Aims

The aims of the course are to:

• Describe mathematically the behaviour of simple mechanical vibrating systems.
• Determine the response of these systems to transient and harmonic excitation.
• Analyse systems with more than one degree of freedom.

Objectives

As specific objectives, by the end of the course students should be able to:

• Obtain differential equations for mechanical systems comprising masses, rigid bodies, rotors, springs and viscous dashpots, noting the analogy with tuned electric circuits.
• Reduce all differential equations to a standard form.
• Solve these standard-form equations for the response to step, ramp, impulsive and harmonic excitation.
• Understand the concept of damping and the meaning of damped natural frequency, damping factor and logarithmic decrement.
• Obtain and solve differential equations in matrix form for mechanical systems with more than one degree of freedom.
• Apply the rudimentary principles of modal analysis to the free vibration of a two-degree-of-freedom oscillator subject to initial conditions.
• Apply these results to the design of a vibration absorber and to methods of vibration isolation.

Content

For each topic, the letter in parentheses is the link to the table at the bottom of the page, giving page numbers in the references.

Introductory material

• The system elements: masses, rigid bodies, rotors, springs and dashpots and their analogies in tuned electric circuits: inductors, resistors and capacitors (a)
• Obtaining differential equations for the motion of linear mechanical systems (b)

First order systems

Go to (c) for book reference pages (c)

• Response to step, ramp and impulsive inputs (d)
• Response to harmonic excitation (e)
• Using the exp(iwt) notation for harmonic response calculations (f)

Second order systems

Go to (g) for book reference pages (g)

• Response to step and impulsive inputs; free vibration and damped SHM (h)
• Response to harmonic excitation (i)
• Damping factor, logarithmic decrement, loss factor (j)

Systems with Two or more Degrees of Freedom

Go to (k) for book reference pages (k)

• Degrees of freedom (l)
• Equations of motion in matrix form, obtaining mass and stiffness matrices (m)
• Natural frequencies and mode shapes (n)
• Eigenvalues and Eigenvectors (o)
• Free vibration and the superposition of modes (p)
• Harmonic excitation (q)
• Vibration isolation and absorption (r)

References

(1) DEN HARTOG, J.P. MECHANICAL VIBRATIONS
(2) HIBBELER, R.C. ENGINEERING MECHANICS: DYNAMICS (SI UNITS)
(3) MEIROVITCH, L. ELEMENTS OF VIBRATION ANALYSIS
(4) MERIAM, J.L. & KRAIGE, L.G. ENGINEERING MECHANICS. VOL.2: DYNAMICS
(5) PRENTIS, J.M. DYNAMICS OF MECHANICAL SYSTEMS

Relevant page numbers are given for each topic in the table. Parentheses indicate an incomplete treatment.

This syllabus contributes to the following areas of the UK-SPEC standard:

Toggle display of UK-SPEC areas.

 
Last modified: 24/05/2022 14:04