Deriving the differential equation for the underdamped case

In summary, the general formula for oscillation involves complex conjugates and the underdamping oscillation formula includes the use of λ, A+, A-, λ+, and λ-. After some operations, the final answer for x(t) is Ae^(-γx)2cos(θ+ωx), which is missing a factor of 2 in the solution given by the teacher.
  • #1
Tony Hau
101
30
The formula for general oscillation is:
1586875856916.png

The formula for underdamping oscillation is:
1586875210286.png

where λ = -γ +- sqart(γ^2 - ω^2), whereas A+ and A- , as well as λ+ and λ-, are complex conjugates of each other.

After some operations, we get:
x(t) = Ae^(-γx)[e^i(θ+ωx) +e^-i(θ+ωx)], where A is the modulus of A+ and A-.

The notes of my teacher give the answer as
1586875556072.png


Obviously, the complex part of e^i(θ+ωx) and e^-i(θ+ωx) cancells each other. However, the real part of the two expressions is the same. So, why isn't the final answer of x(t) Ae^(-γx)2cos(θ+ωx)?
 

Attachments

  • 1586875884559.png
    1586875884559.png
    1.6 KB · Views: 213
  • 1586875891728.png
    1586875891728.png
    1.6 KB · Views: 201
Physics news on Phys.org
  • #2
The solution given by my teacher indeed misses a factor of 2. My guess is correct.
 

What is the underdamped case in differential equations?

The underdamped case in differential equations refers to a type of motion in which the system's response to a disturbance gradually decreases over time. This is characterized by a damped oscillation or "overshoot" in the system's behavior.

How is the differential equation for the underdamped case derived?

The differential equation for the underdamped case can be derived using Newton's Second Law of Motion, which states that the sum of the forces acting on a system is equal to the mass of the system times its acceleration. By considering the forces acting on the system and applying this law, we can derive the differential equation that describes the underdamped motion.

What are the key components of the underdamped differential equation?

The underdamped differential equation typically includes terms for the damping coefficient, the spring constant, and the mass of the system. It may also include an external force term, depending on the specific system being modeled.

What are the applications of the underdamped differential equation?

The underdamped differential equation has many applications in physics and engineering, particularly in modeling systems that exhibit damped oscillations. It is commonly used in fields such as mechanical engineering, electrical engineering, and control systems.

How does the underdamped differential equation differ from other types of differential equations?

The underdamped differential equation is a specific type of second-order differential equation that describes a system's behavior when it is underdamped. It differs from other types of differential equations, such as critically damped or overdamped equations, in the way it models the system's response to a disturbance or input.

Similar threads

Replies
6
Views
1K
Replies
2
Views
2K
  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Differential Equations
Replies
25
Views
2K
Replies
3
Views
371
  • Differential Equations
Replies
4
Views
2K
  • Differential Equations
Replies
1
Views
646
  • Differential Equations
Replies
3
Views
2K
  • Differential Equations
Replies
5
Views
2K
  • Differential Equations
Replies
1
Views
1K
Back
Top