Driven Harmonic Sys. Displacement Amp.

[BadAtMath6]

Homework Statement

I need to find the steady state displacement amplitude for a driven harmonic system.

Homework Equations

mass (m) = .5kg
stiffness (s) = 100 N/m
mechanical resistance (Rm) = 1.4 kg/s
force (f) = 2cos5t

The Attempt at a Solution

What I know:
angular freq (w) = 5 rad/sec
magnitude of force (F) = 2N
necessary diff. eq.: m$\ddot{x}$ + Rm$\dot{x}$ + sx = Fexp(iwt)
complex displacement: x=$\frac{1}{iw}$$\frac{Fexp(iwt)}{Rm + j(wm-s/w)}$

I can ignore exp(iwt) because |exp(iwt)|=1.

So I need to find the magnitude of the complex displacement (while ignoring exp(iwt)) but I just don't remember how to deal with complex numbers. I've basically got all the values for what's in the complex equation I just don't know what to do with it!

 

[KataKoniK]

your approach to solving this problem would involve using mathematical techniques and principles to determine the steady state displacement amplitude.

First, you would need to understand the concept of steady state in a driven harmonic system. In a driven harmonic system, the steady state occurs when the system has reached a constant amplitude and frequency in response to the driving force. This means that the system is no longer accelerating and the displacement and velocity are constant.

To find the steady state displacement amplitude, you would need to solve the differential equation given in the problem. This involves using mathematical techniques such as substitution, integration, and complex numbers. You can use the given values for mass, stiffness, mechanical resistance, and force to solve the equation.

Next, you would need to simplify the complex displacement equation by using the properties of complex numbers. Remember that the real part of a complex number is the part without the imaginary unit (i) and the imaginary part is the part with the imaginary unit. You can also use the properties of exponents to simplify the equation.

Once you have simplified the complex displacement equation, you can find the magnitude of the complex displacement by taking the absolute value of the complex number. This will give you the steady state displacement amplitude.

In summary, as a scientist, you would approach this problem by understanding the concept of steady state, using mathematical techniques to solve the differential equation, and simplifying the complex displacement equation to find the magnitude of the steady state displacement amplitude.