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2015-01-2256

Some Aspects of Rigid Body Dynamics of Power trains using Dedicated Software

with respect to Noise and Vibration.

Author,co-author (Do NOT enter this information. It will be pulled from participant tab in MyTechZone) Affiliation (Do NOT enter this information. It will be pulled from participant tab in MyTechZone)

Copyright 2015 SAE International

Abstract

This paper considers important aspects of rigid body dynamics of

power trains with respect to noise and vibration (by definition a

power train (PT) term here is an engine plus transmission). Flexibility

of PTs and their ancillaries leads to unwanted levels of noise and vibration. By employing rigid body concepts we can assess the levels

of unwanted flexibility of whole PTs and their ancillaries e.g. mounting brackets. Using dedicated software based on rigid body

theory it is possible to define vibration and noise entitlement i.e. minimum vibration and noise that can theoretically be achieved.

Targets can then be to set based upon these entitlements. This can

then lead to better more robust designs to achieve higher levels of

refinement. The use of generic 3 and 4 cylinder one liter in-line PTs modes are used within the software to aid this study. These PTs can be shown to adhere more to rigid body behavior due to their compact

designs and lower (frequency) dominant orders of excitation. This

paper steps through from basic understanding of rigid PT behavior

then shows some rigid vibration and noise results for generic 3 and 4

cylinder PTs at key mount positions it then leads into leads onto how and why rigid vibration theory can help improve refinement with

accompanying graphs and schematics to clarify understanding.

Introduction

In physics, a rigid body is an idealization of a solid body in which

deformation is neglected. In other words, the distance between any

two given points of a rigid body remains constant in time regardless

of external forces exerted on it.

All rotating machinery suffers from this lack of rigidity which leads

to noise and vibration. What separates out refined machinery from

unrefined is robustness of design. A fundamental aspect of robust

design is rigidity. The opposite to this that makes a machine

unrefined is often due to unwanted flexibility.

PTs are no different, there have been many good, and poor designs installed in all manner of automobiles. This paper presents some

basic PT rigid body vibration patterns and how they can be utilized to

set help set targets. This involves the use of dedicated prediction

programs that employ simple rigid lumped mass and inertia and

excitations definitions for prediction of 6 degrees of freedom

movement. The frequency range of this study is primarily from 20+

to 200 Hz. From 0-20+ Hz there are normally 6 rigid body PT modes

that results in often quite complex coupled translational and

rotational modes. These modes are heavily influenced by elastomeric

mounts rates and geometry as much as on the mass and inertia

properties of the PT. This aspect is briefly discussed below but is

beyond the scope of this paper, and moreover not relevant to this

study.

The intention is to concentrate on rigid body behavior in this key

frequency response range 20+ to 200Hz, where flexible modes

ideally do not exist, to satisfy the rigid body ideal. Examples of 3 and

4 cylinder engines are shown where program calculations are made

over the main vibration orders for the 1000-6000 rpm PT excitation

range, which encompasses the above frequency range.

A Basic In-line Power Unit Mounting System:

Many modern in-line 4-cylinder transverse PTs of moderate power and torque in both petrol and diesel variants have adopted the so

called torque roll axis (TRA) mounting strategy. See figure 1 for

clarification.

A good power unit mounting system is one having minimum amount

of coupling between the modes, in other words, in reference to an

XYZ coordinate system, the motion of each rigid body mode is

dominantly along or about the X, Y and Z axes. While keeping the

modes decoupled from each other, the mounting system should have

maximum coupling between its roll mode and the PT TRA. This has

the effect of decoupling or isolating the PT torque variation from

other rigid body modes [1].

The objective is to align the power train mount elastic roll axis (ERA)

to align as close as practically possible with the TRA. Other methods

involve aligning the ERA to the principal axis (PA) [2].

The level of torque de-coupling determines the level of isolation

achieved for passenger comfort during vehicle idling e.g. low seat

rail/steering wheel periodic vibration and transient shake during start

up and shut down of the PT.

Page 2 of 10

Figure 1 Schematic of Power Train on its Mounts.

Power Unit Low Frequency Modal Characteristics

Analysis of the PT shown in figure 1 concerns itself only with rigid

body behavior (grounded on its mounts). This involves 6 degrees of

freedom. It is reasonable to treat a 'real' power train in this way in the

0-20+ Hz region since it will behave as a rigid system producing only

6 natural modes.

Modal - Mass Control Regions

Figure 2 below illustrates the useful concept of modal and rigid non

modal mass controlled regions.

Some three cylinder excitation data is included to aid understanding.

Firstly it is important to note how the lower frequency 1st order

inertia moment extends into the 0-20 Hz modal region for the power

unit, thereby potentially exciting rotational modes. The excitation

frequency range is border line in this respect for say 800 rpm idle

speed (this equates to 20Hz @ 1.5EO).

It can be seen that convergence occurs at c20Hz for the differing

displacement transfer functions (TF) of a given position in space for

two different mounting systems on a given PT. Beyond this

frequency the power unit mass and inertia and excitation level only

determines the response shown here in displacement domain, which

is essentially a flat line for the rigid body condition. Its this flat line response that proves to be a useful concept for assessing higher

frequency response, and the presence or otherwise of unwanted

higher frequency modes from 20+ to 200Hz the main region of

interest.

Rigid Body Vibration Response Program EVRA

Computer modeling CAE is now used extensively to predict power

train vibration and noise behavior. Many of these models involve

enormous computations involving thousands degrees of freedom.

These models however take many man hours to build and are often

more than is needed, certainly for carrying out simple concept

studies. These programs also require other quite complex excitation

definition programs to provide inputs to carry out predicted

vibrations. To counter this, a dedicated program was developed

called EVRA (Engine Vibration Rigid Analysis) [3]. This is a self

contained predictive program utilizing rigid body theory, and in a

free-free state (see appendix A). It is capable of predicting outcomes from 2 to 6 cylinder in-line engines (see also appendix B

for typical program outputs).

Figures 3-4 show output examples of rigid PT translational vibration

velocity for a 1 liter in-line 4 cylinder PT at the RH and LH engine

mounting bracket tips (all analysis based on data contained in

appendix C). It can be seen how the Z response as expected increases

in dominance with increased rpm for both mount positions. It also

shows an increased almost double vertical response levels for the RH

(timing end) mount at mount position1, this is the well known percussion effect (see Appendix D for explanation).

Figure 5 shows the RH mount engine vibration order breakdown, as

expected it is dominated by second order vibration.

TRA/ERA

PA

CRA

GBOX

Where:TRA = Torque Roll AxisERA = Elastic Roll AxisCRA = Crank AxisPA = Principal AxisM1- M3 are typical mount positions.

M2

M1

M3

Figure 2 Modal coupling chart 3 Cylinder engine

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

900.0

1000.0

1.0 100.0 10000.0 Hz

Nm

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

1.5 EO

3 E0

Idle 1.5EO

Idle 3EO

Mxx_1E0

Mxx_2EO

TF1

TF2 modal region

Rigid mass controlled region

3 cyl excitation incursion into modal

region by 1EO moment & idle torque?

20 Hz

3 cyl excitation limit based on c800rpm

for different mount rates/geometry

Excitation orders

3 cylinder example

Flexibility starts

0

5

10

15

20

25

1000 2000 3000 4000 5000 6000 7000

Resp

on

se u

nit

s

rpm

Fig. 3 - RMS Response_RH Mount Posn 1

X mm/s

Y mm/s

Z mm/s

Mtg_total

0

2

4

6

8

10

12

1000 2000 3000 4000 5000 6000 7000

Resp

on

se u

nit

s

rpm

Fig. 4 - RMS Response_LH Mount Posn 2

X mm/s

Y mm/s

Z mm/s

Mtg_total

Page 3 of 10

Figures 6-7 shows a quite different picture for the much more

complex similar sized 3 cylinder in-line PT. Higher levels are seen

due to the lower order higher torque recoil excitation, and at least two

vibration orders play a bigger part.

Rigid Body Noise Response Program ENRA

A derivative of EVRA a noise prediction program called ENRA

(Engine Noise Rigid Analysis) [4] was later developed. Its hybrid in

nature since it takes theoretical rigid vibration output from EVRA

and combines it with elastomeric mount rates and NTFs derived from actual testing. The program also allows target NTFs that can also be incorporated to give theoretical minimum noise entitlement see next section. Figures 8-10 show some typical outputs for the 4

cylinder PT featured. Figures 9-10 shows the breakdown for RH

mount 1, showing clear dominance by Z direction, with a second

engine order dominance (Fig. 10) as expected. (Note: ENRA also

allows mount stiffening with frequency, 0.25N/mm/Hz was used

here).

Figures 11-13 show ENRA outputs for the 3 cylinder power train

featured. Figures 12-13 shows the breakdown for RH mount 1,

clearly dominated again by Z direction but with much more complex

order contribution (Fig. 13). (Mount stiffening of 0.25N/mm/Hz also

used).

0

5

10

15

20

25

1000 2000 3000 4000 5000 6000 7000

Resp

on

se u

nit

s

rpm

Fig. 5 - RMS Response_RH Mount Posn 1

Z mm/s

0.5

1

1.5

2

2.5

3

3.5

4

0

5

10

15

20

25

30

35

40

45

1000 2000 3000 4000 5000 6000 7000

Resp

on

se u

nit

s

rpm

Fig. 6 - RMS Response_RH Mount Posn 1

X mm/s

Y mm/s

Z mm/s

Mtg_total

0

10

20

30

40

50

60

70

1000 2000 3000 4000 5000 6000 7000

dB

(A)

rpm

Fig. 8 - Noise Response_Totals

Mtg1_total

Mtg2_total

Mtg3_total

Total

0

10

20

30

40

50

60

70

1000 2000 3000 4000 5000 6000 7000

dB

(A)

rpm

Fig. 9 - Noise Response_RH Mount Posn 1

X1

Y1

Z1

SUMM1

0

10

20

30

40

50

60

70

1000 2000 3000 4000 5000 6000 7000

Response un its

rpm

Fig. 10 Noise Response_RH Mount Posn 1

Z1

0.5

1

1.5

2

2.5

3

3.5

4

0

5

10

15

20

25

30

35

1000 2000 3000 4000 5000 6000 7000

Response un its

rpm

Fig. 7 - RMS Response_RH Mount Posn 1

Z mm/s

0.5

1

1.5

2

2.5

3

3.5

4

Page 4 of 10

Rigid Body Vibration Entitlement

Programs like EVRA rely on the assumption of rigidity of power trains up to a certain frequency, and the better the designs the higher

the frequency that this assumption can be made. However in reality

most real designs have flexibility e.g. 4x4 long PTs have inevitable bending often well below 200 Hz. Also poor mount bracket design

can add to lower frequency flexing adding further to bracket tip

vibration amplification. The usefulness of rigid body analysis is that

it gives the maximum entitlement i.e. minimal vibration and noise that can be expected in theory as predicated. See figure 14 for a

schematic idealized description of rigid versus flexural vibration for a 4 cylinder PT. It can be seen that this analysis is very useful, since

it enables target levels to be set around ideal rigid body lines.

Figure 15 shows how a typical target envelope of 3.5 dB (50%) could

be based around the rigid response for one or all the key PT vibration

directions for overall and order levels if required. This envelope

would normally account for both PT and mounting bracket tip

combined flexure as shown in Fig.14.

Rigid Body Structural Noise Entitlement

EVRA - Using Non Phased Noise Transfer Functions

(NTFs)

EVRA has a simple noise through mounts analysis option which combines elastomeric mount rates with a simple constant non phased

NTF values, for which energy summation is used for the noise paths.

Figure16 shows this minimum rigid body noise entitlement for the 4

cylinder PT at full load condition based on a 55 dB/N NTF target. An

overlay with a more sophisticated ENRA output involving a phased

set of test NTFs but adjusted to a constant 55 dB/N for comparison is included. The simplified noise estimate does still provide a good albeit smoothed comparable amplitude envelope useful at a concept

stage when NTFs are not available.

0

10

20

30

40

50

60

70

1000 2000 3000 4000 5000 6000 7000

dB

(A)

rpm

Fig. 11 - Noise Response_Totals

Mtg1_total

Mtg2_total

Mtg3_total

Total

0

10

20

30

40

50

60

70

1000 2000 3000 4000 5000 6000 7000

dB

(A)

rpm

Fig. 12 - Noise Response_Mount Posn 1

X1

Y1

Z1

SUMM1

Fig 14. Schematic of 4 Cylinder Inline Power Train showing

typical rigid and flexural vibration patterns

c

Rigid body entitlement

brkt flexures

Power train flexures

cog

Fa

x

0

5

10

15

20

25

30

35

1000 2000 3000 4000 5000 6000 7000

Resp

on

se u

nit

s

rpm

Fig. 15 - RMS RHS Response_Mount Posn 1

Z mm/s

Target

Envelope

0

10

20

30

40

50

60

1000 2000 3000 4000 5000 6000 7000

Response un its

rpm

Fig 13. - Noise Response_Mount Posn 1

Z1

0.5

1

1.5

2

2.5

3

3.5

4

Page 5 of 10

Figure 17 extends this EVRA noise analysis to include a 50%

increased flexing and an added further 3 dB NTF increase. These

could be typically used as simple target envelopes at design concept

stages.

ENRA - Using Phased Noise Transfer Functions

(NTFs)

Figure 18 shows an ENRA noise result for the 4 cylinder PT overlaid

with a simulated flexure increasing from 0 to 50% from mid speed to

high rpm. The NTFs used ranged from 50-58 dB/N which are considered in the lowest range, so for the rigid noise plot shown is the

minimum that could be achieved. For further interest another overlay

was made for a set of NTFs for the main Z directions increased by 3 dB. In summary we can expect easily up to 5-7 dB less noise for well

designed PTs and mounting brackets combined with a robust acoustic body performance.

Conclusions

1. In order to design and develop robust PTs a good understanding of rigid body behavior is essential.

2. Rigid body entitlement is a good concept to help design more robust power trains and mounted ancillaries.

3. The programs EVRA and ENRA utilize rigid body theory in their predictions to understand what vibration and noise

entitlement we can expect.

4. These programs help clarify what to expect for differing PTs in terms of idealized vibration levels, order dominance etc, using the minimum of information.

5. The programs can also separate out vibration, mount forces and NTF levels to ascertain where improvements can be

gained.

6. Equivalent test date can be readily compared to target graphs produced to assess vibration and noise shortfalls.

7. Using EVRA and ENRA can also be very useful for carrying out initial concept studies on mounted PTs e.g. deciding best mount positions and understanding dominant

noise peaks and paths, without the need for highly detailed

and complex full system models.

8. These programs can also aid more detailed CAE modeling and test route tracking to assess against the theoretical rigid

predictions provided, to determine how much entitlement is

actually achieved.

References

[1] Statistical Analysis of Rigid Body Modes of Engine Mounting

System Due to Mount Rates Variability - Mohammad Moetakef and

Bruce Bonhard SAE paper 2006-01-3466

[2] Power train mounting Design Principles to Achieve Optimum

Vibration IsolationJ. Shane Sui, Clarence Hoppe and John Hirshey. SAE paper 2003-01-1476.

[3] EVRA (Engine Vibration Rigid Analysis) Rigid body 3 dimensional Fortran program that computes x/y/z

vibration/force/noise for up to 4 Spatial points (engine mount tips)

using fundamental mass/inertia/geometry/cylinder gas pressures. The

code originally written for mainframes was known as EVA was

developed for Austin Rover, Antony Spillane, Colin Troth, and Derek Gardner.

20

25

30

35

40

45

50

55

60

65

70

1000 2000 3000 4000 5000 6000

dB

(A)

rpm

Fig. 17 - Structural Noise Envelope Target Based on simple non Phased NTF

Rigid 55dB/N

flex vibn 55dB/N

Flex vibn 58dB/N

20 25 30 35 40 45 50 55 60 65 70

1000 2000 3000 4000 5000 6000

dB (A)

rpm

Fig. 18 - Structural Noise Envelope Target

Rigid vibn std ntf flex vibn std ntf flex vibn +3dB NTF

20 25 30 35 40 45 50 55 60 65 70

1000 2000 3000 4000 5000 6000

dB

(A)

rpm

Fig. 16 - Structural Noise Phased vs. non Phased NTF Target

Rigid 55dB/N Rigid 55dB/N phased

Page 6 of 10

[4] ENRA (Engine Noise Rigid Analysis) A compliment Fortran program to EVRA it uses EVRA front end and then computes

structural noise through engine mounts using noise transfer functions

(NTFs) and mount stiffness data. The code originally for mainframes, was known as ETA was developed for Austin Rover

Group, Antony Spillane, Colin Troth, and Elizabeth Bright.

Both the above programs were re-written by the author to perform on

desktops and interface with Microsoft Excel for graphic outputs.

Page 7 of 10

Appendix A Rigid Body Theory employed via EVRA Program:

Program calculates all inherent shaking forces and moments in the time domain, all are then referred to PT centre of

gravity (cog) taking into account any moments produced via COG force offsets.

(The PT is further considered to be in a free-free state. It could be construed as being hung up on very low stiffness slings)

6 DOF matrices are then employed and inverted to produce all six accelerations of the PT COG.

Any PT point in 3d space is then calculated using rigid body transformation equations.

(NB: The results for EVRA have been 100% correlated to commercial software rigid body results e.g. Nastran)

Page 8 of 10

Appendix B EVRA typical output snapshots:

Force Time history snapshot

Typical RMS vibration outputs

Excitation Spectrum snapshot

0

2

4

6

8

10

12

14

16

1000 2000 3000 4000 5000 6000 7000

Resp

on

se u

nit

s

rpm

RMS Response_Mount Posn 2

X m/s^2

Y m/s^2

Z m/s^2

Mtg_total

0

5

10

15

20

25

1000 2000 3000 4000 5000 6000 7000

Resp

on

se u

nit

s

rpm

RMS Response_Mount Posn 3

X m/s^2

Y m/s^2

Z m/s^2

Mtg_total

0

5

10

15

20

25

30

1000 2000 3000 4000 5000 6000 7000

Resp

on

se u

nit

s

rpm

RMS Response_Mount Posn 1

X m/s^2

Y m/s^2

Z m/s^2

Mtg_total

0

5

10

15

20

25

30

1000 2000 3000 4000 5000 6000 7000

Resp

on

se u

nit

s

rpm

RMS Response_Totals

mtg1_total

mtg2_total

mtg3_total

Mtgs_total

0

20

40

60

80

100

120

0 100 200 300 400 500

Nm

Hz

Moments 1- 4 EO's - 1/2 Peak

Mx_1.0

My_1.0

Mz_1.0

Mx_2.0

My_2.0

Mz_2.0

Mx_3.0

My_3.0

Mz_3.0

Mx_4.0

My_4.0

Mz_4.0

CA Fx0 0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

10 0

11 0

12 0

13 0

14 0

15 0

16 0

17 0

18 0

19 0

20 0

21 0

22 0

23 0

24 0

25 0

26 0

-400

-300

-200

-100

0

100

0 200 400 600

Nm

CA Deg

Moments Time History @ COG:

Mx My Mz 1000.rpm

Moments Time History @ COG:

-5000

0

5000

0 100 200 300 400 500 600 700

N

CA deg

Shaking Force Fz Time History @ COG:

1000.rpm

2000.rpm

3000.rpm

4000.rpm

5000.rpm

6000 rpm

0

10

20

30

40

50

60

70

80

90

100

0 200 400

1/2

Peak N

m

Hz

Moments Spectrum @ COG:

Mx My Mz 1000.rpm

Moments Spectrum @ COG:

Import Data

Clear Data

Page 9 of 10

Appendix C:

Key data used in study

Engine type: 1 litre 3 Cylinder In-line 1 litre 4 Cylinder In-line

Reciprocating mass: 0.45 0.42

Engine mass: 145.0 kg 150.0 kg

Engine inertias Ixx...Iyy...Izz: 8.0,-1.3,-0.4,5.0,1.25,7.0 kgm^2 8.0,-1.3,-0.4,5.0,1.25,7.0 kgm^2

Crank radius: 39.72 mm 30.625 mm

Con rod length: 140.0 mm 148.0 mm

Piston bore diameter: 73.0 mm 72.0 mm

Cylinder centre: 80.0 mm 92 mm

Cylinder pressure range used

1000/3000/6000rpm

68/96/75 bar adjusted for 3 cylinder to achieve similar DC torque power

output.

3 point Torque axis mtg system 4 point mass centred mtg system

X-DIST Y-DIST Z-DIST mm CRNK CENTRE 0.0 150.0 0.0 ENG COG 0.0 100.0 120.0 MOUNT 1 -50.0 300.0 250.0 MOUNT 2 50.0 -180.0 140.0 MOUNT 3 200.0 100.0 -150.0 RFOB ref axis system used

RFOB: Rear Face of (engine) Block

X-DIST Y-DIST Z-DIST mm CRNK CENTRE 0.0 150.0 0.0 ENG COG 0.0 100.0 120.0 MOUNT 1 -50.0 300.0 250.0 MOUNT 2 50.0 -180.0 140.0 MOUNT 3 200.0 100.0 250.0 MOUNT 4 0.0 100.0 -200.0 RFOB ref axis system used

3 point Mount rates (10% damping used) 4 point Mount rates

X-dirn Y-dirn Z-dirn % N/mm

MOUNT 1 100.0 240.0 300.0 10.0 MOUNT 2 150.0 150.0 300.0 10.0 MOUNT 3 120.0 10.0 10.0 10.0

X-dirn Y-dirn Z-dirn % N/mm

MOUNT 1 25.0 50.0 50.0 10.0 MOUNT 2 25.0 50.0 50.0 10.0 MOUNT 3 150.0 10.0 10.0 10.0 MOUNT 4 120.0 120.0 400.0 10.0

Page 10 of 10

Appendix D

Centre of Percussion Example of in line 4 Cylinder Power Unit

Zr F Zo Zl 6000 rpm

TOTAL ROTN TRANS 169 kg

Zr = 98 35.4 62.6 Ixx=11.34

Z0 = 62.6 0 62.6

Zl = 33 -29.7 62.6

6000 rpm

TOTAL ROTN TRANS 191 kg

Zr = 97.1 41.7 55.4 Ixx=14.07

Z0 = 55.4 0 55.4

Zl = 20.4 -35 55.4

+

=

Y=77mm

y=113mmYc

Yr

cog

Yl

Ixx

0

10

20

30

40

50

60

Zr = Z0 = Zl =Translation m/s^2

-45

-30

-15

0

15

30

45

Zr = Z0 = Zl =

Rotation m/s^2

Series1

Linear (Series1)

0

10

20

30

40

50

60

70

80

90

100

Zr = Z0 = Zl =Total m/s^2

Series1

Linear (Series1)

1.6 Engine + Transmission