Consider a system of particles enclosed within a region of space, as shown in figure 1.1. No restriction is placed on the manner in which the particles are connected—the system may consist of particles of a fluid or gas, an elastic or rigid body.
In Newtonian dynamics the mass of each of the particles of a system is considered constant. The variation of the mass of a particle with the velocity is considered in relativistic mechanics and is not involved in the problems considered here.
Lagrange’s equations were developed by the great French mathematician J. L. Lagrange and published in his celebrated book Analytical Mechanics, (Paris, 1788). Lagrange’s equations are of great importance and are widely used in engineering and physics.
Figure 4.1 shows a mathematical model of a two-degree-of-freedom vibratory system. The springs and dashpots are assumed light, and all the mass of the system is assumed concentrated in the two bodies each of mass m. All damping in the system is assumed to be of the viscous friction type and concentrated in the dashpots with damping coefficients c.
A rigid body may be supported in such a way that one point in the body is always stationary—the motion of the body is then a rotation about this fixed point. Figures 5.1a and b show two possible arrangements which enable a body to rotate about a fixed point. A body supported as shown in figure 5.1 is generally called a gyroscope and the support shown in figure 5.1b is the most widely used in the application of gyroscopes.
Many elements of machinery consist of a body spinning about a certain axis, and when this axis is forced to rotate about some other axis, gyroscopic effects come into play. In some cases the gyroscopic effects are useful and designed into the system; in other cases the effect is undesirable but unavoidable, and it must therefore be taken into account in the design. In most cases the gyroscopic effect gives rise to pulsating bearing pressures and bending moments in rotors which involves fatigue considerations in the rotors and special consideration in the design of the bearings.
Gyroscopes are now used in a great number of instruments for direction indication, stabilisation and inertial guidance. Some of these applications will be considered in this chapter.
In the international system of units (the Systéme International d’Unités or SI System) the basic units in mechanics are the units for length, mass and time: these units are the metre (m), the kilogram (kg) and the second (s).
All machinery and processes need some means of control to ensure proper functioning. In the past, when machinery was slow functioning, a human operator was able to keep engine speeds reasonably constant by manual changing of the valve gear; as machinery became faster and bigger this soon became impossible and automatic control had to be introduced. Early examples are the automatic valve gear on steam pumps and the centrifugal speed governor applied to steam engines. Many simple control systems are present in modern houses: for instance, the ball-cock for regulating the water level in cisterns and the thermostat in a hot-water supply or heating system. More sophisticated control systems exist in refrigerators, washing machines, radios and television sets.
