Mechanical Vibrations (+88 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.

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.

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.

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.

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.

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.

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)).

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)).

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).

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.