1. Dr. Mian Ashfaq Ali
School of Mechanical and Manufacturing Engineering (SMME)
Mechanical Vibrations ME-421
2. Forced Vibration Analysis
Equations of motion of a general two degree of freedom system under external forces can be
written as
Where X1 and X2 are, in general, complex quantities that depend on ω and the system parameters.
3. Forced Vibration Analysis (cont’d.)
Substituting Fj and xj in first equation will lead to
Like in case of free vibration we can define mechanical
impedance Zrs(iω) as
Using Zrs(iω) in (A)
where
−𝜔2𝑚11 + 𝑖𝜔𝑐11 + 𝑘11 −𝜔2𝑚12 + 𝑖𝜔𝑐12 + 𝑘12
−𝜔2𝑚21 + 𝑖𝜔𝑐21 + 𝑘21 −𝜔2𝑚22 + 𝑖𝜔𝑐22 + 𝑘22
𝑋1
𝑋2
=
𝐹10
𝐹20
→ (𝐴)
𝑍 𝑖𝜔 𝑋 = 𝐹0 → (𝐵)
4. Forced Vibration Analysis (cont’d.)
Now equation (B) can be solved to obtain
Where the inverse of impedance matrix is given by
Above will lead to following solution
By substituting X1 and X2 in solution “xj=Xjeiωt” we can find the
complete solution, x1(t) and x2(t)
𝑋 = 𝑍 𝑖𝜔 −1
𝐹0
5. Example Problem
Find the steady state response of the system shown in fig, when mass m1 is excited by the force
F1(t) = F10 cosωt.
