Node 3:**
- Another spring with a stiffness of 4000 N/m connects node 2 to node 3. An external force of 200 N is applied at node 3 in the direction away from node 2.

Each node is represented by a small circle, and the springs are represented by zigzag lines indicating the connection between the nodes.

Expert Solution

Step 1: Solution:

The stiffness matrix for each element is:

$open square brackets k subscript e close square brackets equals open square brackets table row cell k subscript e end cell cell negative k subscript e end cell row cell negative k subscript e end cell cell k subscript e end cell end table close square brackets$

here, k is the stiffness matrix

For element k₁ is:

$open square brackets k subscript 1 close square brackets equals open square brackets table row 4000 cell negative 4000 end cell row cell negative 4000 end cell 4000 end table close square brackets$

For element k₂ is:

$open square brackets k subscript 2 close square brackets equals open square brackets table row 4000 cell negative 4000 end cell row cell negative 4000 end cell 4000 end table close square brackets$

The global stiffness matrix is:

$open square brackets k close square brackets equals open square brackets k subscript 1 close square brackets plus open square brackets k subscript 2 close square brackets$

Substituting in the equation, we get

$open square brackets k close square brackets equals open square brackets table row 4000 cell negative 4000 end cell row cell negative 4000 end cell 4000 end table close square brackets plus open square brackets table row 4000 cell negative 4000 end cell row cell negative 4000 end cell 4000 end table close square brackets open square brackets k close square brackets equals open square brackets table row cell k subscript 1 end cell cell negative k subscript 1 end cell 0 row cell negative k subscript 1 end cell cell k subscript 1 plus k subscript 2 end cell cell negative k subscript 2 end cell row 0 cell negative k subscript 2 end cell cell k subscript 2 end cell end table close square brackets open square brackets k close square brackets equals open square brackets table row 4000 cell negative 4000 end cell 0 row cell negative 4000 end cell cell 4000 plus 4000 end cell cell negative 4000 end cell row 0 cell negative 4000 end cell 4000 end table close square brackets open square brackets k close square brackets equals open square brackets table row 4000 cell negative 4000 end cell 0 row cell negative 4000 end cell 8000 cell negative 4000 end cell row 0 cell negative 4000 end cell 4000 end table close square brackets$

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