General aim:

This course helps to extend your knowledge of vibrating systems, and lays the foundations for further study in later years.The ideas of periodic and simple harmonic motion are introduced, and free and forced vibrations of mechanical systems are studied.

Objectives

1.1 On completion of this section you should be able to:

• Describe the need for imaginary quantities;
• Explain what is meant by an Argand diagram;
• Define the modulus and argument of a complex number;
• Write down and derive the relations between the real and imaginary parts of a complex number and its modulus and argument;
• Write down the relation between the exponential of an imaginary quantity and the trigonometric functions;
• Interpret a complex number with a time-dependent argument as a rotating vector;
• Add and multiply complex numbers, and relate these operations to translations and rotations on the complex plane;
• Define the inverse of a complex number, and derive expressions for the inverse, given its real and imaginary parts, or its modulus and argument;
• Obtain the result of dividing one complex number by another;
• Explain how phasors allow the addition of several time-dependent complex quantities.

• Write down the differential equation of simple harmonic motion;
• Verify that the solution of this equation is a harmonic function with a phase and amplitude depending on the initial conditions;
• Write the solution in complex notation;
• Draw an Argand diagram showing the relations of position, velocity and acceleration;
• Write an account of energy relations in he harmonic oscillator;
• Derive the quadratic form for the potential energy.

• Describe the general characteristics of the force between atoms, why there is in condensed matter an equilibrium spacing and a binding energy;
• Describe the main features of covalent, ionic, metallic and Van dr Waals bonding;
• Write down the form of the potential energy for a pair of atoms as a function of their separation for ionic and Van der Waals bonding, and;
• Explain which terms have a physical justification and which are used to make a simple approximation;
• Manipulate these potential energy functions to obtain the force between the atoms and the equilibrium spacing;
• Calculate the binding energy of a solid given the interaction energy of a pair of atoms;
• Explain the conditions under which the interatomic potential can be taken to be quadratic;
• Obtain the characteristic frequency of small oscillations of a diatomic molecule;
• Write an account of the infra-red spectroscopy of molecules and solids.

1.4 On completion of this section you should be able to:

• Derive the solution of a second order homogeneous differential equation with constant coefficients;
• Explain the occurrence of two independent solutions with two unknown coefficients;
• Obtain definite values for these coefficients, given boundary conditions;
• Derive the complementary function for an inhomogeneous equation;
• Derive the particular integral for some special inhomogeneous equations;
• Distinguish between transient and steady state behaviour and relate them to the two parts of the solution;

1.5 On completion of this section you should be able to:

• Explain the concept of damping;
• Solve the equation of motion for a damped simple harmonic oscillator;
• Explain the terms weak, strong and critical damping, and find what form of damping is present in a given system;
• Use the solution of the equation of motion to describe and graph the behaviour of a system given particular initial conditions;
• Define quality factor, and obtain an expression for it in terms of the parameters of the system.

1.6 On completion of this section you should be able to:

• Explain the behaviour of oscillatory systems subject to a periodic driving force;
• Solve the steady state equation of motion of undamped and damped simple harmonic oscillators driven by a periodic driving force;
• Discuss the importance of transients;
• Obtain expressions for, and draw graphs of, the amplitude and phase of a forced, damped oscillator;
• Give an account of the effects of varying the amount of damping;
• Explain the relation between the shape of the amplitude resonance curve and the quality factor;
• Identify and solve problems requiring familiarity with the above.