I was simulating the response of a two DoF system with a constant frequency force acting on the second mass. The system was initially at rest. The response obtained through simulation shows that the structure vibrates at three frequencies, the two natural frequencies and the forcing frequency. However the two natural frequencies appear only in the initial part of the response. The system consists of two masses, three springs (one connecting the masses), and two linear viscous dampers. System is fixed on both the sides.
Your results seem plausible. The system oscillates in all 3 frequencies. Due to damping, the oscillations in the eigenfrequencies will vanish quickly, leaving only the steady-state response of constant amplitude in the forcing frequency. Without damping, you should always see a superposition of all 3 frequencies.
The solution of LTI system to harmonic excitation (single frequency) consists of two parts: one is steady state harmonic and other is transient or complementary solution. the steady state part (which has frequency equal to excitation) sustains till excitation where as transient part (combination of natural harmonics) dies down after some short period (?) of time.
Even though the system is at zero initial condition (zero displacement and velocity), the transient part still exist possibly due to the fact that- initially the excitation force imparts energy to the system which tries to excite all the natural modes of the system. This situation you may compare with the free response of muti-dof system to arbitrary initial condition.
You can find a solution to 1–dof problem in any standard text on vibration to see the participation of natural modes in forced vibration at zero initial condition.
Hope this helps
Regards
(You can discuss this problem in detail with Dr Shakti Sing Gupta at IITK. Some seven years back I was enjoying to discuss such problems with him at DIAT, Pune)
Yes, Hauke is correctly said that "Without damping, you should always see a superposition of all 3 frequencies" that I have experienced during simulation of undamped (ideally zero damping) multi-dof systems.
Everything Hauke and Chandrashekhar said is completely correct. I just want to add a few words on the solution of the equation of motion from a more mathematical point of view.
In your example the equation of motion of the two body system is described by a nonhomogeneous second order linear differential equation with constant coefficients. What you actually solve for is a differential equation such as
u'' + au' + bu = f(t)
The solution to this equation is mathematically made up of a complementary solution uc(t) (solution to the homogeneous differential equation) and a particular solution up(t).
u(t) = uc(t) + up(t)
A particluar solution can sometimes be found by guessing but normally a Ansatz for up(t) of the same form as f(t) is made. If you now look at the complete solution you will find that complementary solution die down because of the damping while the particluar solution describes the steady state (stationary) response. Since this response is closely related to the forcing function we obtain the result that Hauke already correctly explained. As already mentioned the solution to your 2 degrees of freedom problem should be included in any text book of engineering mechanics and vibrations.
Thanks to all for your replies. I understand what you are saying. Is there any practical justification like what Chandrashekhar sir mentioned about the force imparting initial energy to the system?
As long as the force actes on the system it always feeds energy to the system (nonconservative system). In the most simple case work is equal to force times displacement. This work is constantly added to the system.
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