rzn972 said:Homework Statement
Suppose a mass- spring is modeled by x''+ cx'+100x = 0. What should the damping constant c be in order that critical damping occurs?
Homework Equations
The Attempt at a Solution
I set the discriminant equal to zero so that there are real repeated roots.
B^2-4ac=0
c^2=400.
c=20 or c=-20.
The answer should only be c=20. For critical damping, the constant must be positive?
rzn972 said:For critical damping, the constant must be positive?
A mass-spring-damping system is a physical system that consists of a mass attached to a spring and a damping element. This system is used to model the behavior of objects that oscillate or vibrate, such as a pendulum or a car suspension. The mass represents the object's inertia, the spring provides the restoring force, and the damping element dissipates energy to reduce the amplitude of the oscillations.
The equation for a mass-spring-damping system is known as the "spring-mass-damper equation" and is given by: m * d^2x/dt^2 + b * dx/dt + k * x = F(t), where m is the mass, b is the damping coefficient, k is the spring constant, x is the displacement of the mass, and F(t) is the external force acting on the system. This equation is a second-order linear differential equation that describes the motion of the system over time.
Damping affects the behavior of a mass-spring-damping system by reducing the amplitude of the oscillations and causing them to decay over time. The damping coefficient, b, determines the amount of damping in the system. A higher damping coefficient leads to a faster decay of oscillations. Without damping, the system would continue to oscillate indefinitely.
The natural frequency of a mass-spring-damping system is the frequency at which the system will oscillate when there is no external force acting on it. It is given by the equation f = 1/(2π * √(k/m)), where k is the spring constant and m is the mass. This frequency is also known as the resonant frequency, as the system will resonate at this frequency if excited by an external force.
A mass-spring-damping system has many real-world applications, including in mechanical engineering, electrical engineering, and physics. It is used to model the behavior of structures such as bridges and buildings, to design vehicle suspension systems, and to study the behavior of electrical circuits. It is also used in the fields of acoustics and vibration analysis to study the resonant frequencies of structures and to reduce unwanted vibrations in machinery.