How to get the critically damped vibration equation

In summary, the critically damped vibration equation is a mathematical formula used to describe the motion of a damped harmonic oscillator at the critical damping point. It can be derived using the equations of motion and includes variables such as mass, spring constant, damping coefficient, and initial displacement and velocity. This equation is commonly used in engineering and physics to analyze and predict the behavior of critically damped systems in real-life situations. However, it may not accurately describe underdamped or overdamped systems and does not account for non-linear effects.

how to u get the critically damped equation

  • x(t)=e^(-wt) [x(0)(1+wt)+ dx/dt(0)t]

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  • j is the xi (greek alphabet which equals C/Cc and w' damping

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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)
 
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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
 

1. What is the critically damped vibration equation?

The critically damped vibration equation is a mathematical formula used to describe the motion of a damped harmonic oscillator when it is at the critical damping point. In this state, the system returns to equilibrium without any oscillations or overshooting.

2. How is the critically damped vibration equation derived?

The equation can be derived using the equations of motion for a damped harmonic oscillator. By setting the damping coefficient to a specific value known as the critical damping coefficient, the solution of the equations simplifies to the critically damped vibration equation.

3. What are the variables in the critically damped vibration equation?

The variables in the equation include the mass of the oscillator, the spring constant, the damping coefficient, and the initial displacement and velocity of the oscillator. These variables are used to calculate the displacement of the oscillator at any given time.

4. How is the critically damped vibration equation used in real-life situations?

The equation is commonly used in engineering and physics to analyze and predict the behavior of critically damped systems, such as shock absorbers, car suspensions, and electrical circuits. It can also be used to design systems that need to return to equilibrium quickly without oscillations.

5. Are there any limitations to the critically damped vibration equation?

While the equation is useful for critically damped systems, it does not accurately describe the behavior of underdamped or overdamped systems. Additionally, it assumes linear behavior and does not account for non-linear effects that may occur in real-life situations.

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