CHAPTER 1- INTRODUCTION amplitude, depending on the applied
force.
Mechanical Vibration – The motion of a
particle or body oscillating around a CHAPTER 2- FREE VIBRATION
position of equilibrium. The result of a
system being displaced from a stable Spring-Mass Model – A system
equilibrium and oscillating due to consisting of a mass attached to a
restoring forces. spring, used to study vibrations.
Equilibrium – The stable position where Static Equilibrium – The position
a system naturally settles when where the forces on a system balance
undisturbed. out, meaning no net motion occurs.
Restoring Forces – Forces that bring a Weight (W) – The force exerted by
system back to equilibrium (e.g., elastic gravity on the mass, given by W=kδs,
forces in springs, gravitational forces in where k is the spring constant and δs is
pendulums). the elongation of the spring in
Period – The time required for a system equilibrium.
to complete one full cycle of motion. Spring Force (T) – The restoring force
Frequency – The number of vibration exerted by a stretched or compressed
cycles per unit time. spring, given by T=kδs.
Amplitude – The maximum Displacement - The deviation of the
displacement of a system from its particle from its equilibrium position.
equilibrium position. Amplitude – The maximum
Free Vibration – Vibration that occurs displacement of the vibrating particle
when a system moves under the from equilibrium.
influence of restoring forces alone, Initial Velocity – The velocity of the
without external forces. particle when it is first set into motion.
Forced Vibration – Vibration that occurs Resultant Force (F) – The net force
when an external periodic force is applied acting on the mass due to weight and
to a system spring force, given by F=W−k(δs+x)=−kx.
Undamped Vibration – An ideal Simple Harmonic Motion (SHM) – A
vibration where energy loss due to type of periodic motion where
friction or resistance is negligible. acceleration is proportional to
Damped Vibration – A vibration where displacement and directed toward
energy is gradually lost, reducing equilibrium.
amplitude over time. Acceleration– The second derivative of
Damping – The effect that reduces displacement with respect to time,
vibration amplitude due to resistance representing how fast velocity is
(e.g., friction or air resistance). changing.
Damped Free Vibration – A free Fundamental Equation of Motion –
vibration where amplitude gradually Given by mx′′+kx=0m x'' + kx = 0mx′′+kx=0,
decreases due to damping. describing the motion of a mass-spring
Damped Forced Vibration – A forced system.
vibration where damping influences the
Sign Convention – A system of degrees of freedom while retaining the
assigning positive and negative signs to dominant motion.
forces and displacements to ensure Degrees of Freedom- The number of
consistency in calculations. independent motions or displacements a
Natural Frequency of Vibration- system can have.
number of cycles described per unit of Multi-Degree of Freedom Systems-
time Systems with multiple independent
Frequency (Hz)- A unit denoting cycles motions or displacements.
per second. 1 Hz corresponds to one Continuous Systems- Systems with an
cycle per second. infinite number of degrees of freedom,
Revolutions Per Minute (rpm)- A such as elastic rods or beams.
measure of rotational speed. Elastic Rods- Structural elements that
Simple Harmonic Motion- A type of can deform under axial tension or
periodic motion where restoring forces compression and possess continuous
are proportional to displacement. degrees of freedom.
Pendulum Model- A bob attached to a Axial Tension/Compression- A type of
cord of length L, swinging under gravity. deformation in which a rod is stretched
Forces acting include tension (T) and or compressed along its length.
weight (W). Static Deflection- The displacement of
Equation of Motion- Governs the a structure under a static load.
angular displacement for small Cross-Sectional Area (A)- The area of
oscillations. the cross-section perpendicular to the
Natural Circular Frequency- axis of the rod.
Frequency at which the pendulum Elastic Modulus (E)- A material
naturally oscillates. property that measures its stiffness,
Period- Time taken for one complete defined as stress divided by strain in the
oscillation of the pendulum. elastic region.
Rigid Body Vibrations- Vibrations Axial Strain- The deformation per unit
involving a rigid body or system with a length in the axial direction
single degree of freedom, similar to Axial Displacement- The displacement
particle vibrations. along the axis at position xx and time tt.
Equation Form- Represents simple Tensile Force- The external force
harmonic motion for rigid bodies. applied to stretch the rod.
Natural Frequency Determination- Membrane Force- The internal force
Calculated by identifying stiffness and within the rod's cross-section due to
mass properties, allowing vibrations to axial deformation
be modeled similarly to simple harmonic Stress- The internal force per unit area
motion. within a material
Cantilever Beam- A beam fixed at one
CHAPTER 3- EQUIVALENT SYSTEMS end and free at the other, capable of
Equivalent Systems- Simplified models supporting transverse loads.
that approximate the behavior of a more Moment of Inertia (I)- A geometric
complex system by reducing the property that quantifies a beam's
resistance to bending or rotation about Torsional Spring Stiffness - Equivalent
an axis. stiffness for torsional resistance
Transverse Point Load- A load applied Floating Body- An object partially or
perpendicular to the beam's longitudinal fully submerged in a fluid that is subject
axis at a specific point. to buoyant forces.
Deflection- Displacement of the beam Buoyant Force - The upward force
in response to an applied load, exerted by a fluid on a submerged
measured perpendicular to its axis. object, equal to the weight of displaced
Spring Constant (k)- A parameter fluid.
representing stiffness in an equivalent Displacement Depth- The depth to
spring system. which a floating body sinks due to
Simply Supported Beam- A structural gravity or applied forces.
element supported at both ends, Pressure- Force exerted per unit area
typically by pins or rollers, allowing within a fluid
rotation but preventing translation. Density- Mass per unit volume of a
Transverse Point Load- A load applied substance, such as fluid density.
perpendicular to the longitudinal axis of Restoring Force- The force that returns
the beam. a displaced floating body to equilibrium.
Equivalent Stiffness- A measure of the Applied Force- An external force acting
beam's resistance to deformation under on the body causing additional
load, represented as an equivalent deflection.
spring constant. Compound Systems- Systems
Bending Stiffness (EI)- A property of composed of multiple interconnected
the beam determined by its material's components, such as beams attached to
modulus of elasticity (E) and its moment other structures.
of inertia (I). Linear Spring System- A simplified
Center-Span Deflection- The vertical representation where stiffness is
displacement at the midpoint of the modeled as linear elastic springs for
beam due to applied loads. analysis purposes.
Elastic Rod- A slender structural Archimedes' Principle- States that the
element that can deform elastically buoyant force on an object submerged
under torsion or axial loads. in fluid equals the weight of displaced
Shear Modulus (G)- A material property fluid.
describing its ability to resist shearing Equilibrium Configuration- The stable
deformation. position of a system under static
Polar Moment of Inertia (J)- A conditions without external disturbances.
geometric property of a cross-section Free-Body Diagram- A graphical
that influences torsional stiffness. representation showing all forces acting
Torque (T0T0)- A rotational force on a body for analysis purposes.
applied to a rod or shaft.
Boundary Conditions- Constraints
applied at the edges of a structure, such
as fixed or free ends.
