How to model vibration damped by applied/contact force? I am computer science student and I am working on a project which needs to know how the vibration of phone is damped by applied force (when human touches the phone). I have read many articles about "forced mass-spring system". The following figure is the model I expect to have for my scenario: (phone is placed at table, vibrated by vibration motor, and damped by hand)

However, when I try to solve the vibration amplitude of this system. The vibration amplitude is not related to the applied force (Fh). I also simulate this system by Simulink and gets the same results:

Even though the math looks right, (the applied force only changes the position of equilibrium), but it is very counter-intuitive in the real-world (i.e., vibration should be decreased when enough force is applied). Moreover, based on my experimental measurement via laser vibrometer. The vibration amplitude does decrease when force is applied by hand.
I have no background at this mechanical system. I have tried my best to read online tutorial and got this system model but it doesn't work (at least not fits real world scenario). Any help to point out what is wrong in my system is very welcome. Any keyword that I should Google is also very helpful. Please help me :(
 A: I think the reason is that you are just adding these two forces i.e. Force by hand and Force by the vibrating mechanism. But you have to understand that the hand comes in contact to the mobile surface only when the surface is going up(towards the hand) i.e. when the positive cycle of the vibration is taking place. When the vibration is moving on the other direction the force on it by hand is not same.
I'm sorry I didn't include any figures but I hope the general idea is clear. If not, please ask.
A: thanks for the reply.
Here are two candidate answers I got from my friends and the other forum:


*

*Vibration amplitude decreases because the system damping factor is increased when the force of hand is applied

*Effective mass is increased when force is applied (because the table is vibrating also with the phone)
Both of those factors are not considered on my current system model, so it can't find the relation between force and vibration amplitude.
However, it is still unknown how to model this "damping factor" and "effective mass" to the relation of "applied force" :(
A: Thank you for your help. Here is a new model built by my friends. Even though this model is not perfect (don't consider the table and hand will also vibrate with phone), I think it is good enough for me to explain it.

In this mode, phone equilibrium changes according to applied force. However, it is impossible to go below the table since the phone is rigid body. Thus, the system spring constant increased to K for meeting the system constraints.
Assume the energy is conserved, the relation between force and amplitude is:
0.5 K0 A0^2 = (FhA^2) / (A0 - A)

