TLDR;
This lecture introduces the fundamental concepts of soil dynamics, focusing on vibration theory. It begins by recapping the previous lecture, then defines key parameters like degrees of freedom (DOF) and their significance in dynamic systems. The lecture explains the basic components of a vibrating system (mass, spring, and damper) and their roles in potential energy, kinetic energy, and energy dissipation. It also covers different types of vibration (free, forced, damped, undamped, periodic, and aperiodic) and their characteristics. Finally, it discusses the equation of motion for a single degree of freedom system and the units of mass, stiffness and damper.
- Degrees of freedom (DOF) are the number of independent coordinates required to define the displaced position of all masses in a system.
- Basic components of a vibrating system are mass, spring, and damper, representing kinetic energy, potential energy, and energy dissipation, respectively.
- Types of vibration include free, forced, damped, undamped, periodic, and aperiodic, each with distinct characteristics and conditions.
Recap of Lecture 1 [0:31]
The lecture begins with a recap of the first lecture, which covered the introduction to soil dynamics. The need for studying the course and its objectives were discussed, along with the definition of dynamic loading as any load varying with time. Examples of dynamic loads include earthquake, wind, moving vehicle, and machinery-induced loads. The key criterion for a load to be considered dynamic is its ability to vibrate the system, involving a continuous exchange of potential and kinetic energy, with energy dissipation through sound or heat.
Degrees of Freedom (DOF) [2:36]
The lecture introduces the concept of degrees of freedom (DOF), defined as the number of independent coordinates (e.g., displacement) required to define the displaced position of all masses relative to their original positions. In dynamic problems, mass properties dictate the degrees of freedom, unlike static loading where stiffness properties are key. Examples, such as a simple pendulum with an inextensible string (1 DOF) and an extensible string (2 DOF), illustrate how different constraints affect the DOF. The lecture continues with examples of two simple pendulums connected by either a rigid rod (1 DOF) or an extensible spring (2 DOF) to further illustrate the concept.
Examples of Determining Degrees of Freedom [9:39]
Several examples are presented to demonstrate how to determine the degrees of freedom for different systems. A mass connected to a spring has 1 DOF, defined by its displacement u(t). A system with two masses and two springs has 2 DOF, represented by the independent displacements x1(t) and x2(t). A combination of a linear spring connected to a mass and a pendulum has 2 DOF, defined by the linear displacement x1(t) and the angular displacement theta(t). If the pendulum is connected via another spring, the system's DOF increases to 3, including the length r(t) of the additional spring. Determining the correct DOF is crucial for modelling any system in practice, as it influences the solution techniques used. The lecture emphasises that DOF is not an intrinsic property of a system but depends on boundary and loading conditions.
Basic Components of Vibration [15:47]
The lecture revisits the three basic components of vibration: potential energy (related to displacement and stiffness), kinetic energy (related to acceleration and inertia), and dissipation or loss of energy (related to damping). For a single degree of freedom (SDOF) system, the basic units are a spring (representing potential energy), a mass (representing kinetic energy), and a damper (representing energy dissipation). These components form the mass-spring-damper (MSD) model, essential for considering all basic aspects of vibration. While dissipation is optional, the potential and kinetic energy components are essential in a simple vibrating system.
Single Degree of Freedom System [18:30]
The lecture describes a single degree of freedom (SDOF) system consisting of a mass (m), a spring (k), and a damper (c). The system's position is defined by the displacement u(t) under an external dynamic load p(t). A free body diagram of the mass shows the weight (mg) balanced by roller reactions, the external force p(t), damper force (cu'), spring force (ku), and inertia force (mu''). D'Alembert's principle is introduced to satisfy the equilibrium of the dynamic system, stating that the externally applied forces, inertial force, and forces of resistance are in equilibrium.
Nature of Forces in a Vibrating System [24:30]
The lecture examines the nature of the inertia force (FI), damper force (FD), and spring force (FS) in a vibrating system. The inertia force is related to acceleration (u'') and is expressed as FI = mu'', valid when mass remains constant. The spring force is related to displacement (u) and is expressed as FS = ku, valid for linear springs. The damper force is related to velocity (u') and is expressed as FD = cu', valid for linear dampers. Non-linear variations of these forces are also discussed, highlighting that the linear relationships are only applicable under specific conditions.
Governing Equation of Motion [33:48]
By considering linear models for inertia, damping, and stiffness, the equation of motion is derived from D'Alembert's principle: m(d²u/dt²) + c(du/dt) + ku = p(t), often written as mu'' + cu' + ku = p(t). This is the basic governing equation of motion for a single degree of freedom mass-spring-damper vibrating system. The lecture also discusses the units of mass, stiffness, and damper in different systems (MLT, FLT, and SI units), with the commonly used SI units being kg for mass, Newton per meter for spring stiffness, and Newton-second per meter for the damper.
Types of Vibration [38:29]
The lecture classifies different types of vibration in a single degree of freedom system. The major categories are free vibration (no external load) and forced vibration (external load present). Free vibration is further divided into undamped (no damper) and damped (damper present). Forced vibration is also divided into undamped and damped, as well as periodic (repeating load) and aperiodic (non-repeating load). Aperiodic forced vibration is sub-classified into transient (finite duration load, e.g., earthquake) and steady-state (infinite duration load, e.g., wind load).
Free Vertical Vibration [47:29]
The lecture discusses free vertical vibration of a single degree of freedom system. It considers a spring initially hanging, then a mass is attached, causing static displacement. The system then vibrates vertically about this new equilibrium position. The equation of motion is derived using D'Alembert's principle, considering the inertia force, spring force, and weight of the body. The dynamic equation of motion remains the same as in the horizontal vibration case, with the displacement denoting only the dynamic component. The lecture concludes by noting that the dynamic equation of motion remains consistent across horizontal and vertical vibration scenarios.