7. Free Vibration of Single Degree of Freedom (SDOF) System

The study of vibrations is a foundational concept in structural dynamics and earthquake engineering. A structure, when subjected to dynamic forces such as those generated by earthquakes, responds with motion that can be understood and predicted through vibration theory. One of the most basic and important systems in vibration analysis is the Single Degree of Freedom (SDOF) system. The understanding of free vibration of an SDOF system, in the absence of any external forces and damping, lays the groundwork for more complex dynamic analysis and is essential in seismic design and assessment of structures.

In engineering, particularly in understanding structures during earthquakes, vibrations play a crucial role. When buildings or structures experience forces from earthquakes, they move in specific ways, and we can predict these movements using vibration theory. The Single Degree of Freedom (SDOF) system is one of the simplest forms of these vibrations and serves as a fundamental building block for analyzing more complicated systems. Understanding how an SDOF system vibrates without any outside influences gives engineers the essential knowledge they need for designing buildings that are safe during seismic events.

Think of a swing at a playground. When you push it (the initial disturbance) and let go, it swings back and forth due to gravity and its own momentum. The swing's motion is a simple version of how structures might vibrate when subjected to forces like earthquakes.

Free vibration refers to the motion of a mechanical system when it is allowed to vibrate naturally without the influence of external forces after an initial disturbance. For a SDOF system, this typically involves a mass-spring system set into motion and allowed to oscillate freely.

Free vibration occurs when a system, once disturbed, moves on its own without any external forces acting on it. A classic example of this is a mass attached to a spring. When you pull the mass down and let go, it will move up and down in a predictable manner due to the spring's characteristics. This is important in engineering as it helps predict the behavior of structures under natural conditions following disturbances like earthquakes.

Imagine a car on a smooth road that suddenly hits a bump (the initial disturbance). After the bump, the car will bounce on its suspension system without any additional pushes from the driver; this bounce is akin to free vibration.

A single degree of freedom system is characterized by:

• One mass (m) which can move in only one direction.
• One spring with stiffness (k).
• No damping or external forcing (for the undamped case).
• A displacement function x(t) which fully describes the motion.

This system is often represented by a mass attached to a spring that can move vertically or horizontally, depending on the setup.

An SDOF system simplifies the analysis of vibrational motion. It consists of just one mass that can only move in one direction (either vertically or horizontally) and is connected to a spring that returns it to its equilibrium position. In theories of vibration, considering this simplified model allows engineers to focus on fundamental behaviors without complicating factors like damping or multiple dimensions. The equation of motion for such a system describes how the mass will change its position over time based on forces exerted by the spring.

Think of a single pendulum – like a swing – that can move back and forth. This pendulum represents an SDOF system where the swing is the mass, and the chains are the spring keeping the swing in motion, moving in just one direction: back and forth.

The general equation of motion for an undamped SDOF system undergoing free vibration is:

mx¨(t)+kx(t)=0 Where:

• m = mass of the system
• k = stiffness of the spring
• x(t) = displacement as a function of time
• x¨(t) = acceleration (second derivative of displacement)

Dividing the equation by m, we get: x¨(t)+ω2x(t)=0 Where ω = √(k/m) is the natural circular frequency of the system.

The equation of motion describes how the mass in an SDOF system behaves over time when it vibrates freely. The left side represents the forces acting on the mass due to its acceleration and the spring, while the right side equals zero, indicating no external forces are involved. By dividing this equation by the mass (m), you arrive at a form that relates the mass, spring stiffness, and acceleration to a characteristic frequency of the system, which is essential for understanding vibration behavior.

Imagine a rubber band: when you pull it and then let it go, it snaps back. The rubber band (spring) tries to pull the mass (your hand) back to its original position, and this behavior can be described mathematically using the equation of motion.

The solution to the second-order differential equation is of the form: x(t) = Acos(ωₙ t) + Bsin(ωₙ t) Where:

• A and B are constants determined by initial conditions (initial displacement x₀ and initial velocity ẋ₀).

Alternatively, using the harmonic form: x(t) = Xcos(ωₙ t + ϕ) Where:

• X = amplitude of vibration
• ϕ = phase angle

These parameters can be related back to the initial conditions using: B/X = √(A² + B²), tan(ϕ) = A/B.

When solving the equation of motion, we find a general solution that describes the displacement of the mass over time. The solution is sinusoidal, meaning it uses sine and cosine functions, which occur frequently in nature, particularly for oscillations. The constants A and B depend on the starting conditions of the system, such as how far the mass was displaced originally and its speed. This mathematical representation helps predict the specific motion of the mass at any time t.

Think of water waves at the beach: they rise and fall rhythmically. The mathematical functions (sine and cosine) describe the height of the water over time, similar to how the displacement of the mass in the SDOF system changes over time.

Natural Frequency (ωₙ): ωₙ = √(k/m) (in radians/sec)

Time Period (T): T = 2π√(m/k) (in seconds)

The natural frequency and time period are intrinsic properties of the system and determine how fast or slow the system oscillates under free vibration.

Natural frequency is a vital concept in vibration analysis; it signifies how many oscillations per second a system can execute without external forces. The time period is the duration of one complete oscillation cycle. Both of these properties are deeply influenced by the mass of the system and the spring's stiffness. Realizing these relationships allows engineers to design structures that do not easily resonate with external forces, such as earthquakes.

If you think of a swing again, the natural frequency would tell you how many times the swing completes a full forward and backward swing in a given timeframe. If the chains are shorter (stiffer), it will swing faster than if they are longer (more flexible).

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