Getaran Mekanik : penyelesaian soal Getaran Bebas sistem 1 derajat kebebasan dengan peredam

Summary

This video discusses the concept of free vibration in a one-degree-of-freedom system with damping, covering underdamping, critical damping, and overdamping conditions. A case study is presented involving a system with a spring, damper, and mass, where the natural frequency, damping ratio, and motion equations are calculated. The video demonstrates how to determine the motion and velocity equations based on given initial conditions, including the use of exponential functions to model the system's behavior. The content aims to guide viewers through the analysis of damped systems in mechanical vibrations.

Takeaways

• 😀 Free vibration systems with one degree of freedom and damping can be classified into three types: under damping, critical damping, and over damping.
• 😀 Under damping occurs when the damping ratio is less than 1, critical damping occurs when the damping ratio is equal to 1, and over damping occurs when the damping ratio is greater than 1.
• 😀 The system in the script includes three components: a spring, a damper, and a mass.
• 😀 The spring constant (k) in the case study is 10,000 N/m, the damping coefficient (c) is 5,000 N·s/m, and the mass (m) is 400 kg.
• 😀 The goal of the analysis is to find the equations for displacement and velocity of the system.
• 😀 The natural frequency (ωₙ) of the system is calculated using the formula ωₙ = √(k/m), resulting in 5 rad/s.
• 😀 To classify the damping type, the damping ratio (ζ) is calculated. The critical damping coefficient is 4,000 N·s/m, and the damping ratio is 1.25, which means the system is overdamped.
• 😀 In an overdamped system, the general solution for displacement is given by an exponential equation that includes two constants, C₁ and C₂, which are determined using initial conditions.
• 😀 The initial conditions are: x₀ = 0 (initial displacement) and v₀ = 4 m/s (initial velocity). These conditions allow for the determination of the constants in the displacement equation.
• 😀 The solution for the displacement of the system is x(t) = 0.53 * (e^(10t) - e^(-10t)). The velocity equation is derived by differentiating the displacement equation, resulting in v(t) = 5.3 * (e^(10t) + e^(-10t)).

Q & A

What is free vibration in a system with one degree of freedom?

Free vibration occurs when a system is displaced from its equilibrium position and left to oscillate without external forces acting on it. In a system with one degree of freedom, this refers to a single mass that oscillates due to its own restoring force, such as from a spring.

What are the three damping conditions for a single degree of freedom system?

The three damping conditions are: 1) Under Damping: The system oscillates with gradually decreasing amplitude (damping ratio < 1). 2) Critical Damping: The system returns to equilibrium as quickly as possible without oscillating (damping ratio = 1). 3) Over Damping: The system returns to equilibrium slowly without oscillating (damping ratio > 1).

How is the natural frequency of the system calculated?

The natural frequency (ω_n) is calculated using the formula: ω_n = √(k/m), where k is the spring constant and m is the mass of the system. For the given example, ω_n = √(10,000 / 400) = 500 rad/s.

What is the damping ratio, and how is it calculated?

The damping ratio (ζ) is a measure of how much damping is present in the system. It is calculated using the formula: ζ = c / (2√(km)), where c is the damping coefficient, k is the spring constant, and m is the mass. For the given system, the damping ratio is ζ = 1.25, which indicates an overdamped system.

How do we classify a system based on its damping ratio?

A system is classified based on its damping ratio (ζ) as follows: 1) Under Damping: ζ < 1, the system oscillates. 2) Critical Damping: ζ = 1, the system returns to equilibrium as quickly as possible. 3) Over Damping: ζ > 1, the system returns to equilibrium slowly without oscillating.

What is the general equation of motion for a damped system?

The general equation of motion for a damped system is: x(t) = C1 * exp[(-ζ + √(ζ² - 1)) * ω_n * t] + C2 * exp[(-ζ - √(ζ² - 1)) * ω_n * t], where C1 and C2 are constants determined by initial conditions, ω_n is the natural frequency, and ζ is the damping ratio.

How are the constants C1 and C2 determined in the equation of motion?

The constants C1 and C2 are determined using the initial conditions of the system, such as the initial displacement (x₀) and initial velocity (v₀). By substituting these conditions into the general equation of motion and solving for C1 and C2, we can find their values.

In the given example, what is the value of the damping ratio, and what does it indicate about the system?

In the given example, the damping ratio is ζ = 1.25, which is greater than 1. This indicates that the system is **overdamped**, meaning it returns to equilibrium without oscillating and more slowly than a critically damped system.

What does the equation x(t) = 0.53 * exp(-10t) - 0.53 * exp(-10t) represent?

The equation x(t) = 0.53 * exp(-10t) - 0.53 * exp(-10t) represents the displacement of the system as a function of time. This is the specific solution to the system's motion, considering the initial conditions and the overdamped nature of the system.

How do you find the velocity equation for a damped system?

The velocity equation is found by differentiating the displacement equation with respect to time. If the displacement equation is x(t), the velocity equation is v(t) = dx/dt, where the rate of change of displacement gives the system's velocity at any point in time.