Multiple Sinusoidal Vibration Test Development for Engine Mounted Components

TECHNICAL ARTICLE—PEER-REVIEWED

Multiple Sinusoidal Vibration Test Development for EngineMounted Components

Liang Wang • Yung-Li Lee • Robert Burger •

Keyu Li

Submitted: 20 October 2012 / Published online: 5 February 2013

� ASM International 2013

Abstract Durability testing is required before vehicle

launching to prevent failure before designed life. Limited

publications were found on specifying vibration testing to

validate durability and reliability of engine mounted com-

ponents. These published test methods did not consider

engine firing order effects and oversimplified the vibration

profiles. In this paper, a new ordered multiple sinusoidal

vibration test method is proposed to improve the existing

procedures. The test method is designed to fulfill an infinite

life durability requirement for engine mounted components

subjected to a four-stroke internal combustion engine vibra-

tion. An innovative test development procedure, based on

engine vibration field data, is illustrated in details in this

paper. The ordered multiple sinusoidal vibration test method

covers the choice of sweep type, sweep time, frequency range,

vibration magnitude and profile, and test duration. Instead of

obtaining the vibration magnitude directly from order anal-

ysis results, vibration magnitudes are determined by fully

analyzing the vibration data in both time domain and

frequency domain. Test profiles designed by enveloping

method are proposed for a better represent of the engine

excitations.

Keywords Durability analysis � Failure prevention �Sinusoidal vibration test � Engine excitation � Development

Introduction

Before been launched to the market, a new vehicle model is

required for passing all reliability and durability testing to

prevent failure before designed life. The testing is usually

conducted in three level sequences such as component

level, system level, and whole vehicle level.

A vibration test such as a sinusoidal sweep test or a

random vibration test is considered the most effective test

to validate durability of a component with the concern of

resonant fatigue. Compared with the random vibration test,

the sinusoidal sweep test is preferable for use in a product

subjected to periodic loading. Also, higher excitation lev-

els, as well as convenience of test setup and monitoring

make the sinusoidal vibration test to be the most popular

laboratory durability test [1–3].

Sinusoidal vibration test has been proposed for studying the

durability and reliability for several components [4–10], how-

ever rarely for engine mounted components. The International

Organization for Standardization (ISO) [11] introduces two

general sinusoidal vibration test profiles for studying the reli-

ability and durability of electrical and electric equipments

which experiences engine excitations. Unfortunately, the

standard fails to explain the procedures and criteria for devel-

oping the vibration profiles and other test parameters.

Moreover, engine mounted components will experience varied

engine excitations depends on the engine type and mounting

location. The presented vibration profiles in the standard may

L. Wang (&) � K. Li

Department of Mechanical Engineering, Oakland University,

Rochester, MI 48309, USA

e-mail: [email protected]

K. Li


Y.-L. Lee � R. Burger

Stress Laboratory & Durability Development, Chrysler Group

LLC, Auburn Hills, MI 48326, USA

Y.-L. Lee


R. Burger


123

J Fail. Anal. and Preven. (2013) 13:227–240

DOI 10.1007/s11668-013-9661-x

not be representative for the actual vibration of the component.

Therefore, documentation with more detailed procedures and

explanations are needed for test engineers to produce their own

vibration profiles based on their test objectives’ field vibration

input. In this paper, a test method with detail procedures for

designing the sinusoidal vibration test is presented. The test

method is designed to fulfill an infinite life durability require-

ment for engine mounted components due to a four-stroke

internal combustion engine excitation. Besides the vibration

magnitudes and profiles, test parameters such as the choice of

sweep type, sweep time, frequency range, and test duration are

also discussed in the test method.

Besides the ISO standard, all proposed sinusoidal vibration

profiles for engine mounted components are single curve

where vibration amplitude sweeps from low frequency to high

frequency. Only one literature is found specifying a multiple

frequency sweep sinusoidal test for engine mounted compo-

nents [12]. However, the multiple sine vibration profiles are

specified by dividing the traditional single sine vibration

profile into several frequency intervals, and then run these

intervals simultaneously to reduce the test duration. This

algorithm mainly relies on the single sine vibration profile and

shaker controls for running the multiple intervals. In addition,

all proposed methodologies did not consider the engine firing

order effect. The order effect is the most distinguished char-

acter for vibrations in rotating machines such as engine. In this

paper, a new ordered multiple sinusoidal vibration test method

is proposed. The test method includes multiple sine vibration

profiles, and each sine vibration profile is determined corre-

sponding to one order of engine vibration. Instead of running

single sine vibration profile used in literatures [4–12], the

multiple sine vibration profiles representing dominant engine

vibration orders run simultaneously during the durability test.

In addition, the acceleration or displacement in the pre-

sented vibration profiles in literatures [4–12] was determined

by enveloping or averaging the frequency spectrum of the

vibration field data. This potentially leads to a non-conser-

vative test because it inherently underestimates the

magnitudes due to fast Fourier transform (FFT). In this paper,

the vibration magnitudes are determined from the time series

field data. The detailed methodology which analyzes the

vibration data in both time domain and the frequency domain

is presented. Finally, innovative vibration profiles based on

the time series vibration magnitudes are proposed for each

engine vibration order following detail procedures.

Multiple Sinusoidal Test Design Procedures

Engine Excitation and Data Processing

Engine mounted components experience mainly the engine

excitations, which are periodic in nature. Considering the

resonant fatigue, sinusoidal vibration test is designed for

studying the durability and reliability of the components.

This paper demonstrates the procedures of an improved test

method, which is designed to fulfill an infinite life durability

requirement for engine mounted components subjected to a

four-stroke internal combustion engine excitation.

To collect the field engine excitations experienced by

the engine mounted components, an engine dynamometer

test was conducted running from idle speed to the maxi-

mum engine speed on the four-stroke four-cylinder engine.

Engine block and cylinder head are the general two places

where accelerometers are installed during the dynamome-

ter test. The vibration measured at engine block indicates

the generic engine excitations. However, for some engine

mounted components, such as throttle body and exhaust

gas recirculation valves which are often cantilevered, the

engine excitations are usually measured at cylinder head.

In addition, installing of the accelerometer is critical when

engine excitation for specific engine mounted component is

required. The Z-axis of the accelerometer should be

installed parallel to the dominant vibration direction of the

component. For demonstrating purpose, an engine excita-

tion measured at engine block is used in this paper. The

engine speed and engine excitations in vertical, lateral and

longitudinal directions versus time are shown in Fig. 1.

Similar to other rotating machines, engine vibration data

shown in Fig. 1 includes higher order frequency contents.

These high-order frequency contents contribute little to

components’ fatigue damage because of the tiny vibration

magnitude. A Butterworth low pass filter is used to remove

them. Different criteria could be used for deciding the fil-

tering frequency. It could be selected as the high end of

components’ vibration frequency range. For example,

automotive vibrations are general low frequency vibra-

tions, whose frequency range is much lower than 500 Hz.

Thus a filtering frequency of 500 Hz is commonly used.

However, considering the fatigue damage due to the

vibration, an empirical criterion is utilized in this paper,

which assumes negligible fatigue damage would be

induced when the peak-to-peak displacement is lower than

0.005 inch.

For a sinusoidal vibration, the filtering frequency (f) is

correlated with the peak-to-peak displacement (Dp–p) and

the peak acceleration (Amax) as shown in Eq 1 considering

the units (peak acceleration in g and peak-to-peak dis-

placement in inch),

Dp�pðinÞ ¼AMax ðgÞ � 32:174 ft=s2 � 12

2p2f 2¼ AMax ðgÞ

0:0511f 2

ðEq 1Þ

As shown in Fig. 1, the peak accelerations (absolute

value) are 23.13, 18.94, and 15.48 g in the vertical, lateral

228 J Fail. Anal. and Preven. (2013) 13:227–240

123

and longitudinal directions, respectively. Thus, for filtering

vibrations whose peak-to-peak displacement is lower than

0.005 inch, the corresponding filter frequencies are 301,

272, and 246 Hz in the three directions.

The filter method includes forward only method and

forward and backward method. The forward only method

can introduce phase lag in the output signal, while the

forward and backward method can eliminate the phase lag.

Therefore, the forward and backward filter method is

selected. To further smooth the signal, a five-order filter is

chosen. Using the five-order forward and backward low

pass filter, the engine block acceleration signals are filtered

shown in Fig. 2.

Determination of Vibration Order and Magnitude

Vibration from rotating machines such as engine contains

order information, which is defined as the normalization of

the rotational speed. The fundamental vibration order (first

order) and corresponding frequency can be obtained

directly from the engine speed. The harmonic orders and

frequencies are integer multiples of the fundamental order

and frequency.

For the engine excitations shown in Fig. 1, the funda-

mental frequency ranges from 27 to 108 Hz, which belongs

to the first-order engine vibration, since the engine speeds

ranging from 1,599 revolutions per minute (rpm) to

6,456 rpm. The second-order harmonic therefore ranges

from 54 to 216 Hz.

Besides the frequency range, vibration magnitudes for

each vibration order should be determined for designing

the vibration profiles. Theoretically, there are infinite

vibration orders. However, high-order vibrations are nor-

mally found to be non-consequential in terms of fatigue

damage due to tiny amplitudes. Therefore, only vibration

magnitudes for dominant orders are considered for

designing the reliability test.

Waterfall analysis is conducted on the filtered time

series vibration data for investigating the vibration orders,

from which the dominant orders are obtained. For waterfall

analysis, parameters such as buffer size, buffer overlap,

window type, averaging method etc., should be carefully

selected. In this paper, buffer size of 4,096, 67% overlap,

Hanning window and peak hold averaging method are

used. Readers can select their own parameters based on

their data type and analysis resolution.

Fig. 1 Engine speed (a) and

engine vibrations in (b) vertical,

(c) lateral, and (d) longitudinal

directions

J Fail. Anal. and Preven. (2013) 13:227–240 229

123

The waterfall analysis results in the vertical, lateral and

longitudinal directions are shown in Fig. 3a–c, respectively.

As shown, the second-order vibration is the most dominant

vibration. In the vertical direction, the dominant vibration

occurs at the high speed portion. While the dominant

vibrations in the lateral and longitudinal directions happen at

the lower and middle speed region. For the four-cylinder,

four-stroke engine, each cylinder is fired once for every two

revolutions of the crankshaft. Therefore, there will be two

firing pulses per engine rotation, which cause the dominant

vibration to occur at the second-order frequency range. The

gas pressure, which is the main engine excitation source, acts

in the vertical direction. Thus, vibrations in the vertical

direction are much higher than the other two directions. The

faster of the engine speed leads to more firing, which induces

higher vibrations. Therefore, dominant vibration is occurred

at the high speed region in the vertical direction as shown in

Fig. 3a. In the other two directions, since they do not directly

experience the firing pressure, vibrations measured in these

two directions do not increase as the engine speed is

Fig. 2 Low pass filtered

accelerations: (a) vertical

(b) lateral (c) longitudinal


123

increasing. They mainly represent the resonant vibration for

the engine mounted components at frequency ranges shown

in Fig. 3b, c.

It is observed that, besides the second-order vibration,

the first-order vibration is also prevalent in the longitudinal

and lateral directions. Vibration magnitudes for other

orders are low and therefore the peak displacements due to

these vibrations are small. Fatigue damages due to these

orders may be negligible. For demonstrating purpose, only

the first- and second-order vibrations are used for designing

the reliability test.

After obtaining the dominant vibration orders, the next step

is to finding the amplitude and frequency information for

designing the vibration profiles. The order analysis, which

extrudes the vibration magnitudes and corresponding engine

speed at each order from the waterfall analysis, is performed.

Order analysis results are shown in Fig. 4a, b, c for the ver-

tical, lateral, and longitudinal directions, respectively.

The selection of the amplitude is demonstrated by picking

the highest amplitude in the second-order vibrations. As

shown in Fig. 4, the highest amplitudes are 12.10, 5.30, and

2.95 g at around engine speed 5,703, 2,087, and 3,543 rpm in

the vertical, lateral, and longitudinal directions, respectively.

Thus, the corresponding frequencies are 190, 70, and

118 Hz, respectively. However, the magnitudes are inher-

ently inaccurate because both waterfall analysis and order

analysis are based on FFT, which always has spectral leak-

age. The spectral leakage, which is result of processing finite

duration records, causes a bias in the amplitude of a harmonic

order [13]. Two options are available for minimize the

problem. The real vibration magnitudes could be estimated

by applying a correction factor on the magnitudes from FFT

analysis. However, the correction factor is critical and

mainly relies on experience. In this paper, another option

which selects the amplitude from the time series vibration

data is used. As shown in Fig. 4, the amplitudes for the

second-order are much higher than those for other orders.

Therefore, the amplitude in the time series can be treated as

the second-order peak magnitude. Also, higher amplitude

makes the reliability test more conservative. Three steps are

followed for determining the magnitude from time series.

From the order analysis shown in Fig. 4, the engine speeds

corresponding to the highest amplitudes are observed. The

time for the engine speed is then determined based on the

linear relationship between engine speed and time shown in

Fig. 1a. The peak amplitude is finally selected at the time

determined at step 2 from the filtered time series vibration

data shown in Fig. 2. Following the steps, the peak magni-

tudes selected from the time series are 14.91, 6.81, and

4.98 g in the vertical, lateral, and longitudinal directions,

respectively.

To obtain the vibration magnitudes and frequencies for

the first-order vibration, a band pass Butterworth filter is

applied to filter the time series vibration data from 27 to

108 Hz. Figure 5 shows the filtered vibration data in the

three directions. An obvious resonant vibration, which is

also a peak, is observed before 60 s, when the engine speed

is below 2,000 rpm. However, whether the resonant

vibration belongs to the first-order vibration is still a con-

cern. To verify this concern, an order analysis is performed

and the results are shown in Fig. 6. As shown, both the

first-order and second-order vibrations are observed in the

frequency range, and the vibration magnitude for the sec-

ond order is much higher than the first order. Thus, the

Fig. 3 Waterfall analysis results for low pass filtered vibration

signals: (a) vertical, (b) lateral, and (c) longitudinal


123

resonant vibration shown in Fig. 5 belongs to the second-

order vibration. However, the second-order vibration

mainly occurs when engine speed is lower than 4,000 rpm,

while the first-order vibration happens between 4,000 rpm

to the fuel cut off speed. Thus, the first-order vibration

between 60 to 135 s shown in Fig. 5 is the range of interest

to get highest magnitudes and corresponding frequencies

for the first-order vibration. Similar to the procedures for

the second-order vibration, the peak amplitude for the first

order is picked from the time series data. The peak mag-

nitudes for the first-order vibration are 2.13, 2.09, and

2.28 g with frequencies 79, 98, and 92 Hz in the vertical,

lateral, and longitudinal directions, respectively.

The highest amplitude and corresponding frequency in

each direction for dominant orders is determined as

described above. In practical, more data points may be

required for constructing a more representative vibration

profile. The amplitudes from time series and corresponding

frequencies for other peaks can be also determined fol-

lowing procedures described above.

Design of Vibration Profiles

Theoretically, the vibration profiles from order analysis

results should be taken as the input for the reliability tests.

The continuous vibration profile is usually accepted by

computer in form of series of discrete data points. How-

ever, in general the vibration profiles as shown in Figs. 4

and 6 are so wavy that intensive data points are required to

represent the vibration profile. In practical, test engineers

would prefer limited data points (for example, ISO stan-

dard [11] uses five data points to construct the vibration

profile) for a simpler vibration profile. The more data

points used, the closer is the designed profile to the actual

vibration profile. Readers can choose an optimized number

of data points considering the work load, the complexity of

the vibration profile and the function of the shaker. In this

paper, the vibration profile design is demonstrated using

limited peaks.

Acceleration, velocity, and displacement are the three

forms of amplitude inputs for the reliability tests. For a

simpler test, they are kept constant during the test. However,

variable amplitude may be used for constructing a more

representative vibration profile. Using the constant ampli-

tudes strategy, three types of vibration profile such as

constant acceleration, constant displacement, and combina-

tion of constant acceleration and constant displacement are

generally performed in automotive reliability tests. While

with the development of the modern shaker and shaker

control technology, vibration profile with variable amplitude

is more desired. This paper demonstrates the design of all

four types of vibration profile for reader’s choice.

It is somewhat common in vibration tests to measure

acceleration first then determine velocity and displacement

by direct integration [14]. Therefore, constant acceleration

is the simplest form of the reliability test. For designing a

conservative test, the highest vibration amplitude deter-

mined from previous section is selected as the vibration

amplitude. The vibration amplitudes are 14.91, 6.81, and

4.98 g for the second-order vibration and 2.13, 2.09, and

2.28 g for the first-order vibration in the vertical, lateral,

and longitudinal directions, respectively. During the reli-

ability test, the vibration amplitudes are kept constant from

the minimum frequency to maximum frequency. Since the

Fig. 4 Order analysis results for low pass filtered vibration signals:

(a) vertical, (b) lateral, and (c) longitudinal


123

peak accelerations are directly gained from the engine

dynamometer test, and they are the extreme response, the

designed constant acceleration test would be severer than

reality.

The constant displacement test is also commonly used in

automotive reliability tests. This type of test requires

constant displacement from the minimum frequency to the

maximum frequency, while the acceleration amplitudes are

changing from the lowest to the highest. For designing a

conservative test, the displacement is determined from

Eq 1 using the highest amplitude and corresponding fre-

quency. Thus, the displacements are 0.0081, 0.0272, and

0.0067 in for the second-order vibration, and are 0.0067,

0.0043, and 0.0053 in for the first order in the vertical,

lateral, and longitudinal directions, respectively.

Other than using only constant acceleration test or

constant displacement test, vibration profile which is

combination of the two are observed in limited vibration

reports [15–17]. However, the reports fail to explain how to

construct the profiles. Combination of the constant

Fig. 5 Band pass filtered

vibration data: (a) vertical,

(b) lateral, and (c) longitudinal


123

acceleration and constant displacement helps constructing

a vibration profile which is close to the actual vibration

profile. Therefore, in this section, design of vibration pro-

files which are schedule of constant acceleration and

constant displacement are demonstrated.

The strategy for designing a profile which is combina-

tion of constant acceleration and constant displacement is

explained as follows. For vibration acceleration signal

which is relatively flat and stable, a constant acceleration

test is desired. For acceleration signal which increases with

time or frequency, the constant displacement is more

desirable. For example, for vibration profile shown in

Fig. 4a, a constant displacement test is desired since

vibration acceleration increases with frequency. For the

first-order vibration in vertical and lateral directions as

shown in the right portion of Fig. 7a, b, a constant accel-

eration test is desired because the vibration acceleration is

stable.

Different from constant acceleration test and constant

displacement test, determination of the cross frequency

(amplitude) at which the profile changes from one type to

another is required for the combined constant displacement

and constant acceleration vibration profile. To determine

the cross frequency, the field vibration profile shown in

Figs. 4 and 7 are divided into several sections, each of

which will be constructed using either constant displace-

ment or constant acceleration. Determination of the cross

frequencies for vibration profile at each direction is dem-

onstrated using limited peaks from the field profile.

Readers can design their own vibration profiles for their

vibration field data using optimal data points following

same procedures described herein. As shown in Fig. 4a, for

the second-order vibration in the vertical direction, vibra-

tion amplitude increases at one rate from idle speed to

around 2,137 rpm. While another rate is observed from

2,500 rpm to the fuel cut off engine speed. Since vibration

amplitude increases with engine speed in both sections, two

constant displacement tests with varied displacement

amplitudes are designed to represent this vibration profile.

The frequency at engine speed 2,137 rpm is selected as the

cross frequency, which is 71 Hz. The section where

vibration drops from 2,137 to 2,500 rpm is simplified by

applying a constant displacement which is back extended

from the second portion of constant displacement. The

cross amplitude is determined from time series field data

following the three steps described previously. The peak

Fig. 6 Order analysis results for band pass filtered vibration signals:

(a) vertical, (b) lateral, and (c) longitudinal

Fig. 7 The designed second-order vibration profile in the vertical

direction using schedule of constant acceleration and constant

displacement


123

acceleration is determined from Fig. 2a by finding the time

corresponding to engine speed at 2,137 rpm. Result shows

that the peak acceleration is 6.368 g. The peak-to-peak

displacement is then calculated from Eq 1 as 0.0323 in,

which is the constant displacement from 54 to 71 Hz. The

constant displacement from 71 to 216 Hz is same as the

displacement corresponding to the highest amplitude,

which is 0.0081 in. The second-order vibration profile in

the vertical direction is shown in Fig. 7. For comparison

purpose, vibration profile developed by using amplitudes

directly from order analysis result is also shown in Fig. 7 in

dash. It is clearly observed that vibration profile by using

time series amplitude is more conservative.

For the second-order vibration in the lateral direction,

the amplitude increases with engine speed until 2,087 rpm,

and tends to stabilize up to 6,400 rpm. Therefore, a con-

stant displacement test is designed for engine speed lower

than 2,087 rpm. The cross frequency is also the frequency

for the highest amplitude as shown in Fig. 4b, which is

70 Hz. The corresponding displacement, which is

0.0243 in, is therefore the constant displacement from 54

to 70 Hz. For frequency range from 70 to 216 Hz, a con-

stant acceleration test is designed since the vibration profile

is somewhat stable. The amplitude at engine speed

2,739 rpm is selected as the constant acceleration, which is

the peak amplitude between engine speed 2,087 to

6,400 rpm. From time series, the constant acceleration is

5.42 g. The second-order vibration profile in the lateral

direction is shown in Fig. 8.

For the second-order vibration in the longitudinal

direction, two constant displacement sections (1,599–

1,836 rpm and 2,740–3,543 rpm) are designed to represent

the two increase trend shown in Fig. 4c, and two constant

acceleration tests (3,543 to 5,301 rpm and 5,301 to

6,400 rpm) are used for the stable trend. The cross fre-

quencies for vibration profile in the longitudinal directions

are 61, 118, and 177 Hz. The peak accelerations picked

from time series corresponding to these cross frequencies

are 3.46, 4.98, and 2.82 g, respectively. Thus, the constant

displacement for the first portion is 0.0181 in, and

0.0070 in for the second portion of constant displacement.

The constant acceleration for the third portion is 3.33 g,

which is the peak between 3,543 and 5,301 rpm. For the

last portion, the peak acceleration is 2.82 g, which is the

peak time series vibration amplitude between 5,301 and

6,400 rpm. Figure 9 shows the designed second-order

vibration profile in the longitudinal direction.

For the first-order vibration, constant acceleration tests

are designed in the vertical and lateral directions. A con-

stant displacement test is designed in the longitudinal

direction. The constant accelerations in the vertical and

lateral directions are 2.13 and 2.09 g. The constant dis-

placement in the longitudinal direction is 0.0053 in from 27

to 108 Hz.

With the development of the shaker and modern shaker

control technologies, reliability tests with variable ampli-

tudes can be accomplished. Compared with the other three

types, vibration profile with variable amplitudes can be

closer to the actual vibration profile. Therefore, it is more

desirable for situation when field vibration profile is so

wavy that it cannot be constructed by constant amplitudes.

For example, none of the three types can represent the

profile that vibration amplitude decreases with increase of

frequency (speed) shown in Fig. 4.

The strategy for designing the variable amplitude

vibration profile is demonstrated as follows. The actual

vibration profile from order analysis is first divided into

several sections, which are differentiated by the vibration

trend such as increase, decrease or stable. In each section, a

line which connects two peaks is made to envelop the

vibration profile. Then, following the same steps described

previously, the amplitudes and frequencies at these two

Fig. 8 The designed second order vibration profile in the lateral


displacement

Fig. 9 The designed second order vibration profile in the longitudinal


displacement


123

peaks are selected, and the equation of this linear curve is

performed for determining the amplitudes at the start and

end points of this section. For the stable vibration profile,

only peak amplitude in this section is needed and a constant

acceleration profile is constructed. Caution is needed for

determining the cross amplitude (frequency), because the

designed adjacent profiles may not have same amplitude at

the intersection. Process of this situation is demonstrated in

the following profile designs in the three directions.

In the vertical direction, the second-order vibration

profile shown in Fig. 4 has three obvious sections

(1,599–2,137, 2,137–2,639, 2,639–6,400 rpm) which are

differentiated as increase, decrease, and increase trend. In

the first section, two peaks at speed 2,086 and 2,137 rpm is

selected to build the envelop line. In the second section, the

two peaks are at speeds 2,137 and 2,486 rpm. And the

peaks are at speeds 3,092 and 5,703 rpm for the last sec-

tion. Following the three steps described previously, the

amplitudes picked from the filtered time series field data

are 8.10, 8.32, 5.33, 6.83, and 14.91 g at these speeds.

Therefore, the linear equation for each section can be

conducted, and amplitude at the start and end points of the

sections are calculated. Using the linear equation of the

first section, the amplitude at 54 Hz is calculated as 4.84 g.

However, at the intersection of section two and three (at

speed 2639 rpm), two amplitude values of 4.08 and 5.43 g

are calculated using the second and third linear equations

of the envelop curves. To make the two lines end at one

point, the line for the decrease section is lifted up so that it

still envelops the section. And the new line is connecting

(2,137 rpm, 8.32 g) with (2,639 rpm, 5.43 g). Therefore,

the vibration profile is designed connecting these points

and shown in Fig. 10. Also, for comparison purpose,

vibration profile developed by using amplitudes directly

from order analysis result is shown in Fig. 10 in dash.

Clearly, enveloping using time series amplitude is more

conservative.

In the lateral direction, the vibration profile is divided

into six sections (1,599–2,087, 2,087–2580, 2,739–3,644,

3,644–4,798, 4,698–5,703, and 5,703–6400 rpm). To avoid

the situation that the amplitudes at intersection are not

same, several simplifications are conducted. In the second

section from 2,087 to 2,580 rpm, instead of a long decrease

line enveloping this section, a shorter portion of decrease

line with same slope is used together with a constant

acceleration section, which has same amplitude as at speed

2,739 rpm. To keep the constant acceleration profile in the

fourth section from 3,644 to 4,698 rpm, the envelop curves

for the third and fifth section are made by connecting the

starting and ending points of each section. However, the

amplitudes for the end point of the third section and the

start point of the fifth section are selected as same as the

amplitude in the fourth section, which is 3.35 g. By doing

this, the problem that varied amplitudes are calculated at

the intersections is solved. Using the amplitudes taken from

the filtered time series data, the vibration profile for sec-

ond-order vibration in the lateral direction is shown in

Fig. 11.

In the longitudinal direction, same strategy is applied for

determine the sections and cross frequencies. However,

different from the other two directions, the second-order

vibration is in same level of magnitude as the first order at

the high frequency portion. Therefore, amplitudes picked

directly from time series are not dominant in second order

and cannot simply treat them as the second-order vibration

amplitudes. In this paper, we assume that the vibration

contents of the orders are same in both time series and FFT

results. For example, if there are only two orders and the

amplitudes of the orders are same in the FFT results, we

assume that the amplitude for each order is half of the

amplitude in the time series field vibration data. Thus, for

the second-order vibration, the amplitude at speed

2,740 rpm is quarter of 3.115 g picked from time series,

and the amplitude at speed 4,950 rpm is two thirds of

4.992 g selected from time series. The vibration profile for

the second-order vibration in the longitudinal direction is

shown in Fig. 12.

Determination of Test Time Duration

After defining the vibration profiles, determination of how

long the reliability tests should be run is demonstrated.

Factors affecting test time duration for sinusoidal vibration

reliability test are frequency range, test bogey, sweep time,

and sweep type.

The frequency range is defined as the range which

includes all possible resonant frequencies for initiating the

resonant fatigue of the reliability test components.Fig. 10 The designed second-order vibration profile in the vertical

direction using variable amplitudes


123

Difference is observed for defining the frequency range in

traditional single curve and ordered multiple sinusoidal

vibration tests. The frequency range is defined as the

minimum frequency to the maximum frequency for tradi-

tional single curve vibration profile. For multiple orders

sinusoidal vibration tests, all the vibration orders run

simultaneously. Thus, the frequency range can be either 27

to 108 Hz or 54 to 216 Hz in this case.

Test bogey is defined as the number of cycles that the

objective component should be run during a reliability test.

In this paper, an infinite life test bogey is designed for the

engine mounted components. The number of cycles rep-

resenting infinite life, which corresponds to the fatigue

limit, depends on test objective’s material and manufac-

turing process. For example, one million cycles test usually

represent the infinite life for steel components, while ten

million cycles are needed for products made from alumi-

num material. For welded joint, five million cycles

represent the infinite life. For demonstrating purpose, a two

million cycles’ reliability test is designed in this paper.

Sweep time per sweep is a critical parameter for the

reliability test. It should not be so long that the physical

effects of the sequence between the larger amplitude and

the small amplitude are not produced. Also, it should not be

so short that the structure cannot reach the steady state

response [18]. The determination of the sweep time is

detailed explained in [18]. In this paper, a 30-min sweep

time is used for an example.

There are general two types of sweep method, the log-

arithmic sweep and the linear sweep. For the logarithmic

sweep, the number of cycles for each frequency interval is:

ni ¼Df

AðEq 2Þ

While for the linear sweep, the number of cycles for

each frequency interval is:

ni ¼Df

A0f ðEq 3Þ

where A ¼ ln fMaxð Þ � ln fMinð Þð Þ=T , A0 ¼ ðfMax � fMinÞ=T ,

Df is the frequency resolution, and T is the duration for

each sweep.

According to Eqs (2) and (3), there are same number of

cycles for given frequency increment using logarithmic

sweep, while more number of cycles are expected at higher

frequency using linear sweep.

Considering logarithmic sweep, the correlations among

the four factors affecting test duration are quantified as

shown in Eq 4:

TD ¼ ST� TBln fMaxð Þ � ln fMinð Þ

Df � T; ðEq 4Þ

Where TD is the test duration, ST is the sweep time, and

TB is the test bogey.

Knowing the frequency resolution Df in Eq 4, the test

duration is determined. For exciting all potential resonant

vibration, the frequency resolution for the sinusoidal

vibration is selected as the half power bandwidth of the

resonance, which is defined as the ratio of resonance fre-

quency and dynamic amplification factor. And the dynamic

amplification factor can be determined by the damping

coefficient. Selection of the damping coefficient is critical

and it depends on the application and experience. For

a system which is consisted of structural components,

a 2–3% damping coefficient value could be used. While for

a system with electronic devices such as the engine

mounted components, a value of 5% is commonly accep-

ted. For demonstrating purpose, the damping coefficient

value of 5% is used. Thus, a value of 10 is used for the

dynamic amplification factor.

The component’s resonant frequency is unknown, thus

the half power bandwidth can range from fMin=10 to

fMax=10. To make sure that the structure is excited, the

Fig. 11 The designed second-order vibration profile in the lateral

direction using variable amplitudes

Fig. 12 The designed second-order vibration profile in the longitu-

dinal direction using variable amplitudes


123

minimum half power bandwidth is used. It is should be

mentioned that a different number of cycles would be

obtained if the first-order frequency range is input in Eq 4.

Since all vibration orders run simultaneously in the ordered

multiple sinusoidal test, only the frequency range for the

dominant vibration order is used to determine the number

of sweeps. Taking the parameters into (4), test duration for

the reliability test in each direction is 174 h.

Design of Accelerated Test

The test duration determined directly from previous section

in generally is so long that it is not economic. In practice,

accelerated vibration tests are designed and conducted. The

principle for accelerating the vibration test is the fatigue

damage equivalence technique which assumes same fati-

gue damage induced for originally designed test and

accelerated test. Factors for designing the accelerated test

are discussed in this section.

Assuming fatigue of the component follows the S–N

curve which relates stress magnitude with number of cycle

to failure. Thus, a scale factor defined as the ratio of the

stress magnitudes between the original designed test and

the accelerated test can be expressed in terms of the fatigue

strength exponent, b, as

Scale ¼ Saccelerated

Soriginal

¼ Naccelerated

Noriginal

� �b

ðEq 5Þ

Therefore, as shown in Eq 5, the reliability test can be

accelerated by scaling the load. At the same time, the

fatigue strength exponent will also contribute to the

acceleration of the test duration.

Assuming a single degree system, the fatigue damage is

proportional to number of cycles and stress amplitude [19].

Thus, stress magnitude should be exaggerated to reduce the

test time duration. However, the load scale should be

carefully selected for various reasons. First, the failure

mode change due to the exaggerated stress magnitude

should be avoided [20, 21]. For a specimen designed for

infinite life, the load in most of the service life is relatively

low. The accelerated test, however, increases these loads to

higher levels, which would cause failure differently from

real applications. Also, the vibration amplitudes such as the

peak-to-peak displacement and peak acceleration for the

accelerated test should not be scaled high enough to go

beyond the shaker limit. Thus, load scaling is generally

determined from experiences. For engine mounted com-

ponents, some companies recommend 94 h test time

duration for determining the scale.

The fatigue strength exponent is a fatigue property of a

material. Instead of a specific value, the exponent for a

material can be a range due to different manufacturing

processes. For example, fatigue strength exponent b ranges

from �0.195 to �0.043 for steel products from different

manufacturing processes [22]. On the other hand, the fatigue

strength exponent is failure mode dependent. For example, a

�0.17 to �0.2 value of fatigue strength exponent is usually

referred to metal fatigue failure. While a �0.33 value of

fatigue strength exponent indicates that the crack growth

behavior is dominant and the typical examples are fatigue of

welded joints, spot weld joints, and soldier joints.

Figure 13 shows how fast the reliability test is acceler-

ated using different load scale and fatigue strength

exponent. As shown, the reliability test is accelerated by

increasing either load scale or fatigue strength exponent, or

both of them. For example, using fatigue strength exponent

b = �0.25 and the load scale of 1.2, the designed test is

accelerated from 174 to 84 h.

Assuming a linear relationship between the stress mag-

nitude and displacement, the scaling of the vibration profile

will be same in value as scaling of the stress since the

displacement is also linearly related with the acceleration.

Thus, for a load scale factor of 1.2, the accelerated sinu-

soidal vibration test profiles should be magnified by a

factor 1.2 for each order in each direction. All the vibration

profiles for the accelerated tests are same except for the

magnitudes. It should be mentioned that for the swept sine

vibrations, it is important to check that the reduction of

time does not lead to too fast a sweep, which as a conse-

quence would have the effect of the response to a

resonance to a value below the steady state response.

Discussions

Following same strategy, the traditional single curve

sinusoidal vibration test where vibration amplitude sweeps

from low frequency to high frequency can be developed.

Fig. 13 Reduced test time duration percentage as a function of scale

and b


123

Instead of conducting the waterfall analysis and order

analysis, the FFT analysis is performed on the filtered time

series vibration data to obtain the vibration profile in the

frequency domain. Then, enveloping method is used to

design the profiles following same procedures described

previously. Unfortunately, the time series amplitude cor-

responding to the peaks in the FFT analysis result cannot

be obtained. Therefore, traditional one curve sinusoidal

vibration test profile can only be gained by enveloping the

FFT analysis results, which will be inherently underesti-

mate the amplitudes. Figure 14 shows the FFT analysis

results for the low pass filtered (filtering frequency

500 Hz) engine excitations. As shown, the highest

amplitude after FFT analysis in the vertical direction is

12.45 g, while the highest amplitude in the time series is

16.73 g. This is also observed when designing the vibra-

tion profiles using time series amplitudes and order

analysis results as shown in Figs. 7–12. Comparing

vibration profiles designed by enveloping time series

amplitudes and order results using variable amplitudes,

the average amplitude differences are 1.89, 1.81, and

1.19 g in the vertical, lateral, and longitudinal direction,

respectively, as shown in Figs. 10–12. Therefore, scaling

must be applied to the vibration profiles made by envel-

oping the FFT results.

There are general two options for verifying a proposed

reliability test method. The first method is to compare the

fatigue damage in the field with that induced at the reli-

ability tests when failure occurs. For example, if the failure

mode after the reliability test is same as that from the field,

we can conclude that the designed vibration test is reliable.

Unfortunately, this verification method cannot be applied

in this method, because infinite life duration is designed for

the engine mounted components. After running for two

million cycles, there is still no failure occurs on the com-

ponents. Therefore, comparing the failure is not possible.

At the same time, there is difference between the reliability

test and the actual vibrations experienced by the compo-

nents. During the reliability test, vibrations in vertical,

lateral and longitudinal directions are running separately,

while they are running simultaneously in the field. There-

fore, different failure mode could be also induced from this

difference even with a reliable reliability test. In this paper,

the second option which compares vibration input between

the field and designed tests is used. When the vibration

input for the reliability test is comparable with the actual

vibration input in the field, the designed vibration test is

considered a reliable test. As described in previous sec-

tions, the multiple sinusoidal vibration test profiles are

designed following exactly the vibration profiles experi-

enced by the engine mounted components. Also, the

vibration amplitudes are closer to the actual vibration

magnitudes for the designed test compared with by FFT

analysis. Therefore, the proposed reliability test in this

paper is reliable.

Before physically conducting the reliability test, another

stage of reliability development process, reliability dem-

onstration testing, is required after the reliability

development testing described in this paper. The reliability

demonstration testing is referred to the test execution plan

to demonstrate that the majority of the tested products

could at least meet the accelerated test bogy with a specific

reliability and confidence target by testing a limited sample

size of the product. Detailed information for developing the

reliability demonstration test can be found in [5].

Conclusion

Detailed procedures for developing multiple sinusoidal

vibration tests for reliability and durability are demon-

strated in this paper. The test method is designed for engine

mounted component fulfilling infinite life duration design

criterion. Also, the proposed test method considers the

engine firing order effect, and the dominant vibration

orders are constructed by the multiple sinusoidal orders

which run simultaneously during the reliability test.

Moreover, design of innovative vibration profiles are

demonstrated by four types of vibration profiles. The

vibration profile using variable amplitude is most close to

the actual vibration profile and can be applied for wavy

vibration profiles. The other three types are used for simple

vibration profiles. Finally, amplitudes for designing the

vibration profiles are picked from filtered time series field

data instead of from order analysis results, which inher-

ently underestimate the vibration amplitude. Following

procedures described in the paper, readers are able to

design a conservative and reliable multiple sinusoidal

vibration test.Fig. 14 FFT analysis results for the low pass filtered engine

excitations


123

Acknowledgments The authors acknowledge the product and

technical support from HBM-nCode’s. L. Wang acknowledges the

scholarship from China Scholarship Council.

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Multiple Sinusoidal Vibration Test Development for Engine Mounted Components