How to get the critically damped vibration equation

[zina]

1. SDOF Systems Governing Equation m(dx^2/dt^2) + c(dx/dt)+ kx = F(t)

how do i get this equation below?

Free Critically damped Vibration x(t) = e^(wt) [x(0)(1+wt) + (dx/dt)(0) t]

(dx/dt is x with 1 dash on top)

 

[zina]

without the 9

 

[nlightner]

To derive the critically damped vibration equation, we start with the general equation for a single degree of freedom (SDOF) system governed by mass, damping, and stiffness:

m(dx^2/dt^2) + c(dx/dt) + kx = F(t)

Where m is the mass of the system, c is the damping coefficient, k is the stiffness coefficient, and F(t) is the external force acting on the system.

Next, we assume that the system is critically damped, meaning that the damping coefficient is equal to the critical damping coefficient, which is given by:

c_c = 2√mk

Substituting this into the governing equation, we get:

m(dx^2/dt^2) + 2√mk(dx/dt) + kx = F(t)

We can then solve this equation using the method of undetermined coefficients, where we assume that the solution has the form:

x(t) = Ae^(wt)

Taking the first and second derivatives of x(t), we get:

dx/dt = Aw e^(wt) and dx^2/dt^2 = Aww e^(wt)

Substituting these into the governing equation, we get:

m(Aww e^(wt)) + 2√mk(Aw e^(wt)) + k(Ae^(wt)) = F(t)

Simplifying, we get:

(Aww + 2√mkAw + kA)e^(wt) = F(t)

Since e^(wt) is never equal to zero, we can equate the coefficients of e^(wt) to zero, giving us the following equations:

Aww + 2√mkAw + kA = 0

This is known as the characteristic equation. Solving for w, we get:

w = (-√k/m) ± √(k/m - c^2/4m^2)

Since we are assuming a critically damped system, we know that c = 2√mk. Substituting this into the equation for w, we get:

w = -√(k/m)

Therefore, the solution for x(t) becomes:

x(t) = Ae^(-√(k/m)t)

To determine the value of A, we can use initial conditions, such as the initial displacement (x(0)) and velocity (dx/dt) at time t=0. Substituting