Structural Engineering & Statics
where the load goes
Dynamics

The modes at zero frequency

A structure held by nothing — a span being launched, a segment on a crane, a pontoon — has a mode in which it moves as one body and stretches nothing, at a frequency of exactly zero. That is a real mode, not a glitch in the stiffness matrix. It carries every kilogram of the structure under a uniform acceleration and leaves the flexible modes none at all. A load that is not uniform is a different matter: pushed suddenly from one end, the structure has no static answer to give, only an acceleration with a vibration riding on it.

Assumes A structure has more than one period, Everything adds to nothing, and that is the whole of statics and Most of the mass moves together.

15 min read 7 figures The statics of things that do not moveThe load must go somewhereThe load that will not hold still

Every structure in the essays on mode shapes so far has been held. A building of several storeys stood on the ground, and one on bearings stood on something soft. Every mode had somewhere to bend from and a frequency above zero, and every quantity the previous essays used — the periods, the shares of mass, the storey drifts, the strain energy — assumed that.

Plenty of structures are not held, at least for a while, and several of them spend that while in the most dangerous part of their lives. A bridge being launched out over a river is pushed by jacks and slides on bearings that resist it only by friction. A segment hanging from a crane is held at a point it can swing about. A floating pontoon is held by water, which resists nothing sideways. A structural model with its supports left off, which every analyst has made by mistake, is the same thing on a computer, and the program’s complaint is that its stiffness matrix is singular. That complaint is correct. The model has a mode at zero frequency, and the mode is not an error.

A mode that stores nothing

The smallest example is a chain of five equal masses joined in a line by four equal springs, with nothing holding it to the ground.

The modes of five masses joined by springs and held by nothing. Five equal masses joined in a line by equal springs, with nothing holding them to the ground. The first four modes, at zero, 0.62, 1.18, 1.62 times the frequency of one mass on one spring. The first is every mass moving together with no spring stretched at all: a mode at exactly zero frequency, a real solution of the eigenvalue problem, which carries all of the mass and none of the strain. Every other mode has the ends moving against each other, and carries none of the mass under a uniform acceleration.
Fig. 1 The first four modes of five equal masses joined by equal springs and held by nothing, at zero, 0.62, 1.18 and 1.62 times the frequency of one mass on one spring. The first is every mass moving together with no spring stretched — a mode at exactly zero frequency, carrying all of the mass. The others have the ends moving against each other and carry none of it under a uniform acceleration.

The first mode is every mass moving by the same amount in the same direction. No spring is stretched, so no strain energy is stored, so nothing pulls the chain back, and it goes on moving. A mode’s frequency squared is its strain energy over its kinetic energy, and here the numerator is zero. The frequency is exactly zero, and the chain in this mode does not oscillate. It drifts.

The other four modes are the chain flexing, at 0.62, 1.18, 1.62 and 1.90 times the frequency of one mass on one spring. Those numbers are twice the sine of a tenth of a half-turn, two tenths, three and four. In each, the two ends move against each other and the chain’s centre of mass does not move at all.

The stiffness matrix says the same thing in the language of algebra. A rigid displacement, every mass moving by one, stretches no spring, so the stiffness matrix times that displacement is zero. A matrix that sends a non-zero vector to zero has a determinant of zero and no inverse. A static analysis asks for the inverse, because it asks what displacement a given force produces. For a free structure there is no such displacement: a force on a free body produces an acceleration. The program that stops with a singular matrix has not failed. It has found a mechanism, in the sense of the count of unknowns: a way for the structure to move without resistance, which for an unrestrained body is simply moving.

Where the zero comes from

A rigid-body mode can look like a special case with its own rules. The next figure shows that it is the ordinary first mode of a structure whose support has been taken away a little at a time.

Taking the support away, one decade at a time. The first two natural frequencies of the five-mass chain as the spring holding it to the ground is made weaker, from as stiff as the others to a millionth of that, both axes logarithmic. The second barely moves, from 0.83 to 0.62. The first falls as the square root of the ground spring — a straight line of slope one half — from 0.28 to 4.5e-4, while its share of the mass rises from 88 to 100.0000 per cent. Nothing happens at the end of the line but that the number reaches zero: the rigid-body mode is the first mode of a structure whose support has gone.
Fig. 2 The first two frequencies of the five-mass chain as the spring holding one end to the ground is weakened from as stiff as the chain’s own springs to a millionth of that, both axes logarithmic. The second barely moves, from 0.83 to 0.62. The first falls as the square root of the ground spring, a straight line of slope one half, from 0.28 to 4.5 × 10⁻⁴, while its share of the mass rises from 88 to 100 per cent.

With the ground spring as stiff as the chain’s own, the chain is an ordinary structure with an ordinary first mode at 0.28, carrying 88 per cent of its mass. Weaken the spring and the first frequency falls, in proportion to the square root of the spring’s stiffness once the spring is soft, because in that mode almost all the deformation is in the ground spring and the chain above moves as a block on it. Its share of the mass climbs toward all of it. The second mode hardly notices, falling only from 0.83 to 0.62 — the free chain’s own first flexing mode.

At the end of the line the ground spring is zero and so is the first frequency, and nothing else has happened. The rigid-body mode is the first mode of a structure whose support has gone. It obeys every rule the first mode obeyed. It is orthogonal to the other modes and has a shape and a share of the mass. It is also what a mass count counts. What it does not have is a frequency above zero, which is the one property a structure gets from being held.

The modes of five masses on a ground spring 0.01 as stiff as their own. Five equal masses joined in a line by equal springs, held to the ground at one end by a spring 0.01 times as stiff. The first four modes, at 0.04, 0.62, 1.18, 1.62 times the frequency of one mass on one spring. The first is every mass moving almost together, the ground spring taking nearly all the deformation, at a frequency 7.2 per cent of the next mode's and carrying 99.99 per cent of the mass: a rigid-body mode with a little support under it. Every other mode has the ends moving against each other, and between them they carry the 0.01 per cent the first leaves.
Fig. 3 The same chain with a ground spring a hundredth as stiff as its own. The first mode is every mass moving almost together, the ground spring taking nearly all the deformation, at 7.2 per cent of the next mode’s frequency and carrying 99.98 per cent of the mass. The flexing modes above it are almost exactly the free chain’s.

With the support a hundredth as stiff as the chain, the first mode is a rigid-body mode in everything but name. The masses move together to within two per cent of each other, and the mode carries 99.98 per cent of the mass at a frequency fourteen times lower than the next. The flexing modes above it are, to the precision of the drawing, the free chain’s.

This is the surprising connection the argument has been heading toward: that picture is base isolation. The six-storey building on bearings had an isolation mode carrying 99.9 per cent of its mass at 2.55 s, against a first structural mode at 0.32 s, and structural modes whose shares of the mass were 0.05, 0.003 and 0.0006 per cent. Those numbers are this figure’s with a building in place of the chain. Isolating a building is making it nearly free: giving it a support soft enough that its first mode becomes a nearly rigid-body mode, far below the frequencies at which the ground has energy. The flexible modes’ tiny shares of the mass, which made the previous essay’s error grow up the modes, are the property of free structures that the next figure shows exactly.

The same fact is used on purpose by people who measure modes for a living. A component tested for its natural frequencies — a turbine blade, a car body, a bridge cable anchor — is very often hung from soft elastic cords rather than bolted to a rig. A rig’s bolted joints are the least certain part of any model: their stiffness is not known and changes each time the component is refitted. A soft suspension instead gives the component rigid-body modes at a small fraction of its lowest flexing frequency, so low that they do not disturb the flexing modes at all. The test measures the free structure’s flexing modes, which a model can reproduce exactly, because a free structure has no boundary conditions to get wrong. Setting a structure free is how its modes are made knowable.

Where the mass goes

A mode’s share of the mass is its participation in a uniform acceleration of the whole structure, which is what an earthquake applies to a building, and what gravity applies to everything.

Where the mass goes when the support goes. Each mode's share of the chain's mass under a uniform acceleration, held to the ground by a spring as stiff as the others and held by nothing. Held: 88.0%, 8.7%, 2.4%, 0.8%, 0.2%. Free: 100.0%, 0.0%, 0.0%, 0.0%, 0.0%. A free structure's flexible modes are orthogonal to its rigid-body motion, so a uniform acceleration — the whole of an earthquake, and gravity — cannot excite them at all: it moves the chain as a block and stretches nothing. Only a load that is not uniform reaches them.
Fig. 4 Each mode’s share of the chain’s mass under a uniform acceleration, with the chain held to the ground by a spring as stiff as its own and held by nothing. Held: 88.0, 8.7, 2.4, 0.8 and 0.2 per cent. Free: 100.0 per cent in the rigid-body mode and nothing in any of the others.

Held, the chain shares its mass among its modes the way a building does: most in the first, a little in each of the rest, all of it accounted for. Free, the rigid-body mode takes every kilogram and the four flexing modes take none. The reason is orthogonality. Every mode is orthogonal to every other in the sense of the mass, and a flexing mode’s orthogonality to the rigid-body mode says that the mass-weighted sum of its displacements is zero. That sum is the movement of the centre of mass. A flexing mode of a free structure does not move its centre of mass, and a uniform acceleration acts only on the centre of mass, so it cannot reach the flexing modes at all.

That is why a floating structure in an earthquake is safe from the earthquake. It is not connected to the ground, so the ground’s acceleration reaches it only through the water, and water carries no shear. A launched girder sliding on bearings is nearly in the same position for horizontal ground motion along its length. And it is why a design response spectrum has nothing to offer a rigid-body mode. The spectrum is read at a mode’s period, and a rigid-body mode’s period is infinite. At the long-period end every spectrum flattens toward the ground’s own peak displacement: the structure stands still while the ground moves beneath it, and no force passes between them.

Pushing a free structure

A uniform load cannot reach the flexing modes. A load applied at one end can, and it is the load a free structure usually meets: the launching jack, the crane’s first pull, the tug.

Pushing a free structure from one end. A force applied suddenly to mass 1 of the free five-mass chain and held, and the force it produces in spring 3, in the middle, as a fraction of the applied force, over four periods of the first flexible mode. The chain as a whole accelerates — the rigid-body mode takes the load, and nothing resists it — and the spring must accelerate the two masses beyond it, which on average takes 0.40 of the force. But it does not settle there: the flexible modes the sudden load reaches make it swing about that value, to 1.22 at its largest — 3.0 times the average. A structure with no support has no static answer to give, only an acceleration and the vibration riding on it.
Fig. 5 A force applied suddenly to the first mass of the free five-mass chain and held, and the force it produces in the middle spring, as a fraction of the applied force, over four periods of the first flexing mode. On average the spring carries 0.40 of the load — the share needed to accelerate the two masses beyond it — and it swings about that value to 1.22 at its largest, three times the average.

The chain accelerates as a whole. That is the rigid-body mode taking the load, and it takes all of it, since nothing resists. For the chain to accelerate uniformly, each spring must push on the masses beyond it with the force their share of the acceleration needs. The middle spring has two of the five masses beyond it, so on average it carries two fifths of the applied force. That average is found by the technique analysts call inertia relief. Apply the rigid-body acceleration to every mass as a load, so that the loads on the structure are in balance, and then solve statically. It is the only static answer a free structure has.

Inertia relief is worth spelling out, because it is the standard way of analysing things that are never supported: an aircraft in a manoeuvre, a ship among waves, a satellite firing a thruster. The applied forces add up to a net force and a net moment. Divide by the mass and the moment of inertia to get the rigid-body accelerations. Put back, at every mass, a force equal to its mass times that acceleration and pointing the other way. The applied forces and those inertia forces together are in equilibrium, so any convenient support can be added — it will carry nothing — and a static analysis gives the internal forces. For the chain: a unit force, an acceleration of one fifth per unit mass, an inertia force of one fifth at each mass, and a middle spring that has to deliver two of those fifths to the two masses beyond it.

It is not the answer the spring gives from moment to moment. A force applied suddenly reaches the flexing modes as well as the rigid-body one, because a force at one end is not uniform, and each flexing mode vibrates about its share of the static answer. The middle spring’s force swings from zero to 1.22 of the applied force, three times its average. A suddenly applied load on a single mass and spring doubles the static answer at most, and that factor of two is the one engineers carry. On a structure with several flexing modes reaching their peaks close together, the factor is not bounded by two.

Pushing a free structure from one end. A force applied suddenly to mass 5 of the free five-mass chain and held, and the force it produces in spring 3, in the middle, as a fraction of the applied force, over four periods of the first flexible mode. The chain as a whole accelerates — the rigid-body mode takes the load, and nothing resists it — and the spring must accelerate the three masses beyond it, which on average takes 0.60 of the force. But it does not settle there: the flexible modes the sudden load reaches make it swing about that value, to 1.35 at its largest — 2.3 times the average. A structure with no support has no static answer to give, only an acceleration and the vibration riding on it.
Fig. 6 The same chain pushed suddenly at the other end. The middle spring now has three masses beyond it and carries 0.60 of the load on average, swinging to 1.35 — 2.3 times the average.

Pushed from the other end, the middle spring has three masses beyond it and carries three fifths of the load on average, now in the opposite sense. It swings to 1.35 of the applied force, 2.3 times its average. The average depends only on how the mass is divided by the spring. The swing depends on how the sudden load projects onto the flexing modes, which depends on where the load is applied.

Pushing a free structure from one end. A force applied suddenly to mass 1 of the free ten-mass chain and held, and the force it produces in spring 5, in the middle, as a fraction of the applied force, over four periods of the first flexible mode. The chain as a whole accelerates — the rigid-body mode takes the load, and nothing resists it — and the spring must accelerate the five masses beyond it, which on average takes 0.50 of the force. But it does not settle there: the flexible modes the sudden load reaches make it swing about that value, to 1.35 at its largest — 2.7 times the average. A structure with no support has no static answer to give, only an acceleration and the vibration riding on it.
Fig. 7 A free chain of ten masses pushed suddenly at one end. The middle spring carries half the load on average, the share needed for the five masses beyond it, and swings to 1.35 of the load — 2.7 times the average.

A longer chain has more flexing modes and more ways for their peaks to line up. With ten masses the middle spring averages half the load and swings to 1.35, 2.7 times its average. The swing does not settle into a fixed factor as the chain lengthens. It depends on the details of how many modes the load reaches and how their periods relate. That is why launching and lifting operations are controlled by the rate at which force is applied rather than by the force alone. A jack that builds its thrust over several periods of the structure’s first flexing mode delivers the inertia-relief answer. One that jerks can deliver two or three times it.

Friction is not a support

A launched bridge seems at first not to belong here, because it slides on bearings that resist it. But what the bearings supply is friction, and friction is not stiffness. A spring pushes back in proportion to how far the structure has moved, which is what gives a mode a frequency. Friction pushes back against the direction of motion by a fixed amount, however far the structure has already gone, so it restores nothing. A girder on sliding bearings has a rigid-body mode along its length at zero frequency, with or without friction. Friction changes the force that reaches that mode — the jack’s thrust less the friction — and it dissipates energy. It does not change the mode.

It also makes the sudden load of the previous figures the normal condition rather than an accident. A pier on sliding bearings moves in jumps, because static friction is larger than sliding friction, so a structure being pushed sticks while the push builds and slips when it breaks through. Each slip is a sudden release of the difference between the two frictions, applied at the jack, to a structure with no stiffness in the direction it is being pushed. Every slip is the force figure again, with its swing to two or three times the average in the members near the jack. Launching procedures that specify a slow, steady thrust and low-friction sliding surfaces are specifying, in effect, that this figure should not happen.

What the figures leave out

The chain moves in one direction only. A free body in a plane has three rigid-body modes — two translations and a rotation — and in space six. Each is at zero frequency, and each is orthogonal to every flexing mode in the corresponding sense: a flexing mode of a free body moves neither its centre of mass nor its angular momentum. A crane lifting a segment off-centre excites the rotation as well as the translation, and the swing that follows is the rotation’s slow return, restored by gravity rather than by stiffness.

Gravity can supply what the structure does not. A segment hanging from a crane is free to translate and rotate as far as its own stiffness is concerned, but its weight, acting below the hook, pulls it back toward hanging straight. That makes its swing a pendulum, with a frequency set by gravity and the length of the hang rather than by any member: a segment ten metres below its hook swings with a period of about six seconds. That is a rigid-body mode given a small frequency by geometry. It sits far below the segment’s flexing modes, and the orthogonality argument applies to it unchanged.

There is no damping. Real structures damp their flexing modes, so the swings in the force figures decay and the force settles to its inertia-relief average within a few dozen cycles. The peak, which comes in the first cycle or two, is hardly affected.

The springs are linear and the masses equal. Neither matters to the zero frequency or to the orthogonality, which hold for any masses and any springs. They matter to the numbers.

The assumption a zero frequency rests on

A mode at exactly zero frequency needs a structure that is exactly unrestrained. Every real structure has some restraint, and the ground-spring figure shows that this changes nothing important: the first mode’s frequency rises from zero as the square root of the restraint, and everything else moves continuously. A launched girder’s bearing friction, a crane’s pendulum stiffness and a pontoon’s mooring lines are small restraints. They give rigid-body modes small frequencies, far below the flexing modes. The results in this essay apply to them with the zero replaced by a small number. The one thing that does change discontinuously is the static analysis, which is impossible with no restraint and possible, badly conditioned, with a little.

An eigenvalue analysis gives a clean test of which case a model is in. A body free in space has six rigid-body freedoms, so a model of it has six eigenvalues at zero and no more; a body free in a plane has three. An analysis that finds exactly that many has found a free structure, and its zero modes will each be a rigid translation or rotation. An analysis that finds more has found something else.

Still open: a mode at zero that bends

Every zero-frequency mode here moved the structure without deforming it. There is another kind. A truss missing a member, or a frame with a hinge in the wrong place, has modes in which the structure changes shape without stretching any member — a mechanism that the count of unknowns may not even see. Those are also modes at zero frequency, and also make the stiffness matrix singular, but a zero there does not mean the structure is free. It means the structure is a mechanism, on the point of folding. Telling the two kinds of zero apart — by the shape of the mode rather than by its frequency — is the question the modes at zero frequency leave open.

Where this goes next

Bearing damping that shakes the roof harder. The roof's white-noise root-mean-square acceleration as the bearings' damping rises from 3 per cent to 58 per cent, exact and classical, both divided by the exact value at 3 per cent, for the six-storey building. The classical analysis falls all the way, to 0.33. The exact one falls to a minimum of 0.44 at about 28 per cent and then rises, to 0.50 at 58 per cent: past the minimum, every extra per cent of damping at the bearings shakes the top floor harder, while the classical analysis says it is still helping. The top floor the bearings shake On a building of several storeys, the damping at the isolation bearings is tied less tightly to each higher structural mode than to the one below. It still does more harm in each, because the ground barely reaches those modes on its own. The error the usual analysis throws away gathers in the upper storeys, and past about a third of critical damping at the bearings the roof shakes harder while the analysis says it is still helping. How much damping is best depends on how tall the building is. Every section is hogged and sagged before the bridge exists. The bending moment envelope of a launched deck, section by section along its own length, taken over every position of the launch. In service each section has one sign; during the launch 100 per cent of them see both, because each passes over every pier and through every span on its way out. The worst launch moment is 42568 kNm against 40500 in service, and with no launching nose at all it would be 185977. That is why a launched bridge is a constant-depth box with symmetric flanges: the design case is not a load, it is a history. Every section was somewhere else A bridge pushed out over its piers subjects each of its cross-sections to a history rather than to a load case. Every one passes over every support and through every span, so the design envelope is the envelope of envelopes — and no in-service condition produces it. 20 kN applied at once and held, on a structure of 0.500 s period. Displacement against time for a single-degree-of-freedom structure of natural period 0.500 s and 2.0% damping, under 20 kN applied at once and held. The static deflection under the same peak force is 12.67 mm and the peak response is 24.56 mm — a factor of 1.94. Twice the deflection, for the same load A weight placed gently on a beam deflects it by one amount. The same weight let go from rest, a millimetre above the same beam, deflects it by twice as much — and the factor of two is exact, for every structure ever built.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

Base isolationDynamic amplificationEigenvalueMechanismModal massMode shapeRigid bodyStiffness matrix