Not all the peaks are always seen however, nor are the amplitudes the same for every engine, even of the same type and model. The "signature" is as unique as an individual. Two people named Anne Smith don't sign their names the same, nor are they the same, even though we might be able to read both names as the same name. We know enough about the first order that we can either fix it by balancing, or instrument the prop enough to tell if it's a crank problem or a prop problem. How common is balance a problem?
About 80% of the time an improvement can be made. Of that 80% there is a standard bell curve distribution, with the center of the bell around .4 IPS. A small number will be just above .2 IPS, the generally accepted limit, and a few will be quite high, around .8 and .9 IPS, but most fall in the .3 to .5 IPS range. For the layman, .5 IPS at 2000 RPM equals about .005 inch radial runout on the crank flange. Point 2 IPS (0.2) equals .002 inch and it's best to get that down to .05 IPS; .0005 inch radial runout 10r a really smooth balance. That's considerably less than the clearance in the front main journal bearing, so now you are running on oil, and not working the oil too hard. To you purists, again, you are running on the oil at .005 inch but you are depending on the oil to cushion the crank, which wears out the oil faster and makes it run hotter. If the propeller is over 1.2 IPS, you are probably scraping the babbit of the bearing a bit. We draw the line at 1.2 IPS as a balance candidate. This puts the vibration correction in the mechanical fix action slot. It is probably not a routine balance problem. It could be anything from a poor static balance to a broken crankshaft. It's not worth finding out in the air!

MOTOR MOUNTS
The half order is probably the most commonly griped vibration, Sadly, it is also one of the most difficult to diagnose and reduce. The motor mounts need to be very soft to attenuate this low frequency vibration.  This has two undesirable side effects in aircraft applications. The engine moves more on the mounts so clearances to allow for abrupt landings and gust loads must be greater. The other unfortunate side effect is that soft mounts do not attenuate high frequency vibration very well. This is very bad for gyros and sensitive electronic gear. The amount of 1/2 order vibration in a given engine varies engine to engine of the same type. Typical ranges might be .2 IPS for a very good engine and up to .8 or .9 IPS on an engine that is otherwise healthy in every respect. Fixing a problem case has three areas of concern. How bad is it on the engine (the forcing function), how well is it being attenuated (damping). and is there anything that is naturally resonant.  A typical upper class balance job will include a spectrum analysis. This will at least identify the levels found on the engine.

The next task is to determine what the levels in the cabin are to find out if it is a transmission problem.
If the levels in the cabin are more than those found on the engine it is likely a resonance problem.
If they are the same or less but still objectionable, the only recourse is to visually inspect the
mounting system (isolators and engine mount) as well as look for hard contact areas in the engine compartment.
I saw one STC'd engine installation that used rubber mounts for the engine but hard mounted the exhaust system to a turbo which was hard mounted to the firewall. This constituted a "hard link" from the engine to the airframe. The engine ½ order was very obvious in the cabin.
About engine isolators/mounts...
Here's a brief primer on how mounts attenuate vibration. The mount is for all practical purposes a spring. What this spring does is absorb vibration by internal resistance to oscillating. As the spring oscillates it absorbs energy. Because of resistance it heats up as it absorbs energy. Spring steel has very little internal resistance and is usually accompanied by an "absorber" which increases its' resistance. The spring rate (constant) of the mount is derived by (1) the frequencies of interest, (2) the amplitudes expected (capacity), and (3) the weight they need to support. Most aircraft use elastomers (compounds of elastic material) that have a spring
rate appropriate to engine frequencies. The dilemma is in choosing one that satisfies all the
frequencies produced by the engine. Usually, the elastomers are only good in a narrow range of frequencies. Since high frequency vibration (lots of cycles) are the most damaging, it is logical to treat the higher frequencies as the more important to attenuate. Unfortunately, a mount that damps high frequency makes a poor mount for low frequencies. What we end up with in the airplane is a mount that damps the high frequency vibration produced directly by the engine, while the lower frequency 1/2 order vibration is passed with less restraint. There are some mounts that actually amplify the 1/2 order! On the higher frequency side, the buzz most people feel in the cabin cannot be attenuated since it is a function of the aerodynamic pulsing of the propeller on the
windscreen and empennage (single engine) or sides of the fuselage (twin). This is acoustic energy converted back to mechanical energy. Some twins have tuning forks in the side wall of the aircraft to absorb (cancel) this vibration and they are quite effective.

I'm going to depart for just a minute to cover a bit of entomology (bug science) to illustrate how tuning forks work. The crane fly (some people call it a daddy long legs) is an air craft of sorts. It has long wings and lots of dangly appendages. This plan form is fraught with possibilities for
resonance in flight but the crane fly has two small appendages immediately behind and below the wings with small masses at the ends. When viewed under a stroboscope so they are viewed in slow motion, the pendulous dampers can be seen to flex in opposition to the wings. This is the way the crane fly damps vibration. Well, if it's good enough for mother nature, it must be good enough for an airplane which is far less complex. The only problem with tuning forks and pendulous dampers is they are generally only good in their "tuned" region. Both mass and spring rate control tuned region. The rule being, the heavier and less resilient, the lower the frequency, and the lighter and stiffer, the higher the natural frequency. The mount depends on its internal resistance to tune it to the desired frequency. This resistance can and will change in operation due to prolonged exposure to heat (bakes and hardens them) and solvents like oil, fuel, and cleaning solvents (breaks them down and softens them ). Both conditions are bad. A hard mount is more likely to put more stress on the engine tubular or sheet metal mount structure because the high frequency (hence more cycles) is passed. A soft one could allow the engine to sag, enough for the prop to hit the cowling during hard bounces. Further, the mount elastomers are sometimes "tuned" by placing one spring rate on the top and one on the bottom, or by placing elastomers of different rates at different positions on the engine.
Failure to adhere to these installation schemes is VERY detrimental to the purpose of the mount! As far as items mounted on the engine mount tubing itself this is a mixed bag. It is possible to "tune" the mount to an engine peak by changing the weight of the mount. Each leg of the mount
has a different resonance, even among mounts of the same part number. By adding weight to a leg or legs, the natural frequency is of course changed. Usually it's not a problem. When in doubt. make it as much like it came from the factory, or as much like a factory installation as you can.

GEARED ENGINES are problematic in that now we have two rotating masses with their associated harmonics. What do you suppose happens when an engine is running at 4000 RPM and the two bladed prop only turns 2000 RPM. Well, you've got propeller one per rev talking to engine half order, and you've got propeller twice order (aerodynamic) talking to the crankshaft. This is a recipe for some very annoying airframe vibrations and one reason most production gear ratios are set up in primes or fractional thirds. The prop might turn .66 time the engine,. In this manner none of the natural peaks of one can influence the natural frequencies of the other. Home builders should take not here. If you are designing a gear reducing unit, say at least 10% away from a half or whole number of the engine RPM. Do not run the engine 1.5, 2.0, 2.5 etc. times the prop speed. Pick an odd and even number of mating teeth or cogs for the driving and driven gears and adequate harmonic separation will eliminate most of these problems.

CABIN MEASUREMENTS
Because there are no accepted norms for readings taken in the cabin, whatever you get is subject to some interpretation. In the helicopter world vibration "footprints. are quite common, in the fixed wing world almost non existent. In any event, frequency will determine what it is. With a baseline signature the very least you will
do is be able to track improvement or deterioration when changes are made. Some people inadvertently hurt themselves. On numerous occasions I have been called on to
take readings on a new engine that vibrates more than the old one did. Turns out that since the airplane would be down for a while the owner decided to revamp the instrument panel too. When the weight of the panel was changed, the natural resonance of the panel changed too, causing
what honestly appeared to be a "bad" engine. Taking a measurement on another airplane of the same type, similarly configured (same engine, same prop) is very valuable in generating a known good sample. Reproduce the running condition as exactly as possible and compare. Since this is not an option on most antiques and difficult on a lot of classic aircraft, it's a good idea to have a signature run when it is running well, or at least before you do anything drastic like replace the engine. This will give you an adequate baseline from which to compare.

A word about props...
Propellers are simple devices but they are one of the most neglected parts of the airplane. Nearly everyone checks the prop and spinner on the preflight but there are some things you just cannot see. As an airplane descends from altitude, air will find its way into every crack, crevice. and hole
as the atmospheric pressure increases. The air in the lower altitudes contains a good deal more moisture than at altitude so what you get is an automatic injection of water. This water accumulation is cumulative. Most airplanes are parked shortly after the descent and this moisture can condense out on cool metal surfaces at night. On propellers, the cavity inside the blade makes a good place for this. This condensate sits in the root of the propeller, or can find its way
on the surfaces of steel parts and cause a good deal of corrosion. Corrosion on propeller parts is very very bad. In the propeller root it can and has caused separation of the blade from the hub, usually followed shortly after by the engine. Devastating is the word. Some propeller manufacturers specify a calendar overhaul period. I read in a lot of the aviation journalism that this is a negative for a potential airplane owner. Not really. It shows a great deal of prudence on the part of the propeller manufacturer in maintaining a customer base (by keeping them alive) and lowering everyone's liability bill. In speaking with a propeller shop owner the other day (1998) he said he received a controllable pitch propeller that had not seen an overhaul
since 1961! The propeller had so many AD's against it almost cost as much for the AD compliance as a new prop. This was neither responsible maintenance or ownership This was a
gamble.

Conclusions
Get the propeller balanced. It's well worth the time and effort it takes (which is not much). The improvement this makes on the engine is cumulative, so a general improvement will result in addition to the immediate improvement. Keep a vibration signature on file in the aircraft. This is part and parcel of a good balance job. Maintain the integrity of the engine mounts with periodic replacement. The rubbers get cooked, bathed in oil and frozen. They deteriorate in ways you cannot see. Operate the engine smoothly. Use a reputable engine overhauler when the time comes.
Fackler

Some of the signature peaks are easily correctable, others are not. What are the big hitters? 
Engine half order -Vibration produced at half engine speed
Engine first order -Vibration produced at prop speed or crank speed Engine second order- Vibration produced at twice engine speed Propeller- Vibration produced at the prop speed "N" per rev Other -Accessories at ratios other than whole numbers or half numbers. Upper order harmonics - for this discussion higher than first order but a ratio of half plus a number, or even or odd numbers times the crank speed.

ENGINE HALF ORDER:
Half order vibration is probably the most complained about vibration in any piston installation. Remember every other stroke there is a power stroke? This is relatively low frequency vibration. One of the characteristics of low frequency anything is it doesn't take much energy to travel very far. Think about radio waves. Low frequency travels very well on very little power, the reason HF radios are used for long range communication on mobile platforms or where there isn't much power available. To move high frequency radio waves takes energy. Like radio waves, the half order travels well, and despite the best intentions of the manufacturers
the isolation mounts on the engine may not substantially reduce the half order levels. In some aircraft they are actually amplified. This is usually felt as a drumming on the floorboards, or shake in the yoke or glareshield. If your ashtray is rattling, it's either the prop out of balance or the 1/2 order Vibration! 

The following affect the half order vibration:
Anything that controls the combustion cycle on the engine
Pretty wide territory for troubleshooting. It could be compression, mixture, induction losses. valve lift, spark timing, or anything that controls the combustion cycle in the engine! Turbocharged engines seem to suffer with this malady regularly due to the lower compression ratios and higher horsepower output. Interestingly, some of the crop of "blueprinted" engines, engines whose parts are held to more
stringent tolerances typically exhibit low half order levels. In fact, they exhibit lower vibration levels throughout their spectrum. Less vibration, smoother flow, more horsepower. Does this mean we should all have blueprinted engines? Buy what you can afford. But be sure you can afford what you buy!

ENGINE FIRST ORDER
Propeller balance and propeller track affect the first order vibration. Remember first order vibration occurs for each revolution of the engine in a direct drive engine. On the surface it seems easy to just balance the propeller and be done with it, but there is more to it than that for a couple
reasons. One, it is not just the balance of the propeller we are concerned with, but that of the crankshaft as well. This being the case, it is prudent to install a velocimeter at the front end of the engine and one at the opposite extreme of the crank, on the accessory section. In a routine balance exercise when the propeller is out of balance, the rear of the engine vibrates out of phase with the front. propeller cone picture The center of the rotation is the nodal point where there is little vibration. For this reason, the center of the engine is a poor location for balance. Anyway, as the propeller is balanced, the rear of the engine also decreases in amount until the whole crank is running on a true line. If the
sensor mounted at the front of the engine reads low. but the sensor at the rear of the engine remains high or goes higher, the propeller mass balance is not the problem. It is either an out of balance of the crankshaft or out of track (thrust distribution inequality) of the propeller.

ENGINE TWICE ORDER:
Engine twice order has as an origin two sources. If the engine has a two blade propeller, obviously the two pulses per rev as the accelerated air will be evident. The other source is
reciprocating mass imbalance. That means the reciprocating masses (pistons and rods) are not canceling each other. Here again, on a "blueprinted" motor, the 2nd orders are generally very low. When top end work is done helter skelter, with no attention to piston size or weight, the engine just will not run as smooth.

OTHER SOURCES
Ignoring the core engine vibration produced for a moment, there are accessories that can create problems, and lets not overlook the airframe. The alternator on most recips is belt or gear driven at some other frequency around 3 to 5 times the crank speed. Since it is not an exact multiple. its pretty easy to pick out in the spectral plot. This is true of air conditioning compressors, pumps and other accessories on the engine. To determine the exact speed of the accessory drive pad. just look at the accessory drive train ratios in the overhaul manual and multiply them by the engine RPM.
Air frame vibrations are as difficult to track down. They're usually airspeed or AOA sensitive. By hand holding the velocimeter sensor inside the cabin, and comparing this plot to the engine signature the engine generated, points can be subtracted form the airframe and yield a starting point. Some strange things can be found, from noisy bearings in a gyro, to wheels that spin in the wells, to weather stripping that vibrates.

Setting up an engine for a health check using the Vibrex 2000.
Note: Other models of Chadwick-Helmuth equipment may be used, please consult your particular operators manual. The reason the Vibrex 2000 is used here is that it is the easiest to use and the data can be manipulated easily for viewing owing to its digital format. A Velocimeter is mounted securely on the engine case where it will measure vibration. The
photocell is mounted where it will time a small retroreflective target on the back of a propeller blade. This provides a timing pulse so the vibration orders can be set up in the instrument. A lead is run from the sensor to the Vibrex 2000 in the cockpit, and the engine is ready to run. The signature is taken at some run condition, say 2000 RPM, and flat pitch on the propeller. The process of acquiring a reading takes about 5 to 10 seconds. The Vibrex 2000 measure vibration signatures via a 400 bin Fast Fourier Transform. Dr. Fourier
reasoned that any complex signal, like a vibration signature, could be taken apart into its constituent parts and represented as individual amplitudes (amount) and frequencies (speed). To do this the signal must be sampled, and then reproduced in some x/y format. Remember the bell earlier? It didn't have just one frequency when it was struck. If it did it would sound no different than an electronic beep from a computer speaker. It consisted of many frequencies and different amplitudes. If it were represented as a spectrum, we could see by looking at the frequencies, many if not most are harmonically related. The same is true of an engine. Most items are harmonically related to the crankshaft. So what are they and what do they mean?
Lets look at a typical four banger flat opposed motor signature. It has a peak at 1/2 order, some at first order (the crank speed), some at 1.5. 2.0, 2.5, 3.0 etc. Since these are ratios of the crankshaft speed, we will see peaks at 1000 RPM intervals for an engine running at 2000 RPM. Generally these diminish at the higher frequencies over about 5th order - five time the engine speed.

Thrust inequalities are more pronounced in changes with pitch so a simple change in load on the prop will identify this quickly enough. If the point remains the same over pitch, by a process of elimination, the only problem left is residual out of balance of the crankshaft. It is either out of balance or bent. To check for a bent crank, change the prop orientation on the crank 180 degrees. If the phase angle of the imbalance changes 180 degrees, it's the crank, not the propeller, because the crank is showing up at the same place relative to the imbalance. Before blaming this expensive piece of machinery though. make one last check. Get the Strobex out and check the track of the propeller. This is done by placing small highly reflective tapes at the blade tips and timing them so they are superimposed. If the track is out, they won't be in line. This is a dynamic track and may differ from the static track performed on the overhaul bench. If there are no track differences as power is changed, it is now almost certainly the crankshaft.