2 DOF oscillator max force reponse

[saxymon]

(NOTE, this is not a homework problem, but it sure seems like one)

Hello,

I am mounting a component onto structure and I need to determine the maximum force input into the component.

My system can be represented by a base driven two degree of freedom oscillator:

http://www.freeimagehosting.net/newuploads/xowmr.png

I need to determine the force applied to m2:

F = k2*(x2-x1) + c2*(x2-x1)

This force needs to be a function of (m1,m2,c1,c2,k1,k2,y) and not of (x1,x2).
Basically, for a given input y, what will the be force response on m2.

Every time I solve the system of equations, my result is a function of x1 and/or x2.


Thank you in advance for your help!

p.s. If it is easier, feel free to remove the dampers from the system. An un-damped system will work for my purposes.

 

[saxymon]

As for an attempt at the solution:

my matrix system of equations is...

[m1 0 ; 0 m2] [x''1 ; x''2] + [k1+k2 -k2 ; -k2 k2] [x1 ; x2] = [0 ; k1] [0 y]

Interestingly enough, the force I need to recover is naturally in the lower portion of the equation giving:

F = k1*y - m2*x''2

I make the reduction to

F = k1*y + Wn^2*m2*x2

but still was not able to remove all of the x2's and x1's (after making a number of substitutions).

 

[rude man]

You have two equations and two unknowns (x₁ and x₂). You can solve for them as functions of m1,m2,c1,c2,k1,k2, and y.

(I would use the Laplace transform, but there is of course a time-domain way to solve them also. I don't have that info on hand.

 

[gneill]

Your mechanical system can be modeled as an electrical circuit (an "analog" system). Use a simulator (Spice) to characterize the maximal force versus various "input signals".

Assuming that voltage represents velocity and current represents force, the mechanical elements have particular electrical analogs:

Spring = Inductor
Damper = Resistor
Mass = Capacitor
Force = Current
Velocity = Voltage

Force inputs become current sources. Initial velocities (of masses) become initial voltages on capacitors. Velocity inputs are okay -- they become voltage inputs, so if you want to determine how your system responds to a given motion driving it, characterize the motion in terms of its velocity.

In the above circuit I_o represents the force that's applied to the mass M2.

 

[DeeNos]

Dear researcher,

Thank you for your inquiry. I understand the importance of accurately determining the maximum force input into a component mounted onto a structure. Your system, represented by a base driven two degree of freedom oscillator, is a complex yet common problem in structural engineering.

To determine the force applied to m2, we can use the equation F = k2*(x2-x1) + c2*(x2-x1), where k2 and c2 are the stiffness and damping coefficients of m2, respectively. This force is a function of the system parameters (m1,m2,c1,c2,k1,k2) and the input y, rather than the displacement values (x1,x2). Thus, the force response on m2 can be calculated for a given input y.

Solving the system of equations for the force response on m2 may seem daunting, but there are various numerical methods and software programs that can assist with this task. It is important to consider the damping effects, as removing the dampers from the system may result in inaccurate or unstable solutions.

I recommend using a simulation software, such as MATLAB or ANSYS, to model your system and determine the force response on m2. These programs allow for the inclusion of damping effects and can provide a more accurate and efficient solution compared to solving the equations manually.

I hope this information helps in your research. Best of luck with your project!

Sincerely,