Complete Mechanical Vibration Online Course RAHME301 Rahsoft

Explore the fundamentals of vibration and its engineering applications. Learn how designers minimize imbalance, address mounting and assembly effects, and apply vibration to improve efficiency, testing, and simulations in engineering.

Explore fundamental concepts of vibration and oscillation, including energy transfer between potential and kinetic energy in pendulums and spring-mass-damper systems, and the role of damping in real-world vibrations.

Determine the number of degrees of freedom by using independent coordinates to describe the system's position over time, noting one, two, three, or infinite degrees of freedom.

Explore discrete and continuous systems in mechanical vibration, distinguishing finite and infinite degrees of freedom, and learn how discrete models approximate continuous structures with springs and dampers.

Classify vibration problems into free and forced vibration, then distinguish damped and undamped, linear and nonlinear, and deterministic versus random excitations, with focus on resonance concepts.

Develop mathematical models and governing equations for vibrating systems, choose appropriate linear or non-linear, single or multi-degree-of-freedom representations, and solve and interpret results under external excitations and initial conditions.

Explore the vibration analysis procedure for a motorcycle rider in the vertical direction, using simple single-degree-of-freedom models that combine rider, wheel, and vehicle mass with spring and damping concepts.

The spring element acts as a massless, damping-free link, with force proportional to relative displacement (f = kx); for small deflections, linearization via Taylor expansion yields a linear stiffness.

Explore spring elements and cantilever beam behavior, modeling a concentrated mass with negligible beam mass, using deflection at the end and stiffness concepts for a single degree of freedom system.

Explore the combination of springs by analyzing springs in parallel and in series, deriving equivalent stiffnesses and static deflection through free-body diagrams and equilibrium equations.

Identify mass or inertia elements and build a spring-mass model to analyze vibrating systems, linking kinetic energy, work, and forces; select the appropriate model for single or multi-degree-of-freedom structures.

Explore translational masses connected by a rigid bar, and derive equivalent masses by equating kinetic energies to have the same angular velocity.

Couple translational and rotational masses to form equivalent kinetic-energy representations, equating translational energy to rotational energy via J0 or equivalent mass Jq.

Determine the equivalent mass of a pulley-link-cylinder system by equating kinetic energies under small displacements, using no-slip relations and translating rotation into translation.

The damping element is essential for accurate vibrational modeling, showing energy converts to heat or sound and the damper is massless, with damping depending on relative velocity.

Viscous damping dissipates energy proportional to velocity in air, gas, water, and oil; dry friction damping provides constant opposing force, while material damping dissipates energy through internal friction and hysteresis.

Demonstrates viscous damping between moving and fixed plates with Newton's law, linking shear stress to the velocity gradient via viscosity, and defines damping constant c as area times shear stress.

Model a lubricated flat-plate bearing using viscous damping to link resistance and velocity, and determine the plate clearance as about 0.86 mm.

Explore piston dashpot dynamics in a liquid-filled cylinder, deriving viscous shear stress, pressure forces, and flow rate in a clearance space between piston and wall.

Explore harmonic motion and periodic motion with a crank and sliding mechanism and a simple pendulum, and derive x = a sin(ωt) along with velocity and acceleration relations.

Derive the equivalent spring and mass concepts for a rigid bar connected to springs, using free body diagrams and moment equilibrium, then analyze dampers in series and parallel.

Explore free vibration in a single-degree-of-freedom mass–spring system with no external force or damping, and note how damping affects energy.

Explore free, undamped vibration of a translational spring–mass system with a single degree of freedom. Derive m x'' + k x = 0 and note gravity's effect on vertical equilibrium.

Derives the equation of motion for a torsional system, relating disk angle to inertia and spring stiffness, and explains natural frequency omega_n for torsional pendulum and clock hands.

Explore free vibration of an undamped torsional system, deriving the equation of motion from torque, mass moment of inertia j0, and torsional spring, and determine the natural frequency and period.

Analyze the stability condition of a uniform rigid bar with end springs under gravity, deriving the angular motion and three cases: stable oscillation, linear growth, and exponential growth.

Use the Rayleigh energy method to find natural frequencies of a single degree of freedom by balancing kinetic and potential energy at equilibrium and maximum displacement, with harmonic motion.

Explore how spring mass affects natural frequency using energy methods, deriving omega_n from kinetic and potential energy, including the mass of the spring and harmonic motion.

Compute the natural frequency of transverse vibration for a water tank modeled as a cantilever with column mass, using a single-degree-of-freedom approach and an equivalent mass from kinetic energy.

Explore viscous damping in a single-degree-of-freedom system, where the damper force is proportional to velocity and opposite to motion, yielding m x'' + c x' + k x = 0.

Assume a solution for the viscous damping vibration equation, derive the characteristic equation with two roots, and form general solution; use initial position and velocity to find C1 and C2.

Explore critical damping, damping ratio, and natural frequency, derive the roots of the characteristic equation, and analyze undamped vibration (zero damping) along with the three damping scenarios.

Explore viscous damping with a damping ratio less than one by deriving the damped vibration solution, natural and damped frequencies, and sine and cosine forms for a damped oscillator.

Case 2 of viscous damping yields repeated roots and non-periodic motion; applying initial conditions determines the constants in the motion equation.

Demonstrates case 3 of viscous damping, where roots become real and negative, yielding exponential decay in motion, unlike cases with imaginary parts.

Explore viscous damping and damping ratio, tracing root movement on the real and imaginary plane from undamped to critically damped, relating to damped vibration and gun recoil examples.

Analyze a single-degree-of-freedom torsional system with viscous damping, where torque opposes angular velocity, governed by inertia, damping, and shaft stiffness, yielding damped frequency and critical damping insights.

Explore the impact response of a forging hammer modeled as a damped mass-spring system, using momentum conservation and the coefficient of restitution to determine post-impact velocities, damping, and natural frequency.

Analyze the cannon recoil as a spring-damper system, linking high-pressure gas and reaction forces to the gun barrel, determine critical damping, natural frequency, and return time from given masses.

Design motorcycle shock absorbers by using damping ratio and natural frequency to achieve a two-second damping period, a 250 mm maximum displacement, and logarithmic decrement.

Describe free vibration of a compound torsional pendulum about a suspension point, derive its linearized equation, and obtain natural frequency, relating it to the simple pendulum via radius of gyration.

Explore how the solution x_c attains maximum and minimum values by differentiating the governing equation, setting the derivative to zero, and deriving sine and cosine relations for the critical points.

Derive the dynamic equation of motion for a pulley-spring-mass system from free body diagrams, relate angular deflection to displacement, and analyze energy and static deflection to determine the Nashar frequency.

Study harmonic excitation of a single degree of freedom system, compare it to free vibration, derive equation of motion, and analyze resonance and transient response under base and displacement excitations.

Explain the equation of motion for a damped spring-mass system under forcing, showing the total response as the sum of homogeneous free vibration and a steady-state particular solution.

Explore undamped systems under harmonic forcing, deriving the equation of motion and both homogeneous and particular solutions, and explain static deflection, magnification factor, and amplitude ratios.

Explore three cases of an undamped system under harmonic excitation, analyzing how the frequency ratio relative to the natural frequency shapes resonance, phase relation, and amplitude behavior.

an example analyzes a fixed-fixed beam under harmonic excitation to determine vibration amplitude, natural frequency, and resonance conditions using beam properties, mass, and stiffness.

Explore the response of a damped system to a harmonic force, deriving the equation of motion, the harmonic particular solution, and the amplitude and phase of the response.

Explore the damped system under general harmonic motion by using complex forcing, extracting the real excitation, and deriving the real and imaginary parts to obtain the particular solution.

Base excitation drives harmonic motion of a mass in a spring–mass–damper system; relate base and mass displacements through the equation, showing it's equivalent to a harmonic force on the mass.

Derive equation of motion for a vibration system with mass and inertia J0, springs k1 and k2, and a sinusoidal input F0 L sin(omega t); obtain particular solution for theta.