19. Introduction to Mechanical Vibration

The lecture begins with an introduction to the importance of studying vibrations, which are fundamental to various phenomena such as speech and music. The instructor emphasizes that while finding equations of motion has been the focus so far, the rest of the term will concentrate on the resulting motion, particularly on how to analyze and control vibrations. The concept of a single degree of freedom system is introduced as a basic building block for understanding more complex vibrational phenomena, with examples provided to illustrate the point.

This section delves into the specifics of single degree of freedom systems, using a Mass-Spring-Dashpot model with two springs as an example. The instructor guides the audience through the process of creating a free body diagram and deriving the equation of motion for the system, taking into account pre-compression in the springs. The natural frequency of the system is discussed, and the audience is encouraged to predict whether pre-compression affects it. The equation of motion is then rearranged to highlight the importance of understanding the behavior of such systems.

The instructor clarifies the concept of natural frequency and its independence from external forces that do not depend on motion coordinates. Using the example of a mass-spring system with pre-compression and a hanging mass system, it is shown that the natural frequency is determined by the stiffness and mass of the system, not by external forces like gravity. The equation of motion for the non-homogeneous system is presented, and the process of finding the static equilibrium position is discussed.

The paragraph focuses on solving the differential equation of motion for a system with a constant term, which represents a non-homogeneous scenario. The static equilibrium position is calculated, and the concept of separating the motion variable into static and dynamic components is introduced. This approach simplifies the equation of motion and allows for a clearer understanding of the system's natural frequency and behavior.

The discussion turns to the role of initial conditions in the motion of a system without external excitation. The solution to the equation of motion is presented in the form of a cosine function, highlighting that any motion is a result of initial displacement or velocity. The natural frequency of the system is derived from the equation, and the solution is expressed in terms of initial conditions, showing how the system responds to them.

Damping is introduced as a factor that can complicate the vibration of a system. The paragraph explains how damping affects the equation of motion and the resulting motion. The solution to the damped system is explored, showing that the motion can be expressed in terms of cosine and sine functions, with the amplitude decaying over time. The importance of understanding the phase angle and its relation to initial conditions is emphasized.

The concept of phase angle is explored in depth, explaining its meaning and relevance in the context of vibration analysis. The instructor uses a graphical representation to illustrate how the phase angle relates to the initial conditions and the time delay to reach the peak of the vibration. The relationship between the phase angle and the time delay is quantified, providing a clear understanding of how these concepts are used in analyzing system behavior.

The lecture revisits concepts from a previous course, 1.803, but this time from an engineer's perspective. The focus is on the homogeneous equation of motion with damping and the use of the quadratic equation to find the system's response to initial conditions. The engineer's approach to solving the quadratic equation is presented, introducing terminology such as the damping ratio and the critical damping concept.

This section discusses the impact of damping on the system's response to initial conditions. The solutions for different cases of damping (zeta less than 1, equal to 1, and greater than 1) are presented, highlighting the differences in system behavior. The importance of the damping ratio in determining whether the system will oscillate or not is emphasized, and the solutions for each case are analyzed.

The paragraph focuses on the characteristics of damped oscillation, particularly when the damping ratio is less than 1. The solution to the equation of motion for this case is detailed, showing how the system's motion is a combination of exponential decay and oscillation. The concept of the damped natural frequency is introduced, and the solution is expressed in terms of initial conditions, illustrating how the system's response changes over time.

The instructor dispels the common myth that damping is the primary cause of frequency change in a vibrating system. Using the example of a pendulum in water, it is explained that the observed change in frequency is not due to damping but rather to added mass, which affects the system's kinetic energy. The importance of distinguishing between the effects of damping and other factors on frequency is highlighted.

The lecture concludes with a practical method for estimating the damping ratio of a system using the logarithmic decrement. The process involves measuring the response of the system over time and calculating the ratio of two peak values separated by a certain number of periods. The formula for estimating the damping ratio is provided, along with a rule of thumb for quick estimation based on a 50% decay in amplitude.

In the final paragraph, the instructor summarizes the key concepts discussed in the lecture, emphasizing their importance for understanding more complex vibrations in future lessons. The practical application of these concepts is highlighted, and students are reminded of their relevance to real-world scenarios. The lecture ends with well-wishes for an upcoming quiz and a reminder of the next class meeting.