Gegeben sei das schwingungsfähige gedampfte Feder-Masse-System (Federpendel) mit den folgenden Kenndaten: $$ m=0,5 \mathrm{~kg}, \quad b=8 \mathrm{~kg} / \mathrm{s}, \quad c=128 \mathrm{~N} / \mathrm{m} $$ a) Wie lautet die allgemeine Lösung der Schwingungsgleichung? b) Berechnen Sie die Kreisfrequenz \(\omega_{d}\), die Frequenz \(f_{d}\) und die Schwingungsdauer \(T_{A}\) der gedämpften Schwingung. c) Wie lautet die spezielle Lösung, die den Anfangswerten \(x(0)=0,2 \mathrm{~m}\), \(v(0)=0\) genigt? Skizzieren Sie den Schwingungsverlauf.

Understanding differential equations is crucial for analyzing a damped harmonic oscillator. A differential equation is a type of mathematical equation that relates a function with its derivatives. In the context of a damped harmonic oscillator, the differential equation describes the motion of the mass-spring system over time.
For a damped mass-spring system, the general differential equation is: \[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + cx = 0 \] , where: - m is the mass, - b is the damping coefficient, - c is the spring constant, - x is the displacement, - t is time.
This equation shows how the mass's motion is influenced by both spring force and damping force. Understanding how to solve this differential equation helps determine the behavior of the system.

Damped frequency, also known as the damped angular frequency, is an essential part of understanding a damped harmonic oscillator. It represents the frequency at which the system oscillates when a damping force is present.
From the characteristic equation, we find the roots which include complex numbers. These complex roots help us identify the damped frequency. In our example, the characteristic equation is: \[ 0.5\beta^2 + 8\beta + 128 = 0 \]. Solving this quadratic equation yields the roots: \[ \beta = -8 \pm 4i \] The damped angular frequency (ωd) is the imaginary part of these roots, which is 4 rad/s. Knowing the damped angular frequency allows us to calculate the damped frequency (fd) using the formula:\[ f_d = \frac{\beta_d}{2\pi} \] This calculation helps us understand the rate of oscillations in the presence of damping.

Decaying oscillations are a hallmark of damped harmonic oscillators. When damping is present, the amplitude of the oscillations decreases over time, eventually bringing the system to rest.
In our system, the general solution to the differential equation: \[ x(t) = e^{-8t} (A \cos(4t) + B \sin(4t)) \] demonstrates this behavior. Here, the exponential term \ e^{-8t} \ suggests that the amplitude diminishes exponentially with time. This term represents the decaying nature of the oscillations, indicating how the system loses energy due to damping.
Understanding this concept helps predict how the system’s vibrations will diminish, which is essential in various engineering and physics applications.

A mass-spring system is a fundamental model in physics, used to illustrate simple harmonic motion and damped oscillations.
The system consists of a mass attached to a spring, where the spring provides a restoring force proportional to the displacement from the equilibrium position. The behavior of such a system is described by Hooke's law: \[ F = -kx \],where F is the restoring force, k is the spring constant, and x is the displacement. In a damped system, additional forces come into play, such as the damping force which opposes the motion and is proportional to the velocity.Analyzing a mass-spring system with damping involves solving the differential equation, \[ {{m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + cx = 0}} \] , to understand how the system behaves over time. This knowledge is crucial for designing systems in engineering that need to control vibrations and ensure stability.