91Ó°ÊÓ

In a spring-mass-damper system, the static equilibrium position is where the system is at rest, meaning no movement or external forces at play. Here, both the velocity (\(\dot{x}\) ) and acceleration (\(\ddot{x}\) ) are zero. To find this position, we simplify the equation of motion by eliminating terms that involve velocity and acceleration.

In our system described by the equation \(100 \ddot{x}+500 \dot{x}+10,000 x+400 x^{3}=0\), simplifying it under static conditions yields:

• \(10,000x + 400x^3 = 0\)

By factoring out \(x\), we have:

• \(x(10,000 + 400x^2) = 0\)

This equation implies two conditions:

• First, \(x = 0\), indicating no displacement at equilibrium.
• Secondly, \(10,000 + 400x^2 = 0\), which isn't possible since \(x^2\) can't be negative.

Hence, the only feasible solution is \(x = 0\), defining the static equilibrium position.

When analyzing a system for small deviations from equilibrium, we use a linearized equation of motion. This approach simplifies the nonlinear characteristics that only slightly affect the system. The original equation \(100 \ddot{x}+500 \dot{x}+10,000 x+400 x^{3}=0\) contains a non-linear term, \(400x^3\), which grows insignificant for small displacements.

Therefore, for tiny deviations, we discard this term to simplify:

• \(100 \ddot{x}+500 \dot{x}+10,000 x = 0\)

Now, in this linear approximation, we have a clearer understanding of the motion. The linearized model is more manageable and suitable to predict the system's behavior near the equilibrium. It allows us to analyze and understand natural oscillations without computational complications of nonlinear terms.

This simplification is crucial because it lets engineers, scientists, and students predict how the system would behave around equilibrium without getting overly involved with complex mathematics.

The natural frequency of a system reveals how it vibrates when disturbed from equilibrium without any damping forces considered. This is crucial for understanding the inherent oscillatory characteristics. Using the linearized equation of motion \(100 \ddot{x}+500 \dot{x}+10,000 x = 0\), we can find this natural frequency.

The standard formula for natural frequency \(\omega_n\) in a simple harmonic motion is:

• \(\omega_n = \sqrt{\frac{k}{m}}\)

Plug in the spring constant \(k = 10,000\) and mass \(m = 100\) into our equation:

• \(\omega_n = \sqrt{\frac{10,000}{100}} = \sqrt{100} = 10\, \text{rad/s}\)

This gives us the natural frequency as 10 radians per second.

Understanding the natural frequency helps in designing systems that either avoid resonance (where the system vibrates uncontrollably) or utilize resonance constructively, such as in musical instruments. It is a fundamental aspect when considering vibrations in machinery, buildings, and numerous other applications in engineering.