Forced Vibration Theory: Deriving Solutions for Displacement

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In forced vibration theory, the solution for displacement is derived from both the complementary and particular solutions. The complementary solution addresses homogeneous linear ordinary differential equations (ODEs), while the particular solution pertains to non-homogeneous linear ODEs. The linearity of the differential operator allows for the combination of these two solutions to form a complete set of solutions for the ODE. Understanding this distinction is crucial for accurately analyzing forced vibrations in systems. The discussion emphasizes the importance of both solutions in deriving comprehensive displacement equations.
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in deriving the solution for displacement in forced vibration theory is derived

for complimentary solution and particular solution.

Can anyone throw some light on the above solution. Why there are two solutions?
 
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Because the complimentary or fundamental solutions are for homogenous linear ODEs, while the particular solution is for non-homogenous linear ODEs, but because of the linearity of the differential operator the combination of both solutions will be the complete set of solutions for the ODE.
 

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