State of the Art of Research in Vibrations
Profs C S Manohar from IISc Bangalore, Anindya Chatterjee and Pankaj Wahi from IITK and Salil Kulkarni from IITB discuss the challenges in vibrations research today. Nonlinearities, uncertainties and reduced order modelling emerge as areas where the focus of further research needs to be.
The luxury of linearity
Many undergraduate engineering courses include a major course on vibrations. Typically, we learn about mathematical techniques for handling single degree of freedom systems of various hues — free, forced, damped and undamped. Highly idealised, lumped parameter systems made up of springs, masses and dampers are translated into differential equations using the principles of mechanics originating from Newton or Lagrange and, occasionally, even Hamilton. Some courses delve deeper into two and multi-dof systems and provide a sneak peek into vibration of some continuous systems (whirling shafts seem to be a popular example).
For multi-dof systems matrix differential equations make an appearance, the language attains sophistication, and reconciling the maths with the physics demands a sounder grasp of linear algebra.
No matter how many dofs you deal with, an overarching assumption lurks un-noticed behind the scenes. The idealised systems you consider are ‘linear’, as are the differential equations you generate from their mechanics. So, to our great advantage, the principle of superposition holds.
It is important to realise that linearity arises by design and not luck. It is an outcome of assuming that deformation remains small, applied forces on the undeformed and deformed structures are not very different, and all springs, dashpots, etc. follow linear constitutive laws. These are not overly restrictive assumptions. In normal operating conditions, most of them hold quite well and are a source of certainty.
But even with linear systems, analysis is not always mathematically simple. Real-world forces or motions are often weird and messy. When we assume the solution of the underlying differential equation to be a Fourier series, we are basically trying to split the messy shape into a stack of sines and cosines with different amplitudes and frequencies. Then, instead of solving the entire puzzle at once, we determine the amount of each frequency that makes up the motion. This is useful because superposition allows us to add all those individual little answers to get the final motion of the system. It is a bit like figuring out a complex musical chord by listening to all the individual pure notes hidden inside it.
The scenario can be complicated by damping which, like a finger lightly touching a vibrating guitar string, leaks energy out of the system and acts as a governor on the amplitudes.
Or by resonance, which occurs when a frequency present in the Fourier series hits the sweet spot and matches a natural frequency of the system. If we stay away from resonance and damping is linear, superposition continues to hold out its reassuring hand. The maths can still be messy but is essentially tractable.
Handling non-linearities
Nonlinearity in vibrating systems often appears in unexpected quarters. A common example that all drivers of vehicles on fast highways have experienced is steering wheel vibrations that appear suddenly in certain speed ranges. The nonlinearities here come from a number of sources. The contact between the tyre and the road is nonlinear. The material of the tyres have nonlinear stiffness, as do the suspension components. Under appropriate conditions, these can produce self excited vibrations of the steering wheel. Similarly, aircraft flutter, machine tool vibrations, squealing of brakes and vortex induced vibrations are all important examples where nonlinearities play a major role.
Modelling nonlinearities and understanding the response of nonlinear systems are a major area of research in vibrations today. The mathematical framework goes far beyond what we learn about linear systems in undergraduate classes. Nonlinearities can lead to resonance that depends on amplitude. Hardening nonlinearities can push resonant frequencies to higher values and softening ones can reduce them. At a single excitation frequency, a nonlinear system can sometimes have not one but multiple stable vibration amplitudes. A sinusoidal input may not necessarily lead to a sinusoidal output but generate harmonic content and, in cases, also sub and super harmonics. The response can depend on history so that the current response may not depend only on the current excitation. Finally, in sufficiently strong nonlinear systems, the response can be extremely sensitive to initial conditions, leading to a situation called chaos.
Nonlinearities are also inherent in so-called moving boundary value problems where the boundary of the vibrating domain changes with time, like in a sitar string contacting the wooden bridge at different points depending on how it is being played.
Solving nonlinear systems require ingenuity. Various techniques have emerged over the years. Direct numerical integration or Finite element based nonlinear dynamics are probably the most straightforward techniques. But perturbation methods, method of multiple scales, harmonic balance and continuation methods are powerful analytical techniques that are often used. Harmonic balance is especially appealing as it considers the response to be a combination of sine waves and, in some way, ensures that each sine wave satisfies the governing nonlinear differential equation. In fact, the combination of continuation methods with harmonic balance can often reveal subtleties like unstable solutions and bifurcations that simple frequency sweeps might miss. Even with lumped parameter models, introduction of nonlinearity continues to produce surprises.
Transmission across joints and interfaces
Real systems like ships, aircrafts and buildings are large and consist of interacting components. Components are glued, bolted, welded with each other across interfaces that are difficult to characterise.
Interacting components can introduce nonlinearities even in otherwise linear problems. Transmission of vibrations across flexible joints with friction, contact or clearence, connecting elastic structures, lead to nonlinear effects. On top of that, the joints are not always represented with all their geometric details and the extent of modelling idealisation governs the effectiveness of the transmission across them.
The problem of transmitting vibration across interfaces also becomes important in case of mega structures. Mega structures are seldom analysed in totality. Take the case of a large aircraft. Different parts like the fuselage, wings and body of the aircraft are often analysed separately. A vibrating wing can be analysed as a cantilever built into a massive wall, but in reality, the vibration of the wings transmits to the body, which, unlike the massive wall, may itself be in a state of vibration. How do you put these disjoint analyses together so that, the whole structure responds in a manner it would if you never partitioned it?
Model order reduction
The problem of stitching the partitions together becomes doubly difficult when each partition itself constitutes a large problem in the dynamic Finite Element method. Many parametric analyses of large systems are expensive and produce data overloads. Model order reduction (MOR) techniques have been developed to systematically identify the degrees of freedom in large system that contribute most to the overall response. Efficient algorithms can reduce millions of degrees of freedom to a few hundred generalised coordinates.
The process of analysing the vibration response may involve sweeps over parameters that govern both excitations and material properties. Most MOR techniques pick out the dominant ‘features’ that describe the response under a range of imposed excitations. But parameter sweeps over material properties are also useful when, given two existing solutions with two sets of material properties a new one with interpolated properties need to be determined.
Techniques of MOR in linear systems are well established. Imagine that you are interested in frequencies below 100Hz. Then, in the simplest techniques for MOR, called modal reduction, you can retain just the first few modes in constructing the part of the response you care about, while jettisoning all others. While this sounds intuitively simple and effective, modal reduction essentially relies on superposition and is not the best way to take in case the system is nonlinear.
When interfaces are present and interface interactions are important, modal reduction must be modified. A workaround is to analyse the individual components in a manner that the combined system essentially ‘discovers’ the interface interactions when put together. This is the idea behind what is called component mode synthesis.
Ordinary modal reduction techniques do not retain arbitrary interface motions as independent co-ordinates because we ask our Finite Element code to determine the modes with the interfaces fixed. In component mode synthesis we still calculate the fixed-interface modes. Additionally, we apply small known displacements to an interface dof, keeping all other interface dofs fixed. Repeating this for all interface dofs gives us an extended set — the fixed-interface modes plus a set of bases that represent the interface motion. When different components are put together, we can now demand that the motion at the interface be the same for both the components. Again, this works for linear systems. Extensions to systems with nonlinearities are possible, but require much more work to be done.
Proper Orthogonal Decomposition (POD) offers a completely different philosophy. In POD, we first run simulations or experiments, collect snapshots at specific time instants and create a snapshot matrix. Now, we demand that we represent the response using a small set of orthonormal basis vectors chosen in a manner that the reconstruction error is a minimum in the least squares sense. The problem involves a mathematical operation called ‘singular value decomposition’ (SVD), which is now a routine procedure packaged with most linear algebra softwares. Obviously, POD is data dependent. If we train the reduced basis using simulations at one excitation amplitude and then use it at a dramatically different amplitude, it may perform poorly. But the idea is extremely powerful for problems with nonlinearities.
The current research thrust is to perform effective model order reductions in nonlinear problems. When the underlying constitutive behaviour of the structure is nonlinear, or joints and interfaces introduce nonlinearity, the internal forces generated cannot be reduced in the same way as they can be for linear systems. The primary reason is that nonlinear forces do not scale well, dilute the efficiency of the reduction process and sometimes completely nullify its advantages. Vibration analysis of large systems, with material nonlinearities and interconnected components, is a subject of intense research.
Handling uncertainties
Consider a building subjected to ground acceleration during an earthquake. The ground motion depends on the magnitude of the earthquake, distance from the fault, soil conditions and its wave propagation characteristics. Additionally, the material parameters involved in the building itself may have uncertainties hidden in them. So an uncertain excitation on a uncertain structure leads to vibrations that are understandably classified as random. Probablilistic methods are required to analyse these responses. Coupling of uncertainty with nonlinearities of the kind discussed above leads to a richer tapestry of interesting engineering challenges.
Probabilistic methods are necessary to analyse random vibrations. We also have to ensure that the quantification of the uncertainties involved are accurate. While this is probably the most difficult part, current research indicates that a data driven approach may be the way to go. Probabilistic models can be continuously informed by real time data acquired from instrumented structures.
But instrumented structures do not give us probabilities. With an array of accelerometers fixed onto a bridge, we simply get responses over long times. These then need to be converted into probability densities and easily answer questions like “what the probability that magnitudes of acceleration on the structure exceed a critical threshold?". The time signals from multiple sensors can also help us determine the covariance between a pair of sensors that tells us if the accelerations measured by those two are highly correlated. Simple Fourier analysis of the time signals can yield frequency domain information, like power spectral densities that tell us how the vibration energy is spread across frequencies, and cross spectral densities that help us validate possible causal relationships between two measured signals.
Bayesian Approaches
While these sophisticated signal processing techniques reveal a lot of information, Bayesian approaches are beginning to play a major role in random vibrations. Let us explain how the well-known Bayes’ theorem comes into all this.
The premise is simple. Suppose that you notice an unusual vibration in a machine. You do not know if the cause of the vibration is a particular loose joint. In this case, you are interested to know that, given that you observe a large unusual vibration, how likely is the particular joint to be loose? Bayes’ theorem allows you to calculate this probability, provided you know the chances of the joint being loose in a machine, the probability that you will see large vibrations when that joint is loose and the probability of seeing large vibrations of the machine for any reason.
For our problem, this translates to calculating how probable a particular combination of elastic properties and random excitation is, given a measured vibration pattern. Now, this is called the posterior in Bayes’ parlance — what you believe after you have seen the data (“the probability of a loose bolt causing the observed vibration is 30 %"). To calculate the posterior, you need the ‘prior’, which is what you believed before you saw the data (“there is a 5 % probability of a loose bolt in a machine") and the ‘likelihood’, i.e. how likely the observed data is if you assume a particular set of elastic properties and impose a particular random excitation.
The task of generating the likelihood information is done through a forward problem. This is essentially a large scale FE model of the structure, subjected to a large set of property combinations and different random excitations. Interestingly, such large computational problems will also harness the powers of model order reduction techniques discussed earlier. Nonlinearities can play a cameo role here too.
The likelihood information is also enriched by structural health monitoring data which alerts us to changes in material properties over time. The statistical framework is elaborate and fraught with exciting possibilities but not yet fully developed. Bayesian model updation techniques are rife with deeper research problems.
Pseudo-dynamic testing facilities are important in the area of earthquake engineering for generating relevant data. These set-ups assume that the ground excitations are applied slowly so that the structure being tested responds quasi-statically. A computer tells the actuators to impart the slow ground actuation.
Such slow-motion earthquake testing provides an important part of the required information — discrete realisations of the nonlinear stiffness of the structure as a function of its deformation. As the real structures are nonlinear, to gather as much information as possible about how the stiffness depends on its deformation, the computer has to continuously tell the actuators how to excite the structure based on how it has been excited till now. In other words, the computer and the actuators are part of a closed loop system which should have as small a delay as possible. Armed with this critical quasi-static information, a computer simulation working in tandem churns out the full dynamic response. The slow excitation makes the experiment tractable (compared to enormous shake tables that rattle structures wildly) but then the translation to a full dynamic response requires careful computational modelling.
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