Mechanical Vibrations (+89 Solved Examples + MATLAB)

Determine dof of massless rod with a ball in the x–y plane and torsional spring. Conclude theta is the dof; x = L sin theta, y = L cos theta.

Learn how chapter two models dynamical systems by deriving equations of motion through dynamics and applying differential equations to analyze vibratory motion.

Select a coordinate system that simplifies modeling of vibratory systems, and define spring length change as a function of mass position and damper velocity as a function of mass velocity.

Explains how to derive the equation of motion using the Lagrange-Euler method and the energy method, detailing kinetic energy, potential energy, dissipative energy, nonconservative forces, and time derivatives.

Describe how a mass to the right of equilibrium experiences the spring force -k x and damping force -c x dot, with spring depending on position and damping on velocity.

Review mass moment of inertia about O and G for a rod, with I_o = 1/3 M L^2 and I_G = 1/12 M L^2, and I = M D^2.

Derive equation for a one dof rotational system with massless rod and particle mass, using Newton-Euler, yielding l theta double dot + g theta = 0 under small-angle approximation.

Derive the kinetic energy of a point mass from its mass moment of inertia about the rotation point, showing kinetic energy equals one-half I theta dot squared.

Derive the equation of motion for a mass-spring system on a frictionless surface using Newton-Euler and energy methods, show energy method fails due to external energy injection, and compare results.

Derive the rolling-disk equation of motion with Newton–Euler method, spring and damping under no-slip, yielding 3/2 m θ̈ + c θ̇ + k θ = 0.

derive equation of motion for a disk rolling without slip using the Newton–Euler method about the center of gravity; 3/2 m theta¨ + c theta˙ + k theta = 0.

Derive the equation of motion for a disc rotating about point O with one degree of freedom theta, using Newton-Euler and inertia IO = 3/2 MR^2, small-angle approximation.

Explore equivalent vibratory systems by defining equivalent mass M, damper C, and stiffness K, and solve the equation M x'' + C x' + K x = 0.

Derive the equation of motion for a damped 1D rotating system using Newton-Euler and Lagrange methods, and linearize for small vibrations to obtain EOM with equivalent mass, damping, and stiffness.

Derive the equation of motion for a mass with springs using the Newton-Euler method, showing m x¨ + (k1 + k2) x = 0 and parallel behavior.

Solve a two-spring parallel system using Newton-Euler and energy methods to derive the equation of motion m x'' + (k1 + k2) x = 0 and the equivalent stiffness.

Explore gravity force cancellation in vibratory systems where gravity and static spring deflection cancel at equilibrium, and learn to omit their potential energies in linear, small-oscillation analyses.

Derive motion from a free body diagram, showing gravity and spring cancel at static deflection, yielding m ẍ + k x = 0 with equivalent stiffness K and mass M.

Derive the equation of motion and equivalent stiffness for a mass on an incline with springs K1 and K2 using the energy method, noting gravity cancels via static deflection.

Define natural frequency as the square root of equivalent stiffness over equivalent mass, rad/s, noting omega_n equals two pi f or two pi over tau and is independent of damping.

Derives the equation of motion for a mass on four springs at 45 degrees, using energy methods to obtain m x¨ + (K1 + K2) x = 0 and equivalent stiffness.

Derive the equation of motion and the equivalent system for a cylinder on a curved track using energy and Newton-Euler methods, with polar coordinates and small-angle natural frequency.

Derive the equation of motion and the equivalent system for two masses connected by a rod at angle alpha using Lagrange and Newton-Euler methods; obtain the small-angle equation in theta.

Apply the energy method to derive the equation of motion for a mass-spring system with a massless pulley; small oscillations yield m_eq=4m, k_eq=k, ω_n=√(k/(4m)).

Use energy method to derive equation of motion and the equivalent system for a pulley with mass and inertia, radius of gyration 0.2 m, yielding equivalent stiffness and 9.45 rad/s frequency.

Using Lagrange, derive equation of motion for the mass–spring–damper system and an equivalent mass M/4; yield (1/4)M y'' + (1/4)C y' + k y = 0 and ω_n = sqrt(4k/M).

Derive the equation of motion for a small-cylinder spring-mass system using x dot = r theta dot and kinetic energy. Find the natural frequency omega_n = sqrt(2k/(3m)).

Derive the equation of motion for a two-mass vibratory system with a neglected pulley, using rope-length constraints and energy methods to obtain the equivalent mass 5/4 m and stiffness k.

Solve example 226 to derive motion equations for massless-rod systems: vertical m x'' + k x = 0; horizontal m v'' + k cot^2(theta) y = 0 via energy method.

Use the energy method to derive the equation of motion and natural frequency for a mass bar with two springs, accounting for gravity and vertical equilibrium in a conservative system.

Using the energy method, derive the equation of motion and natural frequency for a spring–mass–rod system, with kinetic and potential energies from kinematic constraints near equilibrium for small oscillations.

Derive the equation of motion and natural frequency for a two-mass spring system at equilibrium using Newton's law and energy methods, considering inertia about point O.

Use Newton–Euler method to derive motion for a massless-pulley system with springs, yielding m x¨ + (16/3) K x = 0, with x2 = x/3 and x1 = 5 x2.

Derive the equation of motion for a mass with massless pulleys using the energy method. Obtain m x'' + (16/3) K x = 0.

Derive the equation of motion for the mass m with three springs in series and massless pulleys using equivalent springs, as shown in example 2.35, and demonstrate multiple solution approaches.

Using Newton Euler, derive a single DOF equation (m r2^2+J) theta double dot + c r1^2 theta dot + k r2^2 theta = 0 from x, y, theta relations.

Derive the equation of motion using the Newton-Euler method for a one-degree-of-freedom rod-mass system with gravity and a torsional spring, and determine the natural frequency from equivalent stiffness and mass.

Chapter two frames dynamics modeling as essential for vibration analysis, comparing energy methods and Newton's second law, and defining natural frequency as sqrt(K/m).

Explore free vibration of undamped systems defined by initial displacement and velocity, and solve the second-order linear homogeneous differential equation to obtain x(t) with no external forcing.

Solve the free undamped equation m x'' + k x = 0; ω_n = sqrt(k/m); x(t) = C1 sin(ω_n t) + C2 cos(ω_n t).

Derive the mechanical energy of a free undamped system by combining kinetic and potential energy, showing it remains constant for motion x = x sin(ω_n t + φ).

In free undamped systems, maximum kinetic energy occurs at equilibrium with zero potential energy, while maximum potential energy occurs at maximum displacement with minimal kinetic energy.

Describe phase plane analysis for a free undamped system, showing how position and velocity trace an ellipse in the x-xdot plane, with x^2/X^2 + xdot^2/(X ωn)^2 = 1.

Apply the energy method to a vibrational system to derive its equation of motion, using kinetic energy, gravitational and spring potentials, and assess instability via ke = kl − mg.

Determine the minimum static friction coefficient to prevent relative motion using undamped motion, period 0.75 s, 50 mm displacement, a_max = x omega_n^2, and mu_s = a_max / g.

Analyze an undamped free vibration of a mass M released from rest at displacement a, deriving x(t)=a cos(ω_n t) and linking T1=T/6, T2=T/4, T3=T/3 to the period.

Explore the signs of velocity and acceleration in a free undamped vibration through example 3.7, using both derivative methods from x(t)=a cos(ωn t) and Newton's law with spring forces.

Explore undamped free vibration of a 5 kg mass on a 500 N/m spring with 0.15 m initial displacement, and plot kinetic and potential energies versus position in MATLAB.

Chapter three reviews dynamics modeling and differential equations as prerequisites, explains free undamped motion with x(t)=X sin(ωn t+φ), energy conservation, and an elliptical phase space of x and x dot.

Explore the free vibration of damped systems by deriving the second-order linear homogeneous equation m x'' + c x' + k x = 0, and solving it using dynamics modeling and initial conditions.

Solve the equation of motion for a damped mechanical system and derive the characteristic roots. Classify motion as underdamped, critically damped, or overdamped using the damping factor η.

Explore the logarithmic decrement for underdamped vibrations by relating consecutive peaks x1 and x2 to the damping factor eta, via delta = ln(x1/x2) = 2 pi eta / sqrt(1-eta^2).

Solve the damped oscillator with m x'' + c x' + k x = 0, c=300, k=1000, m=10; assess overdamped behavior (η=1.5) and plot x(t) via Matlab.

Analyze a 10 kg mass–spring–damper with c=200 and k=1000 to obtain x(t) from m x''+ c x'+ k x=0, illustrating critically damped behavior and faster return than overdamped via Matlab.

Solve the underdamped mass-spring-damper example with m=10 kg, c=10, k=1000 to obtain x(t) and plot its decaying oscillations toward equilibrium, with eta=0.05, omega_n=10 rad/s, omega_d≈9.9 rad/s.

Demonstrates solving the equation of motion in Matlab with ode45 by converting to two first-order equations, x1 = x and x2 = x dot, and solving 0 to 6 seconds.

Explore phase plane analysis of three damped vibrations by plotting velocity versus position for overdamped, critically damped, and underdamped systems, using examples 4.4–4.6 in MATLAB.

Stability depends on the roots of the characteristic equation. The roots lie on the left of the imaginary axis, so free damped systems are inherently stable over time.

Show that a free system is a special case of a free underdamped system; with η→0, the general x(t)=x e^{-η ω_n t} sin(ω_d t+φ) reduces to x sin(ω_n t+φ).

Use the roots -3 ± 2i of the damped system's characteristic equation to extract the damping factor eta and natural frequency omega_n, showing underdamped behavior.

Use Matlab to plot x(t) and extract t max and x max for example 4.10, yielding t max = 0.17 s and x max = 0.67 m.

Students solve a torsional pendulum problem by comparing natural frequencies in vacuum and oil to compute the damping factor, then use logarithmic decrement to determine the end-of-first-cycle displacement.

Explain example 4.12 for underdamped free vibration, deriving x1/x4 = e^{3 eta omega_n tau_d} and generalizing to x1/x_{n+1} = e^{m eta omega_n tau_d}, with the logarithmic decrement delta.

Explore how the damping factor determines motion in three damped systems, distinguishing underdamped, critically damped, and overdamped responses, and apply the logarithmic decrement to estimate damping.

Explain forced vibrations by introducing external excitation and contrast them with free vibrations, using m x'' + c x' + k x = f(t) to show how f(t) drives motion.

Analyze forced vibrations by solving non-homogeneous equation m x'' + c x' + k x = f_t, where response consists of a general solution x_g and a particular solution x_p.

This lecture explains harmonic excitation, derives the particular solution x_p for f(t)=f0 sin ω_f t, and presents the final solution x = x_g + x_p for underdamped systems.

Define the magnification factor beta as a dimensionless amplitude ratio; show how beta increases with lower damping and certain frequency ratios, and how damping prevents unbounded amplification.

In undamped systems, the particular solution for harmonic excitation is x0/(1 - r^2) sin omega_f t with zero phase; r<1 yields in-phase motion, r>1 yields out-of-phase, r=1 is pivotal.

Solve example 5.3 by plotting mass position x(t) for an undamped 10 kg system with k=1000 N/m under f(t)=0.5 sin(10 t) using Matlab ode45; shows resonance when omega_n equals omega_f.

Derive the equations of motion for a frictionless vibrator with two similar springs using the Lagrange approach, analyze small oscillations, and identify resonance when the frequency matches the natural frequency.

Solve a mass–spring–damper with damping 200 and stiffness 1000 under 0.5 sin(10 t) using Matlab ode45, converting to two first-order equations and showing damping-induced stability with bounded motion.

Explore how the magnification factor depends on damping and the frequency ratio, derive its maximum, and show the peak occurs at R = sqrt(1 - 2 eta^2).

Analyze a 10 kg mass-spring system with Coulomb friction (mu_k=0.1, k=1000 N/m) in Matlab to compute position x(t) from x(0)=0.2 m, x'(0)=0.

Model a mass with stiffness and friction in MATLAB by defining a two-variable ODE, solving with ode45 from 0 to 5 s, and plotting position, velocity, and phase plane.

Compare viscous damping and Coulomb friction damping; viscous damping brings the mass to zero at equilibrium, whereas Coulomb friction can halt at a nonzero position.

Examine the vertical motion of a car on a sine-profile road, derive m x'' + k x = k a sin(omega t), and convert to first-order equations for Matlab.

Examine rotating unbalance in machines caused by an eccentric mass. A mass-spring-damper model experiences a harmonic force of amplitude M e ω_f^2, driving vibration.

Define the degrees of freedom as independent variables needed to determine motion; a two-degree-of-freedom system can arise from two masses or from a single mass with two motions.

Derive the equations of motion for a two-dimensional system by applying the Newton–Euler or energy method, and define two independent variables to fully specify its equation of motion.

Apply the Lagrange method to a two-mass system with a spring between M1 and M2. Define x1 and x2 and derive the motion equations from kinetic and potential energies.

Derive the two-mass vibratory system's equations of motion with the Lagrange method, with two degrees of freedom, forming kinetic and potential energies to yield two coupled differential equations.

Apply the Lagrange approach to a car modeled as a vertical bar with two springs K1 and K2, deriving coupled equations for vertical motion y and rotation theta.

Derive the equations of motion for M in x and theta using Newton–Euler method and linearization to obtain m ẍ + m l θ̈ + m g θ = 0.

Convert the system's equations into matrix form with mass matrix M and stiffness matrix K. Define X as a 2×1 vector and write M x¨ + K x = 0.

Analyze a two-degree-of-freedom mechanical vibration system from example 6.2 using given initial conditions, convert to first-order equations, and solve with MATLAB ode45 to study time responses and phase plane.