Solution of Extraordinary Differential Equations with Physical Reasoning by Obtaining Modal Reaction Series

Table 1

Decomposing the action reaction of second-order mass-spring system.

Mode	Force	Displacement
0	𝑓0=(1/𝑚)𝑓(𝑡)=(1/𝑚)𝛿(𝑡)	𝑥0=𝐷𝑡^−2𝑓0=(1/𝑚)𝑡
Higher modes	Higher modal internal forces	Higher modal internal displacements
1	𝑓1=−(𝑘/𝑚)𝑥0=−(𝑘/𝑚^2)𝑡	𝑥1=𝐷𝑡^−2𝑓1=−(𝑘/𝑚^2)(𝑡^3/3!)
2	𝑓2=−(𝑘/𝑚)𝑥1=+(𝑘^2/𝑚^3)(𝑡^3/3!)	𝑥2=𝐷𝑡^−2𝑓2=+(𝑘^2/𝑚^3)(𝑡^5/5!)
3	𝑓3=−(𝑘/𝑚)𝑥2=−(𝑘^3/𝑚^3)(𝑡^5/5!)	𝑥3=𝐷𝑡^−2𝑓3=−(𝑘^3/𝑚^4)(𝑡^7/7!)
⋯	⋯	⋯