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166 Part HI. The Causes of Vibration in Ships and Their Location by Means of Pallograph Records. A pallograph record of the vibrations of any ship gives us information re- garding the natural periods of vibra- tion of the ship and the frequencies of revolution of the screws; but it is, as a rule, difficult to locate the exciting cause of the vibration, unless there hap- pens to be some simple relation between the frequency of a screw and the na- tural frequency of the ship. This will, however, rarely happen, and the: excit- ing cause can only be located by a proc-' ess of deductive reasoning from our knowledge of all the exciting causes which may be present. Every portion of an elastic structure may vibrate with a natural period of its own, if there is a local exciting impulse of about the same frequency; but these local tremors may sometimes be the exciting cause of the ship's vibration, if the periods of the two should happen to agree. A stretched string, as in a violin or pianoforte, will vibrate in unison with a tuning fork of the same frequency sounded in its neighborhood, It will also vibrate in unison with a tuning fork of double, treble, or times that frequency, the string then having 1, 2, 3 or m nodes in its length. A bar, how- ever, while it will vibrate in unison with a tuning fork having the same fre- quency as the bar, will not vibrate in unison with tuning forks having the harmonic frequencies. -The reason for this lies. im the fact. that the . fre- quencies of vibration of a string in its different modes are in the simple ratios 1, 2, 3...n, while those of a uniform free bar are approximately in the ra- tios of the squares of the odd numbers, a 5, 7) ete. The irequencies of wWibration of a structure. such as.a ship in its different modes cannot pos- sibly follow a simple rule, and cannot be calculated; but the vibrations hav- ing been excited in the ship and record- ed on the pallograph, it lies with us. to analyze all the exciting impu'ses, and to find out which, of these has a_har- monic having the same frequency as the vibration. | The idea of analyzing an impulse may be new to some. If a periodic im- pulse is due to, say, a rotating crank- shaft out of balance, it is of sine form, and has no-harmonics; but if the pe- riodic impulse is of the hammer-blow type, it is very rich in harmonics, and can, by Fourier's analysis, be analyzed into a series of sine impulses whose frequencies are in the ratios 1, 2, 3...m. Thus a periodic hammer blow is equiv- miemt to a. series of shafts out of rotating masses. THe Marine REVIEW balance revolving at frequencies 1, 2 _.n times the frequency of the hammer blow. It is evident, then, that, although the frequency of the blow may not agree with the natural frequency of the ship, the frequency of one of the harmonics may agree with it. Various Periodic Impulses. The various periodic impulses which are applied to a ship may be summar- ized as follows: (1) Inertia forces due to reciprocating masses in the main engines or auxiliaries. (2) Inertia forces due to want of balance in the (3) Longitudinal or transverse vibration of any portion of the ship's structure due to a local ex- citing cause, such as the longitudinal vibration of the mast when used for a boat hoist or derrick, the whirling of a shaft, etc. (4) The reactions on the ship corresponding to the pe- riodic pulsations of speed in the crank shaft. (5) Insufficient clearance between the blades of the screw and the ship's side or A bracket, so that the flow of water to the blade nearest the ship is restricted, and the torque thereby re- duced. The torque is then unsymmet- rically applied to the propeller, with the result that there is a reaction.on the propeller bracket or stern tube. In the 'case of a single screw, this reac- tion would be in the horizontal plane, and would tend to excite horizontal vi- brations; but in the case o1 multiple screws the reaction would generally be in a plane inclined to the vertical, and would tend to excite both horizontal and vertical vibrations. (This reac- tion, if due to horizontal "spectacle" brackets would be vertical.) (6) With multiple screws running at different speeds, the fundamentals or harmonics in the impulses under heading (5) will periodically "beat," corresponding to the beats in music when two notes, slightly out of tune, are sounded together. These beats may give rise to low-fre- quency vibrations. Sea-waves. The vibrations due to rotating or reciprocating out of balance would, as a rule, be easily recognizable by the agreement of the The impulses to which such masses give rise will be almost simple harmonic in character, and will be of the fre- quency of revolution, or double that frequency in the case of the inertia reactions of the connecting rods. The impulses under heading (4) could be obtained by an analysis of the crank- effort diagram, and it may be noted in carrying out such an analysis that any impulse which is symmetrically positive and negative when plotted as a function of time can have no even masses periods. _this June, 1909 harmonics. The graphic method of analysis is very easily carried out, Natural Period of Vibration. Under heading (3) will fall all vibra- tions due to the natural vibration of the tail shaft. We have seen that shaft is subjected to periodic pulsations every time a_ propeller blade passes close to the ship; hence, if this frequency corresponds with the frequency of lateral vibrations of the tail shaft loaded w-th the propel- ler, the shaft will vibrate very con- siderably. The natural period of vi- bration of the loaded shaft is its period of whirling, which is easily calculated. Hence a shaft with 4 three-bladed propeller will vibrate considerably when revolving at one- third of its whirling speed, and with a four-bladed propeller at one-fourth of its whirling speed. In analyzing vibrations it will, therefore, be ad- visable to calculate the whirling speeds of the tail shafts. Under head- ing (4) we should also have to con- sider the natural period of vibration of each bracket loaded as it is with the propeller and a portion of the tail shaft. Since it is subjected to the same periodic pulsations as the pro- peller, it will vibrate considerably if these pulsations correspond with its own natural frequency. There may be other portions of the structure round the stern which are also capa- ble of vibrating in unison with these impulses, but the calculation of their natural periods would not be easy. Let us now apply these considerations to an actual case. The writer is in- debted to W. J. Luke for a pallograph record of the Lusitania. The record shows intermittent vibrations of small amplitude both of low and high fre- quencies, but during one experiment while the ship was turning, the low frequency vibration of consid- erable amplitude, and was well sus- fained for over a minute. A- per tion of this record is reproduced in Fig. 3. There must have been during that interval of time a periodic pul- sation corresponding with the funda- mental frequency of the ship's struc- ture. This pulsation could only arise from the propellers, since the ship is was turbirfe driven. Careful measurement of the pallograph record gives the following data: FREQUENCY OF VERTICAL VIBRATION OF SHIP = 64.1 PER MINUTE Frequency Revolutions of propeller per minute. biade impulses. Propeller No. (1):..... 145.5 436.5 Propeller No. (2).....-- 161.2 483.6 Propeller No. (3)..... 143 429 Propeller No. (4)..... 152.2 456.6