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ENGINEERING TRIPOS PART IA - 2012/2013

PAPER 1 - MECHANICAL ENGINEERING

Mechanical Vibrations

Leader: Prof R.S. Langley

Timing: Weeks 7-8 Lent term and weeks 1-4 Easter term

Structure: 12 lectures, 2 lectures/week

AIMS

The aims of the course are to teach students 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, at 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.

SYLLABUS (Book References)

The five columns refer to the book references as listed below the table. Page numbers are given. Parentheses indicate an incomplete treatment.

	(1)	(2)	(3)	(4)	(5)
The system elements: masses, rigid bodies, rotors, springs and dashpots and their analogies in tuned electric circuits: inductors, resistors and capacitors	2	(212)	(57, 556)	-	25, 27
Obtaining differential equations for the motion of linear mechanical systems	10	212	537	543	25, 27
*First Order Systems*	17	174	-	-	-
Response to step, ramp and impulsive inputs	17	186	-	-	-
Response to harmonic excitation	46	197	-	-	-
Using the exp(iwt) notation for harmonic response calculations	19, 47, 66	-	-	-	11
*Second Order Systems*	18	210	533	521	23
Response to step and impulsive inputs; free vibration and damped SHM	24	216	534	522	31, 37
Response to harmonic excitation	50	219, 306	551	538	42, 47
Damping factor, logarithmic decrement, loss factor	24, 30, 53	(215)	540	(545)	38, 40
*Systems with Two or more Degrees of Freedom*	107	331	-	-	79
Degrees of freedom	107	331	-	-	79
Equations of motion in matrix form, obtaining mass and stiffness matrices	109, 145	-	-	-	-
Natural frequencies and mode shapes	110	335	-	-	79
Eigenvalues and Eigenvectors	161	-	-	-	-
Free vibration and the superposition of modes	123	-	-	-	84
Harmonic excitation	129	-	-	-	130
Vibration isolation and absorption	67, 131	313, 338	-	-	69, 87

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

Please see the Booklist for Part IA Courses for references to this module.

Last updated May 2012

teaching-office@eng.cam.ac.uk