Undamped Free Vibration (Seri Getaran Mekanis part1)

Summary

This video introduces the topic of vibration, starting with a discussion of free and damped vibration. It explains key concepts such as free vibration (which occurs without external force after an initial displacement) and forced vibration (where an external force is applied to the system). The video also covers the basic model of vibration using a mass-spring system, including the differential equation governing the motion. The solution to this equation is sinusoidal, and the concept of natural frequency (ωn) is introduced. The video is part of a broader series on vibration, exploring both single and multiple degrees of freedom.

Takeaways

• 😀 The script starts by introducing the topic of vibration, specifically focusing on the concepts of undamped and damped vibration.
• 😀 The term 'Andam' refers to undamped, while 'Damp' signifies the opposite, damped vibration, indicating whether a damper is included in the vibration model.
• 😀 The script explains free vibration as a scenario where no external force is continuously applied to the object, contrasting it with forced vibration where an external force drives oscillation.
• 😀 Free vibration is initiated by a momentary external force (e.g., pulling a mass to the right and releasing it), causing the object to oscillate without further force.
• 😀 The importance of understanding the equilibrium position when analyzing vibrations is highlighted, as it is the reference point for oscillations.
• 😀 In the analysis, Newton's second law is applied, considering forces acting on the mass-spring system to form a differential equation for the motion.
• 😀 The forces acting on the mass in the horizontal direction (x-axis) include the elastic force (spring force), while forces in the vertical direction (y-axis) include gravitational force and normal force, but these are not relevant for the horizontal motion analysis.
• 😀 The script discusses the mathematical formulation of the system, where the equation of motion is represented as a second-order ordinary differential equation (ODE).
• 😀 Solving the ODE leads to the solution for displacement as a sinusoidal function of time, representing oscillatory motion.
• 😀 The natural frequency (denoted as ωn) is derived from the equation, with its value being a function of the spring constant (k) and the mass (m) in the system, given by ωn = √(k/m).

Q & A

What is the key difference between damped and undamped vibrations?

The key difference lies in the presence of a damper in the system. An undamped vibration occurs when there is no damper, while a damped vibration includes a damper that gradually reduces the amplitude of the oscillations over time.

What does 'free vibration' mean in the context of the script?

'Free vibration' refers to oscillations that occur in an object without any continuous external force acting on it. The motion is initiated by a momentary external force, but no additional forces are applied during the oscillation.

How does forced vibration differ from free vibration?

Forced vibration occurs when an external force continuously acts on an object, causing it to oscillate. In contrast, free vibration happens when an object oscillates on its own after an initial displacement, without any external force being applied during the oscillation.

What is the purpose of momentary excitation in free vibration?

Momentary excitation in free vibration is used to initially displace the object, after which the object oscillates freely without any continuous external force. This momentary force helps set the system into motion.

What role do initial conditions like 'x0' and 'v0' play in free vibration?

Initial conditions, such as initial position 'x0' and initial velocity 'v0', are essential in determining the behavior of the free vibration. These values provide the necessary starting parameters to describe the motion of the object.

What forces are considered in the free-body diagram for the mass-spring system?

In the free-body diagram for a mass-spring system, the only force considered in the horizontal direction is the elastic force (spring force), represented by 'F_k = -k * x'. Vertical forces, such as gravity and normal force, are not relevant for the oscillatory motion in the horizontal direction.

Why is the spring force in a free vibration not constant?

The spring force is not constant because it depends on the displacement 'x'. As the object oscillates, the displacement changes, which in turn alters the magnitude of the spring force.

What is the equation of motion for free vibration?

The equation of motion for free vibration is a second-order ordinary differential equation: '-k * x = m * a', where 'k' is the spring constant, 'x' is the displacement, 'm' is the mass, and 'a' is the acceleration (second derivative of displacement).

What is the solution to the differential equation for free vibration?

The solution to the differential equation for free vibration is a sinusoidal function of the form 'x(t) = A * sin(ω_n * t + φ)', where 'A' is the amplitude, 'ω_n' is the natural frequency, and 'φ' is the phase constant.

How is the natural frequency 'ω_n' related to the spring constant 'k' and mass 'm'?

The natural frequency 'ω_n' is related to the spring constant 'k' and mass 'm' by the equation 'ω_n = √(k/m)', which shows that the natural frequency is directly influenced by the stiffness of the spring and inversely by the mass.