Symmetric Matrix Conditions for a Spring-Mass System

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SUMMARY

The discussion focuses on modeling an undamped spring-mass system with three masses and four springs using differential equations. The equations are expressed in matrix notation as x' = Kx + g(t), where the state vector is x = (x1, x2, x3)T. The matrix K is derived from the coefficients of the masses (m1, m2, m3) and spring constants (k1, k2, k3, k4). It is established that K is symmetric for any values of the masses and spring constants, leading to the conclusion that the conditions for K's symmetry are inherently satisfied.

PREREQUISITES
  • Understanding of differential equations and their applications in mechanical systems.
  • Familiarity with matrix notation and linear algebra concepts.
  • Knowledge of spring-mass systems and their dynamics.
  • Basic grasp of undamped systems and external forces in mechanical contexts.
NEXT STEPS
  • Explore the derivation of matrix K for various spring-mass configurations.
  • Study the implications of symmetry in matrices and its effects on system stability.
  • Investigate the role of external forces in modifying the behavior of spring-mass systems.
  • Learn about the numerical methods for solving differential equations in mechanical systems.
USEFUL FOR

Mechanical engineers, physicists, and students studying dynamics and control systems will benefit from this discussion, particularly those interested in modeling and analyzing spring-mass systems.

MysticalSwan
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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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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:
 

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