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STUDY GUIDE · GKU 1

Mechanics and Vibrations study guide (Mechanical Engineering)

Mechanics and vibrations in three parts, following the ETEC handout. Statics: particles and force systems, rigid-body equilibrium, trusses, frames and machines, internal forces in beams, friction, centroids and area moments of inertia. Dynamics: kinetics of particles (Newton, work–energy, impulse–momentum and impact), kinetics of rigid bodies, kinematics of particles and of rigid bodies, then the machinery built on them: linkages, their position, velocity and acceleration analysis, cams, force analysis, power transmission and balancing. Vibrations: free, damped and forced single-degree-of-freedom systems, the first-order system, isolation, critical speeds and two-degree-of-freedom basics. Each section opens with a quick-revision card, explains the physics, derives the working equations and ends with worked examples graded from easy to exam level.

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Structured by the official ETEC subject standard for this topic: every listed topic and learning outcome, with formulas, figures and worked examples. The full sections are available to members in the app.

SKU 1.1 · Statics

  1. Equilibrium of a particle

    Forces that all pass through one point: how to add them, and the two equations that hold them in balance.

  2. Moments, couples and equivalent force systems

    The turning effect of a force, the couple, how any system reduces to one force and one couple, and how a distributed load becomes a point load.

  3. Equilibrium of rigid bodies and support reactions

    Supports and the reactions they provide, the three equations of plane equilibrium, two- and three-force members, and determinacy.

  4. Plane trusses: method of joints and method of sections

    Member forces in pin-jointed trusses, joint by joint or through a single cut, with zero-force members and the determinacy count.

  5. Frames, machines, pulleys and cables

    Structures with multi-force members: take them apart, use action and reaction at every pin, and let pulleys and cables carry tension around corners.

  6. Internal forces in beams: shear and bending moment

    Cut a beam, keep one side, and the equilibrium of that piece gives the shear force and bending moment at the cut; the diagrams follow from two derivatives.

  7. Dry friction: blocks, wedges, screws and belts

    Coulomb friction, impending slip versus sliding, the angle of friction, and the friction devices the exam returns to: wedges, square-thread screws and belts on drums.

  8. Centre of gravity and centroids

    Where the weight of a body acts and where the geometric centre of an area or volume lies: by integration, by composite parts, and by the theorems of Pappus.

  9. Moments of inertia of areas

    The second moment of area: what it measures, how to integrate it, and how the parallel-axis theorem builds it for composite sections.

SKU 1.2 · Dynamics of particles and rigid bodies

  1. Kinetics of particles: Newton's second law

    Force equals mass times acceleration, written in the coordinates that suit the motion, plus D'Alembert's trick of turning a dynamics problem into a statics one.

  2. Kinetics of particles: work, energy and power

    The work–energy principle links forces to speeds through distances, with no accelerations or times; conservation of energy and power follow from it.

  3. Kinetics of particles: impulse, momentum and impact

    Force integrated over time changes momentum; when no external impulse acts, momentum is conserved, and the coefficient of restitution closes the impact problem.

  4. Kinetics of rigid bodies: equations of plane motion

    Force equals mass times the acceleration of the mass centre, and moment equals mass moment of inertia times angular acceleration: applied to fixed-axis rotation, rolling and general plane motion.

  5. Kinetics of rigid bodies: energy and momentum methods

    Work–energy and impulse–momentum written for a body that both translates and rotates, with the applications the exam returns to: falling rods, rolling bodies, clutches and flywheels.

  6. Kinematics of particles

    Describing motion without asking what causes it: straight-line motion with constant or variable acceleration, projectiles, and curved paths in normal–tangential and polar coordinates.

  7. Kinematics of rigid bodies in plane motion

    Translation, rotation about a fixed axis and general plane motion; relative velocity and acceleration between two points of one body, and the instantaneous centre of zero velocity.

  8. Mechanisms and linkages: mobility, Grashof and synthesis

    The vocabulary of mechanisms, how to count their degrees of freedom, how to tell whether a four-bar can turn fully, quick-return ratios, and how to lay out a linkage for a required motion.

  9. Position, velocity and acceleration analysis of linkages

    Loop-closure equations for position, relative-motion equations for velocity and acceleration, the Coriolis term for sliding joints, and the slider-crank results used in engine work.

  10. Cam design

    Cam and follower types, the displacement diagram, the standard rise motions and their peak velocity and acceleration, and why the acceleration curve must be continuous.

  11. Static and dynamic force analysis of mechanisms

    Finding the joint forces and the input torque of a mechanism, first ignoring inertia, then adding it with D'Alembert's inertia forces; the slider-crank engine worked through.

  12. Power transmission: gear trains, belts and flywheels

    Speed and torque through simple, compound and epicyclic gear trains, power carried by a belt, and sizing a flywheel to hold the speed fluctuation within limits.

  13. Balancing of rotating and reciprocating masses

    Why an off-centre mass shakes a machine, how to balance masses in one plane and in several planes, and what can and cannot be balanced in a reciprocating engine.

SKU 1.3 · System vibrations

  1. Free undamped vibration of single-DOF systems

    The spring–mass oscillator: its equation of motion from Newton and from energy, the natural frequency, equivalent springs, and how to model a beam or a shaft as a single spring.

  2. Damped free vibration

    Viscous damping, the damping ratio and the three kinds of response, the damped frequency, and measuring damping from a decay record with the logarithmic decrement.

  3. Forced vibration, resonance and frequency response

    The steady response of a spring–mass–damper to a harmonic force, the magnification factor and phase, resonance, rotating unbalance, and the frequency response of a first-order (spring–damper) system.

  4. Vibration isolation, critical speeds and two-DOF systems

    Applying the single-DOF results to real design: choosing mounts that isolate, the whirling speed of a shaft, and the two natural frequencies and the vibration absorber of a two-mass system.

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