8.4 An Industrial Vibration Design Problem

Back

In the analysis of vibration characteristics of an engine or an engine-generator set, flexibly supported

rigid-body representations are commonly used. In the case study given here, an engine assembly

system is considered. In designing an engine mounting system, engineers consider a number of

issues. Particularly useful are: static deflection; natural frequencies and spread of these frequencies

relative to the engine speed range; time response under shock or other transient loads; and

frequency response of the system, especially when subjected to unbalanced engine-generated

excitations.

8.4.1 Static Deflection

This problem involves the selection of mounting geometry and mount stiffness parameters. Here, the

design engineer seeks to achieve a pure deflection with as little tilt as possible. In principle, there is

nothing wrong with some mounts deflecting more than the others, but an excessive tilt normally

indicates other problems such as a high degree of coupling between the modes of motion. However,

achieving a pure vertical deflection, at least mathematically, is relatively easy and involves the selection of

spring positions/stiffness values to satisfy the following conditions:

n-sXpring

p¼1

ypkpz ¼ 0

and

n-sXpring

p¼1

xpkpz ¼ 0:

Dominant Oscillation

Direction

Frequencies

Frequency in X 1.01 Hz (60 CPM)

Frequency in Y 1.42 Hz (85 CPM)

Frequency in Z 1.74 Hz (105 CPM)

Frequency in alpha 34.87 Hz (2092 CPM)

Frequency in beta 12.33 Hz (740 CPM)

Frequency in gamma 14.24 Hz (854 CPM)

Computer Analysis of Flexibly Supported Multibody Systems 8-11

© 2005 by Taylor & Francis Group, LLC

Here, xp and yp are spring mount positions and kpz is the spring stiffness in the z direction. In addition,

the designer needs to ensure that the static deflection is well within the allowable deflection range,

especially since it needs to account for any additional deflection due to vibration.

8.4.2 Natural Frequencies

Normally, design application demands that the natural frequencies are kept away from the running

speed of an engine. Such a requirement is easy to satisfy for single-DoF systems. However, in real-life

situations, the number of DoF(s) is six just for a single rigid body. Any change to the mounting

configuration or stiffness parameters (or mass distribution) will affect all six natural frequencies. In

order to be able to modify a natural frequency corresponding to a particular mode shape (e.g.,

resonance in the vertical direction), a vibration design engineer will attempt to decouple various modes

of oscillation. Such a design requirement, however, is unrealistic for many engineering problems, even

for a single rigid body, due to space considerations and is practically impossible for multiple rigid body

systems. However, it is possible to achieve partial decoupling. For example, the condition given above

for pure vertical deflection will also provide decoupling between the rocking motion about twocoordinate

systems (about Ox and Oy) in the horizontal plane and vertical motion.

8.4.3 Transient Response Analysis

Static deflection analysis and modal analysis are normally essential (see Chapter 3). However, there are

two more problem-specific analyses that a design engineer may need to perform depending on the

problem. For example, in many engineering applications, the response of a flexibly supported system to a

shock loading is very important. In this case, coupling among modes as well as stiffness of the system play

an important role in determining the levels of shock transmission to the engine system. As a rule of

thumb, the softer the spring the smaller the shock transferred to the flexibly mounted structure. Making

mounting stiffness elements softer may not be a realistic option (see Chapter 32) as this can result in

unacceptable static deflections and may even be in conflict with the requirements discussed above in

relation to the positioning of a natural frequency relative to the operating speed of the engine and the

static deflection.

8.4.4 Frequency Analysis

Frequency analysis is probably one of the most useful among the various types of analysis listed above. If

the vibration problem is not transient, then it is a problem involving a steady-state vibration. The analysis

of a steady-state vibration problem under sinusoidal excitation is normally referred to as frequency

analysis (see Chapter 2). In mathematical terms, this is the particular solution of the differential equation

of motion. Under a sinusoidal excitation, a vibrating system reaches a steady-state vibration and

frequency analysis provides information of the amplitude of vibration as a function of excitation

frequency. If there is more than one excitation force, then the resulting motion will be a combination of

the results due to each excitation. Multiexcitation force analysis is sometimes referred to as harmonics

analysis (see Chapter 2). Here, the designer needs to ensure that the highest amplitude of oscillation

under harmonic excitation does not exceed the safe limits for the system and, in particular, for the

mounts.

Although design considerations linked to static deflection, positioning of natural frequencies,

decoupling of modes, and response to shock represent a large proportion of vibration design problems,

there are other and more complex design specifications. For example, minimizing vibration at a point on

a flexibly supported body may be considered a design objective. Such a requirement may then cause

problems where the engineer intends to place a drive shaft coupling or an additional mounting at this

position. Because the vibration (or deflection) is at its minimum at the assembly point, a coupling or

additional mount will add a minimum constraint to the system.

8-12 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Obviously, all the design objectives discussed above have to be satisfied within the physical constraints

relating to the problem at hand. In general, these constraints are associated with space and the stiffness

range of industrially available mountings.

8.4.5 A Flexibly Supported Engine — A Numerical Problem

The example considered here involves a study of an engine-mounting configuration. Mounting

configurations are restricted by the geometry and little flexibility exists in modifying these positions.

They are located by the engine manufacturer. To perform vibration analysis, the mass/moments of

inertia values of the system, the coordinates of the mounting positions relative to the COG, and the

forces acting on the system are needed. Although the mass is normally given or easy to obtain,

moments of inertia and the COG are not always supplied by engine manufacturers. Even though it is

possible to build a solid model of an engine in order to calculate moments of inertia, this is a rather

tedious and costly task. The alternative is to define engine moments of inertia in an approximate

manner. This can be done by assuming that the main assemblies of the engine are made of regular

geometrical primitives representing approximate shapes without going into exact geometrical models.

Such an approach works reasonably well, especially since the mass values of these primitives can be

obtained in an exact manner. When calculating the moments of inertia and the overall mass of the

assembly, its COG can also be calculated. Once these are obtained, the mount positions can be

calculated relative to the COG. Having obtained the mass, the moments of inertia, the COG, and the

mount position coordinates relative to the COG, the main step of analysis may be started. This involves

selecting the mount stiffness parameters in such a way that the various conditions and objectives

described above are met. The vibration design, like all engineering problems, involves reconciling many

conflicting requirements.

Engine mass and moments of inertia (symmetry of mass distribution is assumed)

m ¼ 250 kg and moments of inertia Ixx ¼ 45 kg m2; Iyy ¼ 80 kg m2; and Izz ¼ 110 kg m2:

Mounts stiffness values:

ðkx ¼ 150;000 N=m; ky ¼ 150;000 N=m; kz ¼ 300;000 N=mÞ:

Mount positions (all in mm):

8.4.5.1 Satisfying Static Deflection

On running a static analysis under a vertical load of 2500 N (weight), the following results are obtained:

(displacements are in mm and angles are in rad)

In relation to the static deflection, there are two considerations: (i) overall deflection should not be

more than what is allowed by the deflection range of the springs selected for the design, and (ii) the static

position and orientation of the engine should not be outside what is allowed by spatial and other

constraints; i.e., it should not tilt to one side excessively. In either case, stiffer springs will tend to solve

1 0.000 100.000 2 75.000

2 250.000 0.000 2 75.000

3 0.000 2 100.000 2 75.000

4 2 170.000 0.000 2 75.000

X Y Z Alpha Beta Gamma xc yc zc

0.1392 0.0000 2.1205 0.0000 0.0019 0.0000 1142.5000 0.0000 2 75.0000

Computer Analysis of Flexibly Supported Multibody Systems 8-13

© 2005 by Taylor & Francis Group, LLC

the problem. However, such a choice may not be the best for transients, shocks, and vibration

transmission from the supporting frame. “Tilt” level calculated above is assumed to be small

(0.0019 rad).

The results from the program also list the deflection at the mount positions (in mm) as: mass no. ¼ 1

Mount no. Position Deflection

X Y Z x y z

1 0.000 100.000 2 75.000 0.000 0.000 2.120

2 250.000 0.000 2 75.000 0.000 0.000 1.656

3 0.000 2 100.000 2 75.000 0.000 0.000 2.120

4 2 170.000 0.000 2 75.000 0.000 0.000 2.436

The maximum deflection is at the fourth mount position and is 2.436 mm. The mount selected should

give a deflection range that would extend well beyond this to ensure that any additional deflection due to

vibration could be accommodated.

The coordinates xc, yc, zc give the instantaneous center of rotation for this particular static deflection

result. This point (as discussed above) may be used in some design applications as it remains stationary

during the deflection.

8.4.5.2 Eigenvalue Analysis

The eigenvalue analysis (see Chapter 3 and Appendix 3A) will help ensure that the natural frequencies are

not in the vicinity of the idling speed of the engine. It may also help to minimize the number of natural

frequencies in the speed range of the engine.

Since the spring positions and their locations are already specified to satisfy the considerations for

static deflection, it becomes difficult to modify them to satisfy the “natural frequency” requirements as

well. However, all stiffness values could be increased together in the same proportion. This ensures that

the “no tilt” condition is maintained. Of course, stiffening now reduces the static deflection and is likely

to increase the vibration transmission to the frame.

The eigenvalue analysis results are listed below. Here, the natural frequencies spread from 1.98 to

12.09 Hz. The widest gap between these frequencies is between 3.17 and 8.69 Hz. It would be desirable to

have the idling speed in the middle of this range. It is equally important that the cruising speed does not

coincide with the two higher frequencies.

X Y Z Alpha Beta Gamma

Frequency in X ¼ 8.63 Hz (518 CPM)

1.0000 0.0000 2 0.0111 0.0000 2 0.2709 0.0000

Frequency in Y ¼ 8.69 Hz (521 CPM)

0.0000 1.0000 0.0000 0.4403 0.0000 0.0484

Frequency in Z ¼ 12.09 Hz (725 CPM)

2 0.0051 0.0000 2 1.0000 0.0000 0.0685 0.0000

Frequency in alpha ¼ 1.98 Hz (119 CPM)

0.0000 0.0797 0.0000 2 1.0000 0.0000 2 0.0226

Frequency in beta ¼ 3.17 Hz (190 CPM)

0.0869 0.0000 0.0215 0.0000 1.0000 0.0000

Frequency in gamma ¼ 2.13 Hz (128 CPM)

0.0000 2 0.0163 0.0000 2 0.0625 0.0000 1.0000

In order to achieve decoupling between different motions, one practical technique is to minimize

the distance between the COG and the “center of stiffness.” Center of stiffness is a crude term

used in industry to ensure that the coupling between different motions of body is minimized.

The definition of center of stiffness is similar to that of the COG. The Ox axis is located in such a

8-14 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

way that

P

p ypkpz P ¼ 0 holds. Then this axis will pass through the center of stiffness. Similarly,

p xpkpz ¼ 0 will hold for the Oy axis passing through the center of stiffness. If the center of

stiffness coincides with the COG and the axes defined by the first moment of stiffness coincide with

the principal axis of mass, then full decoupling can be achieved. As far as the horizontal plane is

concerned, this also ensures that the assembly is leveled. Note that the relationships are the same as

the “no tilt” condition described above for static deflection. Normally, it may not be possible to

achieve this in all three planes and the designer may choose to achieve this in one plane where the

excitation forces are the greatest. However, it is common among designers to focus on the

horizontal plane alone, purely due to deflection under gravity considerations.

8.4.5.3 Time Domain Analysis — Analysis of the System under a Shock Loading

Suppose that a 10 g, 10 msec, half sine shock is applied in the vertical, z; direction. The shock response in

the z direction is shown in Figure 8.4. The shock response in the y direction is shown in Figure 8.5.

The results show that, in addition to static deflection, if the mounts were to withstand the applied

shock, they should be able to deflect 9 mm in shear and more than 6 mm in the vertical directions.

8.4.5.4 Frequency Analysis

The frequency analysis specifications are given below.

The engine is subjected to an unbalanced force which is known to be proportional to the square of

the engine running speed. In other words, this is given as Aw2: It is measured that when the engine

speed is 300 rpm the unbalanced force is 250 N. A simple calculation shows that A ¼ 1:013: The force

menu option 20 in VIBRATIO provides the required excitation, which increases with the square of

the running speed. The option 20 allows the A value to be linearly increased between the start and

end frequencies during which the excitation is active. In our case, this is to be taken to be the same

(frequency is independent of the A value). The vertical amplitude vs. frequency results are given in

Figure 8.6. The amplitudes in other directions are much smaller and are not shown here. The analysis

is not carried out beyond 15 Hz as we know already that the maximum resonance is at 12.09 Hz.

According to the result, the selected mount should allow a 16 mm deflection on top of static

deflection. Now, the designer should be in a position to make a decision on whether the selected

FIGURE 8.4 Shock response in the z direction.

Computer Analysis of Flexibly Supported Multibody Systems 8-15

© 2005 by Taylor & Francis Group, LLC

spring type is acceptable or not. If the answer is no, then the whole analysis process has to be

repeated to find an acceptable solution.

Навигация