5.6.1. Undamped Free Vibration

The governing equation for undamped free vibration is given by:

u¨(t)+ω²u(t)=0

The equation, u¨(t) + ω²u(t) = 0, describes undamped free vibration. Here, u(t) represents the displacement of the system over time, and ω (omega) is the natural frequency of the system. The term u¨(t) refers to the acceleration (the second derivative of displacement with respect to time). This equation reflects a situation where there is no damping present, meaning that energy losses (like friction or air resistance) do not affect the motion of the vibrating system. The absence of damping allows the system to oscillate indefinitely at its natural frequency unless acted upon by an outside force.

Think of a child on a swing. When pushed, if there is no one to slow it down (like someone getting off the swing or friction with the air), the swing will continue to move back and forth. The speed and frequency of these swings depend on how far it was pushed and the swing's design (i.e., the properties of the system). In this analogy, the swing system oscillating represents an undamped free vibration.

The general solution to the equation is expressed as:

u(t)=Acos(ω t)+Bsin(ω t)

The general solution for the undamped free vibration equation, u(t) = Acos(ω t) + Bsin(ω t), represents the displacement of the system as a function of time. Here, A and B are constants determined by the initial conditions of the system (like the initial position and velocity). This solution indicates that the motion is periodic and will repeat itself, showing a combination of cosine and sine functions, which are fundamental to oscillatory motion. Essentially, Acos(ω t) accounts for the part of motion that starts from a maximum position, whereas Bsin(ω t) accounts for the part of motion that starts from rest.

Imagine a pendulum swinging. At the highest point of the swing (maximum displacement), the pendulum can be represented by the cosine function starting at its peak, and as it moves through the lowest point (maximum speed), the sine function kicks in. The constant A represents how far the pendulum swings (its amplitude), while B relates to how quickly it starts moving from that peak position. Regardless of the pendulum's starting point or speed, summarizing its motion using these functions allows for predicting future positions over time.

The natural frequency (ω) is defined as:

ω = √(k/m)

The natural frequency, represented by the symbol ω, is crucial in understanding the dynamics of oscillatory systems. It is defined as the square root of the stiffness of the system (k) divided by its mass (m). This relationship illustrates that a stiffer system (higher k) or a lighter system (lower m) will oscillate more quickly, meaning it has a higher natural frequency. If either parameter changes, the natural frequency adjusts accordingly, impacting how the system responds to dynamic loads. The insight into natural frequency is invaluable; it helps engineers design structures that can withstand dynamic forces like earthquakes.

Think about a trampoline: if it's very tightly stretched (high stiffness), it will bounce back more quickly when you jump on it (higher natural frequency). Conversely, a heavier person (increasing mass) will experience a different bounce characteristic than a lighter person. This illustrates the mass-stiffness relationship: the balance between how 'tight' the system is against how much 'weight' it has directly determines the speed of its vibrations.

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