7.7. Graphical Representation of Motion

The motion of an undamped SDOF system is sinusoidal and periodic. The displacement-time plot shows a smooth oscillation about the equilibrium position.

In this chunk, we discuss how an undamped Single Degree of Freedom (SDOF) system behaves when it is in motion. When it's allowed to vibrate, the motion follows a sinusoidal pattern, which is a smooth wave-like form that goes up and down over time. Periodic motion means that this motion repeats itself over regular intervals. The displacement-time plot visually represents this pattern, showing how far the mass moves from its rest position at any given time. The equilibrium position is where the mass would sit if it weren’t moving—the center point around which the motion oscillates.

Think of a playground swing. When you push it, it moves back and forth in a smooth, predictable way, following a regular pattern. Just like the swing, if you let a mass on a spring oscillate without any external interference (like air resistance or friction), it will show a sinusoidal pattern, going back to its resting point repeatedly.

The frequency remains constant and amplitude does not decay over time (in the ideal undamped case).

In this chunk, we explain two important characteristics of the motion of an ideal undamped SDOF system: frequency and amplitude. Frequency refers to how often the motion repeats itself in one second, and it stays the same throughout the motion. Amplitude, on the other hand, is the maximum distance the mass moves from the equilibrium position. In an undamped system, this amplitude does not decrease over time; the system continues to vibrate with the same energy indefinitely, maintaining that maximum distance. This is contrasted with real systems that lose energy and experience decay in amplitude over time due to internal friction and damping effects.

Imagine a perfectly elastic rubber band. If you stretch it and release it, it bounces back to its original shape without losing any energy. In an ideal undamped system, it behaves similarly—no energy is lost, so it keeps stretching and bouncing back at the same rate and with the same maximum stretch, just like the rubber band that doesn’t tire over time.

Plots include:

• Displacement vs. time
• Velocity vs. time
• Acceleration vs. time
• Phase diagram (x vs. ẋ)

Here, we discuss different types of plots that illustrate the motion of an SDOF system. Each type of plot provides valuable insights into different aspects of the motion:

• Displacement vs. time: This plot shows how far the mass moves from the equilibrium position at any time, indicating the oscillation pattern.
• Velocity vs. time: This plot reveals how fast the mass is moving at any moment, showing maximum speed at the equilibrium position and zero speed at the extremes of motion.
• Acceleration vs. time: Here, we see how the acceleration varies—the system experiences maximum acceleration at the extremes and zero acceleration at the equilibrium position.
• Phase diagram (x vs. ẋ): This diagram plots displacement against velocity, illustrating how these two quantities are interrelated in a circular motion-like format, demonstrating the overall dynamics of the system.

Consider a roller coaster. As it climbs up and moves down, you can visualize different aspects of its motion. The displacement plot is like marking its height over time, the velocity plot captures how fast it moves at any point, the acceleration plot shows how much it speeds up or slows down, and the phase diagram is like tracing its overall path, connecting each point of height with its corresponding speed in a visual representation. Each graph gives a different view of the thrilling roller coaster ride!

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