Dinamika Struktur #3: SDOF Getaran Bebas Dengan Redaman

Summary

In this video, Sandi explains the concept of free vibration with damping in structural dynamics. Building upon previous discussions of undamped vibrations, the video introduces the equation of motion for damped free vibrations (m x'' + c x' + kx = 0), covering various damping cases such as underdamped, critically damped, and overdamped systems. The focus is on the underdamped system, typical in civil structures like buildings and bridges. The video also delves into logarithmic decrement, damping ratios, and real-world applications like displacement calculations and plotting motion. It concludes by comparing damped and undamped responses and setting up the next topic of forced vibrations.

Takeaways

• 😀 The video discusses SDOF (Single Degree of Freedom) free vibration with damping, building on a previous video about free vibration without damping.
• 😀 The equation of motion for damped free vibration is presented as m x'' + c x' + kx = 0, where m is mass, c is damping coefficient, and k is the stiffness of the system.
• 😀 The concept of damping ratio (zeta) is introduced, describing the ratio of the system's damping coefficient to the critical damping coefficient.
• 😀 Critical damping is defined as the damping that prevents oscillation, with a damping ratio of 1, and occurs when c = 2mω.
• 😀 The solution for damped free vibration varies based on the damping ratio, with three possible cases: underdamped, critically damped, and overdamped.
• 😀 In an underdamped system (with damping ratio less than 1), the system oscillates, but the amplitude gradually decays over time.
• 😀 The solution for the underdamped system is a combination of exponential decay and sinusoidal functions, involving parameters like omega_d (damped natural frequency).
• 😀 For practical applications like buildings and bridges, damping ratios are usually less than 10%, meaning that the damped and undamped frequencies and periods are nearly the same.
• 😀 Logarithmic decrement is introduced as a measure of the reduction in amplitude over time, and the number of cycles needed for a 50% reduction in amplitude is calculated based on damping ratio.
• 😀 The script concludes with a practical example, showing how to calculate the displacement of a system with 5% damping and compare it to an undamped system.
• 😀 The damping effect on the displacement response of a system is shown in a plot comparing undamped and damped systems, demonstrating the reduced oscillation in the damped system.

Q & A

What is the main topic of the video?

The video focuses on the concept of damped free vibration in a Single Degree of Freedom (SDOF) system. It discusses the governing equation of motion for damped vibration and its various cases, including underdamped, overdamped, and critically damped systems.

What does 'underdamped' mean in the context of vibration systems?

'Underdamped' refers to a system where the damping ratio (zeta) is less than 1, causing the system to oscillate with decreasing amplitude over time. This is common in civil engineering structures such as buildings and bridges.

How is the damping ratio defined?

The damping ratio (zeta) is the ratio of the actual damping coefficient (c) to the critical damping coefficient (c_cr), which is the value of damping that would prevent oscillation. A smaller damping ratio results in more oscillations before the system comes to rest.

What is the equation of motion for a damped free vibration system?

The equation of motion for a damped free vibration system is m * x'' + c * x' + k * x = 0, where m is mass, c is the damping coefficient, k is the stiffness, x is the displacement, and x' and x'' are the first and second derivatives of displacement with respect to time, respectively.

What are the different cases for damping in a system?

The three cases for damping in a system are: 1) Critically damped, where the system returns to rest as quickly as possible without oscillating. 2) Overdamped, where the system returns to rest slowly without oscillating. 3) Underdamped, where the system oscillates with decreasing amplitude over time.

How is the natural frequency and period of a damped system different from an undamped system?

In a damped system, the natural frequency is lower than in an undamped system due to the energy dissipation caused by damping. The period of a damped system is longer compared to the undamped case.

What is the significance of the damping ratio for civil engineering structures?

For civil engineering structures such as buildings, bridges, and dams, the damping ratio is typically less than 10%. This low damping ratio means that these structures exhibit minimal oscillation, and the damped frequency and period are often nearly the same as the undamped system.

What is logarithmic decrement in the context of damped vibrations?

Logarithmic decrement refers to the rate at which the amplitude of oscillations in a damped system decreases over time. It is the natural logarithm of the ratio of consecutive peak amplitudes.

How can we calculate the number of cycles required for a system's displacement to reduce by 50%?

The number of cycles required to reduce the displacement by 50% can be calculated using the formula: Number of cycles ≈ 0.11 / damping ratio (zeta). This indicates that with a damping ratio of 10%, about one cycle is needed for a 50% reduction in amplitude.

What is the formula for the displacement response of an underdamped system?

The displacement response for an underdamped system can be written as: x(t) = rho * e^(-zeta * omega * t) * cos(omega_d * t - theta), where rho is the amplitude, zeta is the damping ratio, omega is the natural frequency, omega_d is the damped natural frequency, and theta is a phase angle.