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

Table 2

Modal force and displacements for second-order system with fractional order damping.

Mode	Force	Displacement
0	𝛿(𝑡)	𝑡
1	−𝑡	−𝑡^5/2−𝑡Γ(3.5)Γ(^34)
2	𝑡^5/2+𝑡Γ(3.5)Γ(^34)	𝑡^4𝑡Γ(5)+2^9/2+𝑡Γ(5.5)Γ(^56)
3	−𝑡^4𝑡Γ(5)−2^9/2−𝑡Γ(5.5)Γ(^56)	−𝑡^11/2−𝑡Γ(6.5)^6−𝑡Γ(7)^13/2−𝑡Γ(7.5)Γ(^78)
⋯	⋯	⋯