Contoh Soal Damped Free Vibration (Seri Getaran Mekanis part5)

Summary

This video script explains how to solve a physics problem related to mass-spring-damper systems using free vibration analysis. The problem involves calculating the displacement of an 88 kg mass that moves 0.2 meters from equilibrium. The video walks through the steps of determining the system’s parameters, solving differential equations, and applying initial conditions to find the displacement at a given time. It also demonstrates how the solution can be verified using computational tools like Octave. The tutorial is detailed, covering both the mathematical approach and the use of engineering software for quick verification.

Takeaways

• 😀 The mass of 88 kg is displaced 0.2 meters to the right from the equilibrium position and then released from rest (initial velocity = 0).
• 😀 The problem is about finding the displacement at t = 2 seconds for a mass-spring-damper system.
• 😀 The governing equation for the system is a second-order differential equation: Mx'' + Cx' + Kx = 0, where M is mass, C is damping, and K is stiffness.
• 😀 The natural frequency (ω_n) is calculated as √(K / M) = 2 rad/s using the provided values for stiffness (K = 32) and mass (M = 8).
• 😀 The damping ratio (ζ) is calculated as 0.625, indicating an underdamped system (ζ < 1).
• 😀 The solution for displacement in an underdamped system takes the form: x(t) = a * exp(-ζω_n * t) * sin(ω_d * t + φ), where ω_d is the damped frequency.
• 😀 The damped frequency (ω_d) is calculated as ω_n * √(1 - ζ^2), resulting in ω_d = 1.56 rad/s.
• 😀 To solve for the constants 'a' and 'φ', initial conditions (x(0) = 0.2 m and v(0) = 0) are applied, leading to a system of equations for these unknowns.
• 😀 After solving the system, the values for 'a' and 'φ' are found: a = 0.256 and φ = 0.896 radians.
• 😀 The final displacement solution is: x(t) = 0.256 * exp(-1.25 * t) * sin(1.56 * t + 0.896), which describes the motion of the mass over time.
• 😀 The same problem can be solved using Octave and engineering vibration tools, which provide the same results more efficiently, confirming the correctness of the manual solution.

Q & A

What is the initial condition of the system described in the problem?

The initial condition of the system is that the mass starts at rest with an initial displacement of 0.2 meters. This is indicated by the statement 'released from rest' which implies that the initial velocity (v0) is zero.

What physical quantities are needed to solve for the displacement of the mass at a given time?

To solve for the displacement, we need to determine the equation of motion for the system, which depends on the mass (m), damping coefficient (c), spring constant (k), and initial conditions. We also need the natural frequency (ωn) and the damping ratio (ζ).

How is the equation of motion for a mass-spring-damper system formulated?

The equation of motion for a mass-spring-damper system is given by: m * x''(t) + c * x'(t) + k * x(t) = 0, where m is the mass, c is the damping coefficient, k is the spring constant, and x(t) is the displacement as a function of time.

What are ωm and ζ, and how are they related to the system's characteristics?

ωm (the natural frequency) is calculated as the square root of k/m, and ζ (the damping ratio) indicates the system's damping behavior. The damping ratio helps categorize the system as underdamped (ζ < 1), critically damped (ζ = 1), or overdamped (ζ > 1).

What does it mean if ζ is less than 1 in this context?

If ζ is less than 1, the system is underdamped, meaning it will oscillate around the equilibrium position with gradually decreasing amplitude until it eventually comes to rest.

What is the solution for displacement in an underdamped system?

The solution for displacement in an underdamped system is given by: x(t) = A * e^(-ζωn * t) * sin(ωd * t + φ), where A is the initial amplitude, ωd is the damped natural frequency, and φ is the phase angle.

How do you calculate the damped natural frequency ωd?

The damped natural frequency ωd is calculated using the formula: ωd = ωn * sqrt(1 - ζ^2), where ωn is the natural frequency and ζ is the damping ratio.

What is the significance of the initial conditions in solving for the constants A and φ?

The initial conditions, such as displacement (x(0)) and velocity (v(0)), are used to solve for the constants A (amplitude) and φ (phase angle) in the solution equation. For instance, x(0) = 0.2 meters and v(0) = 0 help determine these constants.

How did the values for A and φ come out in this specific problem?

By applying the initial conditions to the equation for displacement, the values for A and φ were found to be A = 0.256 and φ = 0.896 radians, respectively.

How does the solution change when using computational tools like Octave for this problem?

Using Octave or similar engineering tools can simplify the process by directly calculating the values of A and φ, as well as providing the displacement response over time. This helps in visualizing the behavior of the system more easily without manual calculations.