6.2. Free Vibration of Undamped SDOF System

• Assumptions:
  • No external force.
  • No damping.

In this section, we begin with the basic assumptions of a free vibration scenario for an undamped Single Degree of Freedom (SDOF) system. First, we assume there are no external forces acting on the system. This means that the only forces present are the internal forces due to the system's mass and spring. Second, we assume that there is no damping, which means the system does not lose energy over time and continues to oscillate indefinitely after being displaced from equilibrium.

Imagine a swing at a playground that has been pushed and let go. If there are no kids sitting on it (no external force) and it’s perfectly smooth (no damping), it will keep swinging back and forth forever without losing height, similar to the undamped SDOF system.

• Governing Equation:
  • mu¨(t) + ku(t) = 0

The governing equation for the free vibration of an undamped SDOF system is expressed as 'mu¨(t) + ku(t) = 0'. Here, 'm' represents the mass of the system, 'u(t)' is the displacement, 'k' is the stiffness of the system, and the double dot over 'u' indicates the second derivative with respect to time, which represents acceleration. This equation reflects that the restoring force provided by the spring (which is proportional to the displacement) is equal to the inertial force due to the mass (which is proportional to its acceleration).

Consider a rubber band being stretched. If you let it go, the band pulls back to its original length. The equation mathematically describes how the forces work: the further you pull (displacement), the more the rubber band (spring) wants to pull you back to the center (equilibrium) by exerting a force proportional to how far you pulled.

• Solution:
  • u(t) = Acos(ωn t) + Bsin(ωn t)
  • where ωn = √(k/m) is the natural frequency.

The general solution to the governing equation takes the form 'u(t) = A * cos(ωn t) + B * sin(ωn t)'. Here, 'A' and 'B' are constants determined by initial conditions, and 'ωn' stands for the natural frequency of the system, calculated as the square root of the stiffness 'k' divided by the mass 'm'. The natural frequency represents how quickly the system oscillates when displaced from its resting position. The solution indicates that the motion is a combination of cosines and sines, which is characteristic of harmonic motion.

Think about a pendulum. If you pull it to one side and release it, it will swing back and forth in a regular pattern—that's harmonic motion! The terms 'A' and 'B' define the amplitude or height of the swing, while 'ωn' determines how fast it swings back and forth.

• Interpretation: Pure harmonic motion.

The motion described by the solution is identified as pure harmonic motion. This means the motion is sinusoidal and periodic, where the system oscillates around a central equilibrium position. The concept of pure harmonic motion is essential in dynamics as it represents the idealized behavior of many mechanical systems that can oscillate.

Think of a music instrument, like a guitar string. When you pluck it, it vibrates in a sinusoidal pattern, producing sound waves. This continuous vibratory motion is similar to how the undamped SDOF system behaves—smooth, predictable, and periodic.

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