Symmetric Matrix Conditions for a Spring-Mass System

In summary, the conversation discussed a system of 3 masses and 4 springs with external forces, modeled by a set of differential equations. The system can be expressed using matrix notation, with the state vector x and the input g(t). The question in part (b) asked about the conditions for the matrix K to be symmetric, but it seems that K is already symmetric for any values of the masses and spring constants.
  • #1
MysticalSwan
3
0
The differential equation that model an undamped system of 3 masses and 4 springs with external forces acting on each of the three masses is

m1x1''=-k1x1+k2(x2-x1)+u1(t)
m2x1''=k2(x1-x2)+k3(x3-x2)+u2(t)
m3x3''=k3(x2-x3)-k4x3+u3(t)​

a)express the system using matrix notation x'=Kx+g(t) for the state vector x=(x1,x2,x3)T. Identify the matrix K and the input g(t).

b) Give conditions m1, m2, m3, k1, k2, k3, k4 under which K is a symmetric matrix.




I am pretty sure I have gotten the first part but I am having trouble even figuring out what the second part means. When I created my matrix K it seems like it is already a symmetric matrix. Any help would be great.
 
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  • #2
I don't undestand part (b) either.

The equations you are given will be symmetric for any values of the m's and k's - so what was the question really asking you about :confused:
 

1. What is a spring-mass system matrix?

A spring-mass system matrix is a mathematical representation of a physical system consisting of a mass and one or more springs. It is used to model and analyze the behavior of the system, such as its displacement, velocity, and acceleration.

2. How is a spring-mass system matrix constructed?

The spring-mass system matrix is constructed by representing the mass and springs as nodes, and connecting them with elements that represent the stiffness and damping properties of the system. The resulting matrix is then solved using mathematical methods to obtain the system's dynamic response.

3. What are the applications of a spring-mass system matrix?

A spring-mass system matrix has many applications in various fields, such as mechanical and civil engineering, robotics, and control systems. It can be used to design and optimize structures, analyze the response of vibrating systems, and develop control strategies for complex systems.

4. How does a spring-mass system matrix differ from a mass-spring-damper system?

A spring-mass system matrix and a mass-spring-damper system are similar in that they both model the behavior of a mass and springs. However, a mass-spring-damper system also includes a damping element, which represents the energy dissipation in the system. The spring-mass system matrix does not include this element, making it a simplified version of the mass-spring-damper system.

5. Can a spring-mass system matrix be used for nonlinear systems?

Yes, a spring-mass system matrix can be used for both linear and nonlinear systems. For linear systems, the matrix can be solved using analytical methods. For nonlinear systems, numerical methods such as finite element analysis can be used to approximate the solution. However, the accuracy of the results may be affected by the level of nonlinearity in the system.

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