Thanks to Curiosity Stream for sponsoring this video. A good understanding of how structures behave when vibrating is what allows engineers to build rotating machinery, to launch sensitive instruments into space, and to safely design buildings in seismic areas, to name just a few of the many applications. Systems like these can be very complex, so to study their vibrating behaviour engineers usually start by building a simple model that approximates the dynamics of the system, but is easier to assess.
The two most important parameters in any vibrating system are its mass and its stiffness. A common modelling approach is to lump all of the different contributions to mass and stiffness together, and represent them using a point mass with a mass m and a spring with a stiffness k. This is called the lumped parameter modelling approach.
This kind of simplified model might seem quite abstract, but can actually represent the dynamic behaviour of a lot of real systems quite accurately. The beauty of this simplicity is that we now have something that we can analyse mathematically. But first we have to make a few assumptions.
We'll assume that the mass can only move up and down. Since the system behaviour is defined by a single output, the x coordinate of the mass, this is what's called a single degree-of-freedom model. We'll also neglect the effects of gravity, and for now we'll assume that there's no damping, meaning that no energy is lost from the system as it vibrates, by friction or other means.
No external loads are acting on the system - the purpose of the model is to understand how the system behaves in free vibration, or in other words how it will oscillate when it's displaced, and then released. Since we've assumed there's no damping, the mass will continue to oscillate like this indefinitely. The way the system vibrates is defined by its equation of motion, which can be determined by applying Newton's second law.
The second law states that the sum of the forces acting on the point mass is equal to the product of its mass and its acceleration, F=ma. We can figure out the sum of the forces acting on the mass by drawing a free body diagram. There is only one force, the force exerted by the spring, which is equal to the displacement x multiplied by the spring stiffness k.
And so we obtain the equation of motion for the system. This is an ordinary differential equation, and the solution is a sinusoidal function. t is time, Phi is the phase angle and A is the amplitude of vibration.
We can determine A and Phi by considering the initial position and velocity of the mass. Let's look at an example where the system has a mass of 5 kilograms and a spring stiffness of 20 Newtons-per-metre, and vibration is triggered by applying an upwards velocity of 2 centimetres per second to the mass. Since the displacement x is initially zero, the phase angle Phi must also be equal to zero.
And then we can differentiate the equation for x to calculate the amplitude of vibration A. An important property that can be calculated from a mass-spring model is the system's natural frequency, the frequency at which it will oscillate naturally when in free vibration. It's given by this term in the solution to the equation of motion, and is denoted using the Greek letter Omega.
It depends only on the mass and the spring stiffness, so no matter what the initial conditions are a system will always oscillate at the same frequency. It has units of radians per second, so is called the angular natural frequency. But it's sometimes more practical to think of the natural frequency as a number of cycles per second, in which case it's denoted using the letter f and has units of Hertz.
The inverse of the natural frequency is the period T, which is the duration of each cycle in seconds. Let's compare how two different systems oscillate. Both of these models have the same spring stiffness, but different masses, and so different natural frequencies.
The heavier mass oscillates at a much lower frequency. A neat demonstration of the natural frequency is the tuning fork. When the fork is struck it vibrates at its natural frequency, which is much faster than shown here.
This causes the air molecules to vibrate at that same frequency, which produces a corresponding tone. By assuming that the prongs behave like cantilever beams in bending, beam theory can be used to derive a formula for the natural frequency of the fork. The density, length and cross-section of the prongs can be calibrated to obtain the desired tone.
Of course when a mass oscillates freely it doesn't do so indefinitely. Energy within the system is dissipated as heat over time, so the oscillations progressively decrease in magnitude and eventually stop altogether. This loss of energy is called damping, and it occurs in all real mechanical systems.
There are several different mechanisms that can contribute to the overall damping of a system. With structural damping, energy in a vibrating structure is dissipated due to the relative motion of components at structural joints. And material damping is damping provided by the material itself, where energy dissipates in a vibrating material due to interactions occurring at the molecular level.
To improve our spring-mass model, we can lump the damping from all of the different sources together, and represent them by a single dampening device called a dashpot, which is essentially a plunger that moves through a liquid-filled cylinder. Whenever the plunger moves a force will act to oppose its displacement, and the magnitude of this damping force is proportional to the velocity of the displacement - the faster the plunger moves, the larger the damping force. C is the viscous damping coefficient - it defines the total amount of damping in the system.
This model of damping is called viscous damping, because it behaves in a similar way to viscous forces in a fluid, which are proportional to the fluid velocity. There are other damping models, but viscous damping is commonly used because of its simplicity. If we include the dashpot in our spring-mass model, the equation of motion is the same as for the undamped system, but also includes the damping force.
It's a little more difficult to solve this equation, and the solution will depend on the amount of damping. If a system is underdamped, it will oscillate, and the magnitude of each successive oscillation will decrease until it stops. If the damping of the system is increased significantly, which you can think of as the dashpot being filled with a far more viscous fluid, any oscillation will be completely suppressed by the damping.
This is called an overdamped system. And a critically damped system occurs right at the limit between these two cases - it has just enough damping to suppress vibration. Each of these cases has a different function that defines the displacement of the system, obtained by solving the equation of motion.
The ratio of the actual damping coefficient of the system to the damping coefficient that would result in a critically damped response is the damping ratio. Most engineering systems and structures have a damping ratio of less than 1, so they’re underdamped. Of course if we're modelling a real system we need a way of figuring out which value to use for the damping coefficient.
It usually has to be determined experimentally, and one way of doing this is by measuring the displacement of the system as it oscillates. A parameter called the logarithmic decrement can be calculated based on this test data, as the natural logarithm of the ratio of any two successive amplitudes. The damping ratio can be calculated from the logarithmic decrement, providing an estimate of the overall damping in the system.
So far we've looked at free vibration, where oscillation is only caused by the initial conditions - there are no externally applied loads. But another scenario is forced vibration, where oscillation is driven by an external force. This type of loading often occurs in rotating machinery.
A common problem with turbines and motors occurs when a rotating component is unbalanced, meaning that its mass is unevenly distributed. This introduces a load that has a sinusoidal component in the vertical direction, and can cause vibration. Unbalance can occur because components were poorly fabricated or have been distorted.
But eccentric masses are sometimes added to motors on purpose - intentionally unbalanced motors are how phones and video game controllers are able to vibrate. We can analyse this type of forced vibration using the spring and dashpot model, by adding a sinusoidal external load. The resulting equation of motion is similar to the free vibration case, but is a non-homogeneous differential equation.
Its solution is the sum of two functions - a complementary solution and a particular solution. The complementary solution is the solution to the homogeneous form of the equation, where the right hand side is equal to zero. This is just the solution to the equation of motion for an underdamped system in free vibration that we saw earlier.
And the particular solution captures the effect of the external loads and is given by this expression. R is the ratio of the frequency of the external force to the natural frequency of the model. Since there's damping in the system, the complementary solution that represents free vibration will eventually reduce to zero.
At this point the motion of the system is defined by the particular solution only. For this reason the particular solution describes what's called the steady-state response. It has the same frequency as the forcing function, but is offset by a certain phase angle, meaning that the response of the system lags the external force.
Something interesting happens to the steady-state response when the frequency of the forcing load is very close to the natural frequency of the system. R approaches 1, and so the first term in the square root is close to zero. And if the system has very little damping the second term will also be close to zero, which means that the displacement will become very large.
Let's plot the normalised maximum displacement against the frequency ratio. For an undamped system the damping ratio is equal to zero, so when the forcing and natural frequencies match, the displacement becomes infinite. All systems have some level of damping, so this is just a theoretical case - here's what the response looks like for different damping ratios.
We can see this effect if we adjust the speed of the unbalanced motor. As the frequency of the force caused by the eccentric mass approaches the natural frequency of the system, the displacements become very large. This happens because when the natural frequency and forcing frequency are the same, the energy added to the system by the external force is timed just right so that it increases the amplitude of the displacement with each cycle.
This is called resonance. Resonance can be very dangerous and needs to be assessed carefully. It's one of the reasons it's so important to be able to calculate the natural frequency of a system.
If the natural frequency of a bridge matches the frequency of wind loading acting on it, or of pedestrians crossing it, the results can be catastrophic, particularly if the structure has little damping. Devices called tuned mass dampers are sometimes installed in buildings and bridges to control the dynamic response at resonant frequencies. With a rotating eccentric mass the external force acting on the system is a simple sine wave function.
This makes it easy to solve the equation of motion, since we can obtain a neat closed-form solution. But if the loading is defined by a complicated function, or is completely arbitrary, which might be the case if it's based on test data, it might not be possible to solve the equation of motion directly, and numerical integration methods will have to be used instead. Designing a structure to withstand seismic events is difficult because the loading caused by an earthquake is random and can't be predicted ahead of time.
And so engineers have to use special probabilistic techniques like the response spectrum method to design structures to withstand seismic loads. Single degree of freedom models are really useful, but sometimes the dynamics of a system are better modelled using multiple degrees of freedom. Say we want to model the dynamic response of a three storey building.
If we assume that the columns between the floors are axially rigid but can bend laterally, we can model it as a system with three degrees of freedom, the x coordinate of each floor. Each of the masses in the model has its own equation of motion, and if we rewrite this system of equations in matrix form we can see it has the same familiar form as the equation of motion for an undamped single degree-of-freedom system. A single degree-of-freedom system has one natural frequency and can only vibrate in one way.
But since our model has three degrees of freedom it will also have three natural frequencies, and at each natural frequency the system will vibrate in a specific way, which is called a mode shape. Exactly how the system vibrates in practice will depend on the initial conditions that are applied. For the three storey building the three modes of vibration will look like this.
As the number of degrees of freedom increases it becomes necessary to use numerical techniques like the finite element method to determine the natural frequencies and associated mode shapes. We've only covered mass-spring models in this video, which oscillate by translating. But there are other types of vibration too, like pendulums, which oscillate by rotating.
This video is long enough already, but I've published a short companion video that covers the motion of pendulums over on Nebula, where we take a look at how the motion of a pendulum differs from a mass-spring system and how to derive the equation of motion. Nebula is a streaming service built by a group of educational creators, which means it's completely free of ads, and revenue generated by the project is distributed directly to the creators. It's full of curated independent content, including Nebula originals and bonus content you won't find anywhere else.
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