This series of tutorials under the heading **Mechanics of machines** is concerned with the mechanics of simple mechanisms, primarily the **slider and crank mechanism** for which methods of analysis for displacement, velocity, acceleration, inertia forces and associated input and output torques can be applied to a wide range of other mechanisms.

The level is broadly equivalent to the early years of an engineering first degree. The tutorials are not a substitute for a text book or formal lecture notes but might help in understanding tricky points of detail not necessarily mentioned elsewhere.

Tutorials covering aspects of Robotics and Mechanical vibrations, which can also be considered under the broad heading of *mechanics of machines*, are in separate series.

Tutorials in the **Mechanics of machines** series are as follows.

Crank mechanism kinematics - classic analysis

Derivation of general expressions for displacement, velocity and acceleration of the slider in a slider and crank mechanism using geometry and differential calculus; plots to illustrate cyclic characteristics for a specified mechanism.

Crank mechanism kinematics - velocity and acceleration diagrams

General principles of velocity and acceleration diagrams; explanation of plane motion of a rigid body as a combination of translational and rotational motion; explanation of instantaneous centre of rotation (velocity pole) of a rigid body; construction of velocity and acceleration diagrams for elements of a slider and crank mechanism at a specific crank angle; construction of the velocity pole diagram for the mechanism's connecting rod; calculation of velocities and acelerations directly from the diagrams.

Crank mechanism kinematics - vector equations

Construction of vector diagrams and corresponding vector equations for velocity and acceleration of elements of a slider and crank mechanism; plots and analysis of angular velocity and angular acceleration of the connecting rod over one crank cycle.

Crank mechanism statics - free body diagrams

Construction of free body diagrams of forces and moments in static mode for elements of slider and crank mechanisms operating (a) as an engine and (b) as a compressor; this tutorial is an introduction to succeeding tutorials covering forces and torques in crank and slider mechanisms.

Crank mechanism - inertia forces and crankshaft torque

Incorporation of inertia forces and moments on machine elements in free body diagrams using D'Alembert's principle; use of acceleration diagrams to derive the inertia force and moment on a connecting rod; worked example to resolve forces and moments on the individual elements and the crankshaft output torque when acting as an engine; alternative graphical derivation of crankshaft torque ignoring inertial forces; plot illustrating variation of crankshaft torque with crank angle (crank effort diagram); method of *equivalent dynamical systems* to account for the inertial force and moment acting on the connecting rod.

Crank mechanism - crank effort and balancing

Deriving mean engine torque from a crank effort diagram; application of a flywheel to minimise cyclical fluctuations in angular velocity and energy delivered to the crankshaft; reducing flywheel dimensions by using multiple cranks with phased crank angles; general principles of static and dynamic balance of a rotor; balancing the rotational inertial force generated by the crank arm; balancing primary and secondary inertial forces generated by reciprocal motion of the slider; balancing inertial forces and moments in the connecting rod using an *equivalent dynamical system.*

Crank mechanism - offsets and eccentrics

Illustration of offset configurations of slider and crank mechanisms with eccentricity of axes; derivation of general expressions for displacement and velocity of the slider in an offset mechanism using geometry and differential calculus; plots illustrating cyclic characteristics of a specified mechanism for a range of eccentricities; illustration of the construction and operation of an *eccentric crank mechanism*.

I welcome feedback at: