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. Approximate1 the diesel generator as a large, lumped mass (2000 Kg, resembling a large 6 cylinder Diesel engine and a 177 KVA generator) attached to the hull via four springdamper 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. 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. Plot the magnitude of the Ft/F frequency response functions2 of the system 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. a. Would you use a mount with higher or lower damping ratio (same stiffness as before), if you like to lower the transmitted vibration to the hull? 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? 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. 2 Of force transmissibility, motion response, and relative transmissibility as presented in Appendix A In a four stroke engine each cylinder fires every other revolution, therefore the lowest frequency of vibration occurs at 1/2 the engine RPM often called the 1/2 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. Contrary to vibration perturbation that is viewed as a sustained, repetitive forcing input that makes the structure to vibrate at the forcing frequency, shock perturbation is classified as a transient, abrupt, occasional 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 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? 7. With the isolation system switched to shock isolation, we need to make sure the isolators are capable of withstanding the shock loads without bottoming out. Explore the relative motion of the isolated equipment in response to a 10g ½ sine wave shock pulse hitting the boat; see Appendix A for the shape of such pulse. 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. 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. 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. 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. Base perturbation Perturbing force, F M K C x_rel isolator Ft x 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, 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., Ft/F is The motion response transfer function mapping the force F perturbing the machine to the motion of the machine x, i.e., x/F is. 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. 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.

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