Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Transcribed Image Text:**Problem Statement:**
Determine the equations to find the nodal displacements using the finite element direct method for the system as shown below. Assume that all free nodes undergo translation in \( x \) (1 DOF at each node) and that external forces \( F_1 \) and \( F_2 \) are known. Nodes 3 and 4 are fixed. The spring constants are known too. Assume \( +x \) is to the right.
Derive all equations and apply boundary conditions—leave your answer in terms of all known variables. **Do not rename the nodes or elements!**
**Diagram Explanation:**
The diagram illustrates a mechanical system consisting of four nodes and three springs, subjected to external forces:
- **Nodes:**
- Node 1 and Node 2 are free to move in the \( x \)-direction.
- Node 3 and Node 4 are fixed.
- **Springs:**
- Spring \( k_1 \) connects Node 1 to Node 3.
- Spring \( k_3 \) connects Node 1 to Node 2.
- Spring \( k_2 \) connects Node 2 to Node 4.
- **Forces:**
- \( F_1 \) is applied to Node 1 in the \( +x \) direction.
- \( F_2 \) is applied to Node 2 in the \( +x \) direction.
- **Displacements:**
- \( u_1 \) is the displacement of Node 1.
- \( u_2 \) is the displacement of Node 2.
The objective is to derive the equations for \( u_1 \) and \( u_2 \) using the finite element method, reflecting the system behavior under the given forces and constraints. Boundary conditions at Nodes 3 and 4 (fixed nodes) must be applied in the solution.
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