Sinusoidal Vibration

by Lalanne; Christi

• ISBN13: 9781560329855
• ISBN10: 1560329858
• Format: Hardcover
• Copyright: 2002-03-29
• Publisher: CRC Press

Summary

About the Series: This important new series of five volumes has been written with both the professional engineer and the academic in mind. Christian Lalanne explores every aspect of vibration and shock, two fundamental and crucially important areas of mechanical engineering, from both the theoretical and practical standpoints. As all products need to be designed to withstand the environmental conditions to which they are likely to be subjected, prototypes must be verified by calculation and laboratory tests, the latter according to specifications from national or international standards. The concept of tailoring the product to its environment has gradually developed whereby, from the very start of a design project, through the to the standards specifications and testing procedures on the prototype, the real environment in which the product being tested will be functioning is taken into account. The five volumes of Mechanical Shock and Vibration cover all the issues that need to be addressed in this areaof mechanical engineering. The theoretical analyses are placed in the context of the real world and of laboratory tests - essential for the development of specifications. Volume I: Sinusoidal Vibration The relative and absolute response of a mechanical system with a single degree of freedom is considered for arbitrary excitation, and its transfer function defined in various forms. The characteristics of sinusoidal vibration are placed in the context both of the real world and of laboratory tests, and transient and steady-state response of the single-degree-of-freedom system. First viscous damping and than non-linear damping is considered. The various types of swept sine and their properties are described and, for the one degree-of-freedom system, the consequences of an inappropriate choice of sweep rate are considered. From the latter, rules governing the choice of suitable sweep rates are developed.

Table of Contents

Foreword to series	xi
Introduction	xv
List of symbols	xvii
Basic mechanics	1	(40)
Static effects/dynamic effects	1	(2)
Behaviour under dynamic load (impact)	3	(3)
Tension	5	(1)
Elements of a mechanical system	6	(26)
Mass	6	(1)
Stiffness	7	(1)
Definition	7	(1)
Equivalent spring constant	8	(3)
Stiffness of various parts	11	(3)
Non-linear stiffness	14	(1)
Damping	15	(1)
Definition	15	(2)
Hysteresis	17	(1)
Origins of damping	18	(5)
Specific damping energy	23	(1)
Viscous damping constant	24	(1)
Rheology	25	(1)
Damper combinations	26	(1)
Non-linear damping	27	(2)
Static modulus of elasticity	29	(1)
Dynamic modulus of elasticity	30	(2)
Mathematical models	32	(9)
Mechanical systems	32	(1)
Lumped parameter systems	33	(1)
Degrees of freedom	34	(1)
Mode	35	(2)
Linear systems	37	(1)
Linear one-degree-of-freedom mechanical systems	37	(1)
Setting an equation for n-degrees-of-freedom lumped parameter mechanical system	38	(3)
Response of a linear single-degree-of-freedom mechanical system to an arbitrary excitation	41	(48)
Definitions and notation	41	(2)
Excitation defined by force versus time	43	(4)
Excitation defined by acceleration	47	(2)
Reduced form	49	(4)
Excitation defined by a force on a mass or by an acceleration of support	49	(1)
Excitation defined by velocity or displacement imposed on support	50	(3)
Solution of the differential equation of movement	53	(10)
Methods	53	(1)
Relative response	53	(1)
General expression for response	53	(2)
Subcritical damping	55	(1)
Critical damping	56	(1)
Supercritical damping	56	(1)
Absolute response	57	(1)
General expression for response	57	(1)
Subcritical damping	58	(1)
Critical damping	59	(1)
Supercritical damping	60	(2)
Summary of main results	62	(1)
Natural oscillations of a linear single-degree-of-freedom system	63	(26)
Damped aperiodic mode	64	(4)
Critical aperiodic mode	68	(3)
Damped oscillatory mode	71	(1)
Free response	71	(4)
Points of contact of the response with its envelope	75	(1)
Reduction of amplitude: logarithmic decrement	75	(8)
Number of cycles for a given reduction in amplitude	83	(2)
Influence of damping on period	85	(1)
Particular case of zero damping	86	(2)
Quality factor	88	(1)
Impulse and step responses	89	(42)
Response of a mass-spring system to a unit step function (step or indicial response)	89	(13)
Response defined by relative displacement	89	(1)
Expression for response	89	(4)
Extremum for response	93	(2)
First excursion of response to unit value	95	(2)
Response defined by absolute displacement, velocity or acceleration	97	(1)
Expression for response	97	(1)
Extremum for response	98	(3)
First passage of the response by the unit value	101	(1)
Response of a mass-spring system to a unit impulse excitation	102	(10)
Response defined by relative displacement	102	(1)
Expression for response	102	(4)
Extremum for response	106	(2)
Response defined by absolute parameter	108	(1)
Expression for response	108	(2)
Peaks of response	110	(2)
Use of step and impulse responses	112	(7)
Transfer function of a linear one-degree-of-freedom system	119	(11)
Definition	119	(2)
Calculation of H(h) for relative response	121	(1)
Calculation of H(h) for absolute response	122	(3)
Other definitions of transfer function	125	(1)
Notation	125	(1)
Relative response	125	(1)
Absolute response	126	(1)
Summary tables	126	(4)
Measurement of transfer function	130	(1)
Sinusoidal vibration	131	(12)
Definitions	131	(8)
Sinusoidal vibration	131	(1)
Mean value	132	(1)
Mean square value -- rms value	133	(2)
Periodic excitation	135	(3)
Quasi periodic signals	138	(1)
Periodic and sinusoidal vibrations in the real environment	139	(1)
Sinusoidal vibration tests	139	(4)
Response of a linear single-degree-of-freedom mechanical system to a sinusoidal excitation	143	(54)
General equations of motion	144	(11)
Relative response	144	(3)
Absolute response	147	(2)
Summary	149	(1)
Discussion	150	(2)
Response to periodic excitation	152	(1)
Application to calculation for vehicle suspension response	153	(2)
Transient response	155	(4)
Relative response	155	(4)
Absolute response	159	(1)
Steady state response	159	(2)
Relative response	159	(1)
Absolute response	160	(1)
Responses	161	(14)
Variations of velocity amplitude	162	(1)
Quality factor	162	(2)
Hysteresis loop	164	(1)
Energy dissipated during a cycle	165	(3)
Half-power points	168	(2)
Bandwidth	170	(4)
Variations in velocity phase	174	(1)
Responses	175	(14)
Variation in response amplitude	175	(1)
Dynamic amplification factor	175	(4)
Width of H(h) for HRD = HRDmax/√2	179	(1)
Variations in phase	180	(9)
Responses, y/xm, y/xm, y/xm and FT/Fm	189	(5)
Movement transmissibility	189	(1)
Variations in amplitude	190	(2)
Variations in phase	192	(2)
Graphical representation of transfer functions	194	(3)
Non-viscous damping	197	(30)
Damping observed in real structures	197	(1)
Linearization of non-linear hysteresis loops -- equivalent viscous damping	198	(4)
Main types of damping	202	(8)
Damping force proportional to the power b of the relative velocity	202	(1)
Constant damping force	203	(2)
Damping force proportional to the square of velocity	205	(1)
Damping force proportional to the square of displacement	206	(1)
Structural or hysteretic damping	207	(1)
Combination of several types of damping	208	(1)
Validity of simplification by equivalent viscous damping	209	(1)
Measurement of damping of a system	210	(14)
Measurement of amplification factor at resonance	210	(2)
Bandwidth or √2 method	212	(1)
Decreased rate method (logarithmic decrement)	213	(7)
Evaluation of energy dissipation under permanent sinusoidal vibration	220	(4)
Other methods	224	(1)
Non-linear stiffness	224	(3)
Swept sine	227	(24)
Definitions	227	(2)
Swept sine	227	(1)
Octave -- number of octaves in a frequency interval (f1, f2)	228	(1)
Decade	228	(1)
`Swept sine' vibration in the real environment	229	(1)
`Swept sine' vibration in tests	229	(2)
Origin and properties of main types of sweepings	231	(20)
The problem	231	(3)
Case no1: sweep where time Δt spent in each interval Δf is constant for all values of f0	234	(11)
Case no2: sweep with constant rate	245	(1)
Case no3: sweep ensuring a number of identical cycles ΔN in all intervals Δf (delimited by the half-power points) for all values of f0	246	(5)
Response of a one-degree-of-freedom linear system to a swept sine vibration	251	(30)
Influence of sweep rate	251	(1)
Response of a linear one-degree-of-freedom system to a swept sine excitation	252	(17)
Methods used for obtaining response	252	(1)
Convolution integral (or Duhamel's integral)	253	(2)
Response of a linear one-degree-of-freedom system to a linear swept sine excitation	255	(10)
Response of a linear one-degree-of-freedom system to a logarithmic swept sine	265	(4)
Choice of duration of swept sine test	269	(1)
Choice of amplitude	270	(1)
Choice of sweep mode	271	(10)
Appendix. Laplace transformations	281	(14)
A.1 Definition	281	(1)
A.2 Properties	282	(3)
A.2.1 Linearity	282	(1)
A.2.2. Shifting theorem (or time displacement theorem)	282	(1)
A.2.3. Complex translation	283	(1)
A.2.4. Laplace transform of the derivative of f(t) with respect to time	283	(1)
A.2.5. Derivative in the p domain	283	(1)
A.2.6. Laplace transform of the integral of a function f(t) with respect to time	284	(1)
A.2.7. Integral of the transform F(p)	284	(1)
A.2.8. Scaling theorem	284	(1)
A.2.9. Theorem of damping or rule of attenuation	285	(1)
A.3 Application of Laplace transformation to the resolution of linear differential equations	285	(2)
A.4 Calculation of inverse transform: Mellin-Fourier integral or Bromwich transform	287	(2)
A.5 Laplace transform	289	(3)
A.6 Generalized impedance -- the transfer function	292	(3)
Vibration tests: a brief historical background	295	(2)
Bibliography	297	(12)
Index	309