Dynamic Systems and Controls

06 Aug Dynamic Systems and Controls

It is desired to abate the transmitted vibration and shock of a diesel generator installed in the engine room

of a large boat, from the machine to the hull and from the hull to the machine. This should be done without

making the diesel generator experience excessive motion. The

machine runs at 1500 rpm.

In a four stroke engine each cylinder fires

every other revolution, therefore the

Approximate

1

the diesel generator as a large, lumped mass

lowest frequency of vibration occurs at

1/2 the engine RPM often called the 1/2

(2000 Kg, resembling a large 6 cylinder Diesel engine and

a 177 KVA generator mated to it) attached to the floor via

four spring-damper combinations resembling elastomeric

(rubber like) mounts. In your one degree of freedom

(dof) approximation, consider only one mount carrying

¼ of the mass of the diesel-generator.

order vibration.

The higher order

harmonics of the1/2 order vibration, 1

order, 1-1/2 order, 2 order, … etc. will

also be present in the vibration

signature. Generator imbalance also

contributes to the 1 order vibration.

1. Size the mount (find its stiffness) if the resonant

frequency of the engine+mount system is not to exceed 50% of the lowest excitation frequency of

the engine; use damping ratio of 5% (typical for natural rubber used in most mounts) to size the

damping coefficient.

2. Construct the model of the engine+mount 1 dof approximation.

3. Considering the combustion force as the input to the engine, plot the frequency response function

(FRF) of the engine (mass) displacement as well as the transmitted force from the engine to the

floor.

4. Make the FRFs2 of the systems equipped with mounts having damping ratios of 5%, 2.5% (less

than nominal damping of 5%), and 10% (more than nominal damping of 5%) damping. Plot the

magnitude of the FRFs on the same coordinates.

Contrary to vibration perturbation that is

a. Would you use a mount with higher or

viewed as a sustained, repetitive forcing

lower damping ratio (same stiffness as

input that makes the structure to vibrate at

before), if you like to lower the

the forcing frequency, shock perturbation is

transmitted vibration to the hull?

classified as a transient, abrupt, occasional

b. Would you use a mount with higher or

lower damping ratio (same stiffness as

before), if you like to lower the

transmitted shock from the hull to the

Diesel generator?

input that makes the structure to exhibit

transient (decaying) vibration at its natural

frequency. Shock is normally defined by

a pulse, with a half-sine, amongst others,

defining the pulse shape. For example, the

Diesel generator running at a constant rpm

vibrates at the frequencies corresponding to

the harmonics of the running rpm. On the

other hand when the engine is turned on/off

or a wave hits the boat, the isolated

5. Repeat the experiment of part 4, but this time use

a mount with the stiffness of K/2, K, and K*2

where K is the nominal stiffness calculated in

step 1; use the same damping ratio of 5% for all

three scenarios.

machine vibrates at its natural frequency.

6. As discussed in Appendix A, shock isolation

requirement in terms of damping and stiffness conflicts with the vibration isolation requirement.

Propose a feedback control scheme to address the conflict?

2

Of force transmissibility, motion response, and relative transmissibility as presented in Appendix A

Appendix A

Vibration and Shock Isolation

Vibration and Shock isolation systems lower the transmission of vibration and shock between

two interconnected objects. Such systems are commonly realized by placing a set of resilient

elements such as elastomeric (rubber), steel, or air springs between the two objects isolated

from each other (e.g., a piece of equipment and its support structure/base).

In addition to load-supporting (resilience), an isolation scheme has energy dissipating

attributes. In elastomeric isolators, made of natural or synthetic rubber, the load-supporting

and energy-dissipating tasks are commonly performed by a single element, i.e. the material

itself.

Perturbing force, F

If an isolator has the resilience but lacks sufficient

energy-dissipating characteristics, e.g., metal

springs; then separate energy-dissipating means

(e.g., viscous dampers) are paired with the

resilient element.

x

M

x_rel

isolator

C

K

The spring-mass-damper system of Figure 1 is

commonly used as the one degree of freedom

representation of an isolated machine/equipment.

The mass M resembles a machine or equipment

being isolated and the pair of spring K and damper

C resemble the isolator.

Ft

Base perturbation

Figure 1 Schematic of an isolated

system

VIBRATION ISOLATION

The goal of a vibration isolation system is to isolate the support structure (base) from the vibration

of the mass caused by the perturbation force F, i.e., lowering the force transmitted to the base Ft,

while avoiding excessive motion of the mass, x. In addition, the vibration isolation system is to

isolate the mass (isolated machine/equipment) from the perturbing motion of the base.

The effectiveness of a vibration isolation system intended to reduce the transmission of the

perturbation force (F) generated by the

machine/equipment to the base (Ft) is evaluated by

transmissibility. The design goal of an isolation

system is to reduce the magnitude of

Transmissibility, defined by Ft/F , is a measure

of the reduction in a) transmitted force (from

the equipment to the base), provided by an

isolator.

transmissibility, over the frequencies of interest,

without inducing too much motion into the machine/equipment, itself; in other words, reducing the

magnitudes of transmissibility Ft/F and motion response x/F (or its dimensionless

representation, x/(F/K) ).

The transmissibility transfer function mapping the perturbation force F to the transmitted force Ft,

i.e., F /F is

t

Ft

F

Bs  K

2

 Bs  K)



(Ms

The motion response transfer function mapping the force F perturbing the machine to the motion of

the machine x, i.e., x/F is.

x 

F

1

 Bs  K)

(Ms

2

The effectiveness of a vibration isolation system

used to reduce the vibratory motion transmitted

from a vibrating base (x_base) to the

machine/equipment (x_rel ) is characterized by

Relative transmissibility x_rel/x_base . Note

that x_rel is the motion of the mass/equipment

relative to the perturbing motion of the base.

Relative transmissibility, defined by

x_rel/x_base is the ratio of the relative motion

of the isolated machine/equipment with respect

to the base to the displacement of the base.

Note that the relative motion x_rel is also the

‘deflection’ of the isolator; it is a measure of the

working space required for the isolator.

Shock Isolation

Shock perturbations excite all resonances in an isolated system. Therefore a shock isolation

system must be designed to dissipate considerable amounts of energy in a minimal amount of

time which can be done by incorporating a sizeable amount of damping into the isolation system.

That is, to enhance the shock isolation effectiveness, it is desirable for the isolation system to be

heavily damped.

Shock and Vibration Isolation

Soft isolators, such as air springs, perform very well as shock isolators; this is despite the

misconception that a good shock isolation system must be mechanically “stiff”. Note that a soft

system behaves as a mechanical low pass filter abating the high frequency components of shock

inputs making a negligible amount of this high frequency energy to get to the isolated mass.

However, the problem with a low frequency device is that shock inputs tend to a) excite the

resonance of the isolator and b) being a low frequency device results in significant deflections

that might be unacceptable due to working space limitations.

Increasing damping is unfavorable to both the force and displacement transmissibility

under vibration inputs. However, large amounts of damping are required in a shock

isolation system. This is the classic trade-off in shock and vibration isolation system design.

A good compromise is to use a controllable damper that provides light damping in vibration

isolation mode, but can be turned on during a shock event to provide the large amounts of

damping necessary for controlling the effect of the shock input.