《内燃机学》课程教学资源（书籍文献）Pounder's Marine Diesel Engines and Gas Turbines，9th ed.

15VibrationPropeller QNodePropellerFlywheel@FlywheelNodeNodeWCylindersURelative angular displacements1Elastic curveatmaximumamplitudeFIGURE1.7bTwo-nodemodeA node is found where the deflection is zero and the amplitude changessign. The more nodes that are present, the higher the corresponding naturalfrequency.The problem arises when the forcing frequencies of the externally applied,or input, vibration coincide with, or approach closely, one of these naturalfrequencies. A lower frequency risks exciting the one-node mode; a higherfrequency will possibly excite the two-node mode and so on.Unfortunately,the input frequencies or-to give them their correct namethe forcingfrequencies'are notsimple.As far as the crankshaft is concerned, the forcing frequencies are caused bythe firing impulses in the cylinders. But the firing impulse put into the crank-shaft at any loading by one cylinder firing is not a single sinusoidal frequencyat one per cycle. It is a complex waveform which has to be represented forcalculation purposes by a component at 1x cycle frequency; another, usuallylower in amplitude,at 2x cyclefrequency;anotherat 3xand so on,up to atleast 10 before the components become small enough to ignore.These com-ponents are called the first, second, third up to the tenth orders or harmonicsof the firing impulse.For four-stroke engines,whose cycle speed is half therunning speed, the convention has been adopted of basing the calculation onrunning speed.There will therefore be"half'orders as well,for example, 0.5,1.5, 2.5 and so on.Unfortunatelyfromthepointof view ofcomplexity,butfortunatelyfromthepointofviewofcontrol,thesecorrespondinggimpulseshavetobecombinedfrom all the cylinders according to the firing order. For the first order the inter-valbetweensuccessiveimpulsesisthesameasthecrankanglebetweensuc-cessivefiring impulses.For most engines,therefore,and for our six-cylinderengine in particular, the one-node first order would tend to cancel out, as shownin the vector summation in the centre of Figure 1.8.The length of each vector

Vibration 15 A node is found where the deflection is zero and the amplitude changes sign. The more nodes that are present, the higher the corresponding natural frequency. The problem arises when the forcing frequencies of the externally applied, or input, vibration coincide with, or approach closely, one of these natural frequencies. A lower frequency risks exciting the one-node mode; a higher frequency will possibly excite the two-node mode and so on. Unfortunately, the input frequencies or—to give them their correct name—the ‘forcing frequencies’ are not simple. As far as the crankshaft is concerned, the forcing frequencies are caused by the firing impulses in the cylinders. But the firing impulse put into the crankshaft at any loading by one cylinder firing is not a single sinusoidal frequency at one per cycle. It is a complex waveform which has to be represented for calculation purposes by a component at 1 cycle frequency; another, usually lower in amplitude, at 2 cycle frequency; another at 3 and so on, up to at least 10 before the components become small enough to ignore. These components are called the first, second, third up to the tenth orders or harmonics of the firing impulse. For four-stroke engines, whose cycle speed is half the running speed, the convention has been adopted of basing the calculation on running speed. There will therefore be ‘half’ orders as well, for example, 0.5, 1.5, 2.5 and so on. Unfortunately from the point of view of complexity, but fortunately from the point of view of control, these corresponding impulses have to be combined from all the cylinders according to the firing order. For the first order the interval between successive impulses is the same as the crank angle between successive firing impulses. For most engines, therefore, and for our six-cylinder engine in particular, the one-node first order would tend to cancel out, as shown in the vector summation in the centre of Figure 1.8. The length of each vector Propeller Node Flywheel Cylinders Relative angular displacements at maximum amplitude Elastic curve Node Flywheel Node Propeller No. 1 cyl No. 6 cyl Figure 1.7b Two-node mode

16 Theory and General Principles shown in the diagram is scaled from the corresponding deflection for that cylinder shown on the elastic curve such as is in Figure 1.7. On the other hand, in the case of our six-cylinder engine, for the sixth order, where the frequency is six times that of the first order (or fundamental order), to draw the vector diagram (right of Figure 1.8), all the first-order phase angles have to be multiplied by 6. Therefore, all the cylinder vectors will combine linearly and become much more damaging. If, say, the natural frequency in the one-node mode is 300 vibrations per minute (vpm) and our six-cylinder engine is run at 50 rev/min, the sixth harmonic (6  50  300) would coincide with the one-node frequency, and the engine would probably suffer major damage. Fifty revolutions per minute would be termed the ‘sixth order critical speed’ and the sixth order in this case is termed a ‘major critical’. Not only the engine could achieve this. The resistance felt by a propeller blade varies periodically with depth while it rotates in the water, and with the periodic passage of the blade tip past the stern post, or the point of closest proximity to the hull in the case of a multi-screw vessel. If a three-bladed propeller were used and its shaft run at 100rev/min, a third order of propellerexcited vibration could also risk damage to the crankshaft (or whichever part of the shaft system was most vulnerable in the one-node mode). The most significant masses in any mode of vibration are those with the greatest amplitude on the corresponding elastic curve. That is to say, changing them would have the greatest effect on frequency. The most vulnerable shaft sections are those whose combination of torque and diameter induce in them the greatest stress. The most significant shaft sections are those with the steepest change of Firing order Resultant 1st order summation scaled from Figure 1.7(a) 2–5 Enlarged 1–6 Resultant 6th order summation scaled from Figure 1.7(a) 2–5 3–4 1 4 5 2 6 3 3–4 1–6 4 2 6 3 5 1 Figure 1.8 Vector summations based on identical behaviour in all the cylinders

17Vibrationamplitude on the elastic curve and therefore the highest torque. These are usu-ally near the nodes, but this depends on the relative shaft diameter. Changing thediameter of such a section of shaft willalso have a greater effect on the frequency.The two-node mode is usually of a much higher frequency than the one-node mode in propulsion systems, and infact usually only the first two or threemodes are significant. That is to say that beyond the three-node mode, the fre-quencycomponents of the firing impulsethat could resonatein the runningspeed range will be small enough to ignore.The classification society chosen by the owners will invariably make itsown assessment of the conditions presented by the vessel's machinery, and willjudge by criteria based on experience.Designers can nowadays adjust the frequency of resonance, the forcingimpulses and the resultant stresses byadjusting shaff sizes,number of propel-ler blades, crankshaft balance weights and firing orders, as well as by usingviscous or other dampers, detuning couplings and so on.Gearing,of course,creates further complications—and possibilities. Branched systems, involvingtwin input or multiple output gearboxes,introduce complications in solvingthem; but the principles remain the same.The marine engineer needs to be aware, however, that designers tend torely on reasonably correct balance among cylinders. It is important to realisethat an engine with one cylinder cut out for any reason, or one with a seriousimbalance between cylinder loads or timings, may inadvertently be aggravat-inga summation of vectors which thedesigner, expecting it to be small, hadallowed to remain near the running speed range.If an engine were run at or near a major critical speed, it would soundrough because, at mid-stroke, the torsional oscillation of the cranks with thebiggest amplitude would cause a longitudinal vibration of the connecting rod.This would set up, in turn, a lateral vibration of the piston and hence of theentablature.Gearing.if on a shaft section with a high amplitude,would alsoprobably be distinctly noisy.The remedy, if the engine appears to be running on a torsional critical speed,would be to run at a different and quieter speed while an investigation is made.Unfortunately,noise is not always distinct enough to be relied upon as a warning.It is usuallydifficult, and sometimes impossible,to control all the possiblecriticals, so that in a variable speed propulsion engine, it is sometimes neces-sary to‘bar a range of speeds where vibration is considered too dangerous forcontinuous operation.Torsional vibrations can sometimes affect camshafts also. Linear vibra-tions usually have simpler modes, except for those which are known as axialvibrations of the crankshaft. These arise because firing causes the crankpin todeflect, and this causes the crankwebs to bend. This in turn leads to the settingup of a complex pattern of axial vibration of the journals in the main bearings.Vibration of smaller items, such as bracket-holding components,or pipe-work, can often be controlled either by using a very soft mounting whose natu-ral frequencyisbelowthat of thelowestexcitingfrequency,orby stiffening

amplitude on the elastic curve and therefore the highest torque. These are usually near the nodes, but this depends on the relative shaft diameter. Changing the diameter of such a section of shaft will also have a greater effect on the frequency. The two-node mode is usually of a much higher frequency than the onenode mode in propulsion systems, and in fact usually only the first two or three modes are significant. That is to say that beyond the three-node mode, the frequency components of the firing impulse that could resonate in the running speed range will be small enough to ignore. The classification society chosen by the owners will invariably make its own assessment of the conditions presented by the vessel’s machinery, and will judge by criteria based on experience. Designers can nowadays adjust the frequency of resonance, the forcing impulses and the resultant stresses by adjusting shaft sizes, number of propeller blades, crankshaft balance weights and firing orders, as well as by using viscous or other dampers, detuning couplings and so on. Gearing, of course, creates further complications—and possibilities. Branched systems, involving twin input or multiple output gearboxes, introduce complications in solving them; but the principles remain the same. The marine engineer needs to be aware, however, that designers tend to rely on reasonably correct balance among cylinders. It is important to realise that an engine with one cylinder cut out for any reason, or one with a serious imbalance between cylinder loads or timings, may inadvertently be aggravating a summation of vectors which the designer, expecting it to be small, had allowed to remain near the running speed range. If an engine were run at or near a major critical speed, it would sound rough because, at mid-stroke, the torsional oscillation of the cranks with the biggest amplitude would cause a longitudinal vibration of the connecting rod. This would set up, in turn, a lateral vibration of the piston and hence of the entablature. Gearing, if on a shaft section with a high amplitude, would also probably be distinctly noisy. The remedy, if the engine appears to be running on a torsional critical speed, would be to run at a different and quieter speed while an investigation is made. Unfortunately, noise is not always distinct enough to be relied upon as a warning. It is usually difficult, and sometimes impossible, to control all the possible criticals, so that in a variable speed propulsion engine, it is sometimes necessary to ‘bar’ a range of speeds where vibration is considered too dangerous for continuous operation. Torsional vibrations can sometimes affect camshafts also. Linear vibrations usually have simpler modes, except for those which are known as axial vibrations of the crankshaft. These arise because firing causes the crankpin to deflect, and this causes the crankwebs to bend. This in turn leads to the setting up of a complex pattern of axial vibration of the journals in the main bearings. Vibration of smaller items, such as bracket-holding components, or pipework, can often be controlled either by using a very soft mounting whose natural frequency is below that of the lowest exciting frequency, or by stiffening. Vibration 17

18TheoryandGeneralPrinciplesBALANCINGThe reciprocating motion of the piston in an engine cylinder creates out-of-balanceforces acting along the cylinder,whilethe centrifugal force associatedwith the crankpin rotating about the main bearing centres creates a rotatingout-of-balance force.Theseforces, if not in themselves necessarily damaging.create objectionable vibration and noise in the engine foundations, and throughthem to the ship (or building) in which the engine is operating.Balancing is a way of controlling vibrations by arranging that the overallsummation of the out-of-balance forces and couples cancels out,or is reducedtoamoreacceptableamount.The disturbing elements are in each caseforces,each of which acts in itsownplane,usually including the cylinder axis.Theessence of balancingisthat a force can be exactly replaced by aparallel forceacting in a referenceplane(chosento suitthe calculation)anda couplewhosearm isthedistanceperpendicularlybetween theplanes in which these two forces act (Figure 1.9)Inasmuch as balance is usuallyconsidered at and about convenient referenceplanes,all balancing involves a consideration offorcesand of couples.In multi-cylinder engines, couples are present because the cylinder(s) cou-pledtoeach crank throwact(s)in a differentplaneThere are two groups of forces and couples. These relate to the revolving andreciprocating masses.(This section of the book is not concermed with balanc-ing the power outputs of the cylinders. In fact, cylinder output balance, while itaffects vibration levels,has no effectonthebalancethatweareabouttodiscuss.Revolving masses are concentrated at a radius from the crankshaft, usuallyat the crankpin,butarepresumed to include a proportion of the connectingrodshank mass adjacent to the crankpin. This is done to simplify the calculations.Designers can usually obtainrotating balance quite easily by the choice ofcrank sequence and balance weights(a)FIGURE 1.9 Principle of balancing: (a) force (P) to be balanced acting at A(b)equalandoppositeforces ofmagnitude (P)assumedtoactat B,adistance ()arbitrarily chosen to suit calculation (c) system is equivalent to couple (P × I) plusforce (P)nowactingatB

18 Theory and General Principles Balancing The reciprocating motion of the piston in an engine cylinder creates out-ofbalance forces acting along the cylinder, while the centrifugal force associated with the crankpin rotating about the main bearing centres creates a rotating out-of-balance force. These forces, if not in themselves necessarily damaging, create objectionable vibration and noise in the engine foundations, and through them to the ship (or building) in which the engine is operating. Balancing is a way of controlling vibrations by arranging that the overall summation of the out-of-balance forces and couples cancels out, or is reduced to a more acceptable amount. The disturbing elements are in each case forces, each of which acts in its own plane, usually including the cylinder axis. The essence of balancing is that a force can be exactly replaced by a parallel force acting in a reference plane (chosen to suit the calculation) and a couple whose arm is the distance perpendicularly between the planes in which these two forces act (Figure 1.9). Inasmuch as balance is usually considered at and about convenient reference planes, all balancing involves a consideration of forces and of couples. In multi-cylinder engines, couples are present because the cylinder(s) coupled to each crank throw act(s) in a different plane. There are two groups of forces and couples. These relate to the revolving and reciprocating masses. (This section of the book is not concerned with balancing the power outputs of the cylinders. In fact, cylinder output balance, while it affects vibration levels, has no effect on the balance that we are about to discuss.) Revolving masses are concentrated at a radius from the crankshaft, usually at the crankpin, but are presumed to include a proportion of the connecting rod shank mass adjacent to the crankpin. This is done to simplify the calculations. Designers can usually obtain rotating balance quite easily by the choice of crank sequence and balance weights. P P P P P A B A B A P P (a) (b) (c) Figure 1.9 Principle of balancing: (a) force (P) to be balanced acting at A (b) equal and opposite forces of magnitude (P) assumed to act at B, a distance (I) arbitrarily chosen to suit calculation (c) system is equivalent to couple (P  I) plus force (P) now acting at B

《内燃机学》课程教学资源（书籍文献）Pounder's Marine Diesel Engines and Gas Turbines，9th ed. - D. Woodyard（B-H, 2009）