Single Degree of Freedom Systems in Vibration Analysis

Single Degree of Freedom Systems

Single degree of freedom (SDOF) systems are a fundamental concept in vibration analysis, allowing the study of the vibration of a system with a single degree of freedom, typically a linear spring-mass system. These systems can be used to model various real-world applications, such as the vibration of a car suspension system or a building subjected to external forces. This article will discuss the equation of motion, forced vibration, damping, resonance, and free vibration in the context of SDOF systems.

Equation of Motion

The equation of motion for a SDOF system can be derived using Newton's second law, which states that the force acting on an object is equal to the mass of the object times its acceleration. For a SDOF system, the equation of motion can be written as:

$$m\ddot{x} + c\dot{x} + kx = F(t)$$

where:

• $$m$$ is the mass of the system
• $$k$$ is the stiffness of the system
• $$c$$ is the damping coefficient
• $$F(t)$$ is the external forcing function
• $$\ddot{x}$$ and $$\dot{x}$$ represent the second and first derivatives of the displacement $$x$$ with respect to time

Forced Vibration

Forced vibration occurs when an external force is applied to the system. In this case, the displacement of the system as a function of time can be determined by solving the equation of motion. The solution can be expressed in terms of the vibration amplitude, which depends on the frequency and amplitude of the applied force, as well as the damping and natural frequency of the system.

A example of a forced vibration problem involves a car suspension system with a natural frequency of 0.5 Hz, a damping coefficient of 0.2, and a harmonic force of amplitude 500 N at a frequency of 0.5 Hz. To solve this problem, one can calculate the steady-state amplitude of vibration, which represents the steady-state response of the system to the applied force.

Damping

Damping is a crucial parameter in SDOF systems, as it determines the energy dissipation and the steady-state response of the system. A heavily damped system will have a smaller amplitude of vibration compared to a lightly damped system when subjected to the same external force. The damping coefficient $$c$$ in the equation of motion represents the damping in the system.

Resonance

Resonance occurs when the frequency of the external force applied to the system is equal to the natural frequency of the system. In this case, the system can exhibit oscillatory behavior, and the amplitude of vibration can be significantly larger than when the force is not at resonance. Resonance can lead to undesirable effects in various applications, such as increased vibrations in buildings or vehicles.

Free Vibration

Free vibration occurs when the system is initially displaced from its equilibrium position, and the external force is removed. In this case, the system will oscillate without the influence of an external force. The natural frequency of the system, which is the frequency at which the system oscillates when no external force is applied, is a crucial parameter in determining the behavior of the system during free vibration.

In conclusion, single degree of freedom systems provide a fundamental understanding of vibration analysis and can be used to model the behavior of various real-world applications. By understanding the equation of motion, forced vibration, damping, resonance, and free vibration in SDOF systems, engineers and physicists can design and optimize systems to minimize unwanted vibrations and improve performance.