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Vibration

Noise, vibration and harshness

Slide 2 of 63

AIM Introduce the basic concepts and importance of vibration theory to vehicle design Consider the role of the designer in vibration control Demonstrate methods for the control of vibration to help the elimination of noise and harshness Indicate methods by which the designer can control vibration and noise to create an equitable driving environment

Slide 3 of 63

Basic concepts Vibration sources are characterized by their time and frequency domain characteristics Categorized principally as Periodic originate from the power unit, ancillaries or transmission simplest form of periodic disturbance is harmonic In the time domain this is represented by a sinusoid and in the frequency domain by a single line spectrum

Random disturbances from terrain inputs to wheels only statistical representations are possible commonly represented by its power spectrum

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Basic concepts All mass-elastic systems have natural frequencies For linear system these frequencies are constant related only to the mass and stiffness distribution

Non-linear effects require special treatment

A few of the lower order frequencies are of interest because the higher ones are more highly damped. For one frequency, a system vibrates in a particular way, depicted by the relative amplitude and phase at various locations - mode of vibration

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Reason for vibration analysis Lightly damped structures can produce high levels of vibration from low level sources if frequency components in the disturbance are close to one of the systems natural frequencies. This means that well designed and manufactured subsystems, which produce low level disturbing forces, can still create problems when assembled on a vehicle. In order to avoid these problems, at the design stage it is necessary to model the system accurately and analyze its response to anticipated disturbances

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Approach to vibration analysis Develop a mathematical model of the system and formulate the equations of motion Analyze the free vibration characteristics (natural frequencies and modes) Analyze the forced vibration response to prescribed disturbances and Investigate methods for controlling undesirable vibration levels if they arise

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Mathematical models Provide the basis of all vibration studies at the design stage. Represent the dynamics of a system by one or more differential equations. Distributed-parameter approach - distributed mass and elasticity of some very simple components such as uniform shafts and plates by partial differential equations. not generally possible to represent typical engineering systems (which tend to be more complicated) in this way.

Lumped-parameter approach - a set of discrete mass, elastic and damping elements, resulting in one or more ordinary differential equations. Masses are concentrated at discrete points and are connected together by mass less elastic and damping elements. The number of elements used dictates the accuracy of the model To have just sufficient elements for natural vibration modes and frequencies while avoiding unnecessary computing effort.

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Mathematical models

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Formulation of System Equations of motion determined by applying Newtons second law to each free-body For complicated geometry, the equations can be formulated by energy methods

Figure from Smith,2002

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System characteristics Equation of motion mx + cx + kx = F(t ). Characteristics taken at free vibration [F(t) =0] x = X cos (nt ) no damping Similar formulation of a SDOF system to obtain response as given by graph A(w) =

Slide 11 of 63

Multi Degree of freedom*M+,x- + *K+,x- = ,0Assuming {x} = {A} est ([M]s2 + [K]){A} = {0} The non-trivial solution of these is the characteristic equation (or frequency determinant) Set of roots si2 = = 2 i is the i-th eigenvalue and i the i-th natural frequency *M]{u} = [K]{u} It is solved using standard procedures used for vibration problems

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Multi Degree of Freedom Viscous Damping In (stable) lightly damped systems the frequency determinant is | [M]s2+ [C]s + [K] | = 0 For an n-DOF system this produces n complex conjugate roots having negative real parts providing information about the frequency and damping associated with each mode of vibration Damping must be considered in the analysis of response of the system when one or more components of a periodic excitation is at or near to one of the systems natural frequencies

Slide 13 of 63

Multi Degree of Freedom Forced-damped vibration (harmonic) Many of the features of harmonic response analysis of SDOF systems extend to MDOF ones, e.g.: When subjected to harmonic excitation an MDOF system vibrates at the same frequency as the excitation. The displacement amplitudes at each of the degrees of freedom are dependent on the frequency of excitation and The dynamic displacement at each DOF lags behind the excitation

The frequency response functions (and hence the frequency responses) are complex if damping is included in the analysis.

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Multi Degree of Freedom Forced damped vibration (random excitation) Random excitation arises particularly from terrain inputs and is important in the analysis and design of vehicle suspensions. Form of excitation is non-deterministic in that its instantaneous value cannot be predicted at some time in the future. Some properties of random functions which can be described statistically. The mean or mean square value can be determined by averaging and the frequency content can be determined from methods based on the Fourier transform (Newlands, 1975).

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Vibration control Control at source Engine firing and reciprocating unbalance combine to produce a complex source of vibration which varies with engine operating conditions Reciprocating unbalance arises at each cylinder because of the fluctuating inertia force associated with the mass at each piston no such thing as perfect balance

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Vibration isolation k* = k(1 + i) where k is the dynamic stiffness and the loss factor

mx + k(1 + i)x = mer2 sin wt = F f (t )

SDOF vibration isolation model and free-body diagram Up to 1.4

Figure from Smith,2002

Slide 17 of 56

Tuned absorbers useful for reducing vibration levels in those systems in which an excitation frequency is close to or coincides with a natural frequency of the system The principles of undamped and damped tuned absorbers can be understood by outlining first the analysis of the damped absorber Undamped absorber as a special caseFigure from Smith,2002

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Tuned absorbers

For C = 0 ( undamped) [ideally k2-m2w2=0 ie w1= w2]

Figure from Smith,2002

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Tuned absorbersA1 = K *X1 / F A2 = K *X2 / F Dimensionless numbers With damping Wider operating range Reduced fatigue of absorber spring

Damped

Figure from Smith,2002

Un damped

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Untuned viscous dampers Devices consist of an inertia (seismic) mass which is coupled to the original system via some form of damping medium (usually silicone fluid)

Not tuned for particular resonance At infinite damping both masses move together as one

Models for analyzing the untuned viscous damper

Response of an untuned viscous damper (m = 1.0) Figure from Smith,2002

Slide 21 of 63

Engine Isolation Fluctuating torque at the crankshaft Shaking forces and moments additional dynamic inertial loads arising from vehicle maneuvering and terrain inputs to the wheels The primary components of engine vibration at idling are integer multiples of engine speed Idle speeds for four cylinder engines range from 820 Hz producing dominant frequency components in the range from 1640 Hz. Since the primary bending mode of passenger cars can be less than 20 Hz it is obvious that it is easy to excite body resonance at idle if engine isolation is not carefully designed

The problem isolate the chassis from the excitations and restrain the engine against excessive movement due to the engine torque

Select appropriate mounts and position them correctly

Slide 22 of 56

Formulation of the vibration equations Denoting the position of the ith mount attachment point relative to the powertrain in the equilibrium position as r ri 0 (in chassis coordinates) and the translation and rotation of the powertrain relative to the chassis axes as rG and , the deflection of the mount is given by dci = ri ri0 i is the location of the mount attachment point relative to the powertrain axes

Figure from Smith,2002

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Formulation of the vibration equations The elastic potential energy for a set of mounts typically having orthogonal complex stiffness components represented by the diagonal stiffness matrix [km]i is

For chassis

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Formulation of the vibration equationsIs a non-diagonal matrix Kinetic energy of the system is

{h}G is the momentum denoted by

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Formulation of the vibration equations [M], the symmetrical mass/inertia matrix, is of the form

Lagragnes Equations

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Mount requirements and types a low spring rate and high damping during idling and a high spring rate and low damping for high speeds, manoeuvring and when traversing rough terrains.

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Mounting types [1/4]Simple rubber engine mounts These are the least costly and least effective forms of mount and clearly do not meet