When considering a situation where a vibrating system is initially set in motion by some external force and subsequently experiences no external force, the vibration of the system will involve regular oscillatory motion, with the amplitude gradually decreasing until it eventually comes to a stop.
This type of motion is referred to as free vibration, and mathematically expressing this situation is equivalent to setting the right-hand side term in equation (2.1) to zero.
We assume the solution to the linear second-order ordinary differential equation expressed in equation (3.1) as follows.
If you differentiate this once and then twice, it becomes equation (3.2).
Since equation (3.2) is referred to as the solution to equation (3.1), substituting equation (3.2) into equation (3.1) and rearranging yields equation (3.3).
For this equation to always be 0, the term on the right-hand side cannot be a time-varying value that becomes 0, and the expression inside the parentheses must be equal to 0.
And solving the quadratic equation in S for Equation (3.4), we obtain Equation (3.5).
Therefore, the solution to Equation (3.1) is expressed as Equation (3.6).
In the special case where s = s1 = s2, the form of the solution is expressed as Equation (3.7).
Let's not delve into why the particular solution is represented in this way for now. The author is also not certain.
Here, D1 and D2 are arbitrary constants of integration.
If we assume an undamped oscillator, setting c=0 in Equation (3.1) and then substituting c=0 into Equation (3.5), we obtain Equation (3.8).
Here, ω represents the natural frequency of the undamped oscillatory system. As intuitively understood, the vibration frequency increases with stronger structural stiffness and decreases with greater mass, reflecting well-known physical characteristics.
Applying the value of equation (3.8) to equation (3.6), we get equation (3.9):
However, while expressing the equation in complex numbers provides conciseness, it may be inconvenient for understanding the physical meaning of the formula. Therefore, the expression can be transformed into the following form:
Using Euler's formula and introducing arbitrary constants of integration D1 and D2, the expression can be modified as follows:
When expressed in the form of complex numbers, and representing the expression of equation (3.9) differently, the following equation is obtained.
Here, the integration constants can be replaced with arbitrary new constants.
If we set new arbitrary constants of integration A and B as mentioned above,
Equation (3.9) becomes equation (3.11), representing the most fundamental form of oscillation.
In some textbooks, this process might be omitted, and many textbooks directly start the solution of free vibration with equation (3.11).
Here, the integration constants A and B are values determined by the initial conditions of the oscillatory system. In other words, finding the solution to the differential equation involves determining the values of these integration constants.
For example, with initial conditions such as initial displacement (initial position) and initial velocity:
To find the solution for the given initial conditions, you can differentiate equation (3.11) once with respect to time, yielding:
If you substitute the initial value t=0 into equations (3.11) and (3.12),
Integration constants A and B can be expressed in terms of the initial conditions.
Therefore, substituting the initial values of the vibrating system into Equation (3.11) instead of the integration constants A and B, the free vibration under initial conditions becomes Equation (3.13).
Equation (3.13) represents the simplest form of harmonic oscillation, where the natural frequency of x(t) with a phase angle determined by initial conditions varies periodically according to the following formula, as illustrated in Figure-3.2.
This T is referred to as the natural period of the oscillatory system, and the natural angular frequency expressed as the reciprocal of the period is given by the following equation.
Expressed as the natural angular frequency, it has the following relationship in the given equation:
Furthermore, the natural period T, expressed using mass m and the stiffness k of the oscillatory system, can be represented by the following equation.
It becomes the most fundamental physical quantity representing the vibrational characteristics of the structure.
Moreover, to express equation (3.13) more simply, the trigonometric identities learned in high school, specifically the cosine double-angle formula, can be used.
By utilizing the representation of the triangle shown in Figure 3.2, equation (3.14) can be expressed.
Where,
And equation (3.15), when represented on an Argand diagram as shown in Figure 3.2, can be expressed as the real values of two rotating vectors representing the initial displacement and initial velocity.
Take a moment here, while looking at the graph in Figure 3.1, to marvel at how elegantly the solution to the differential equation for undamped free vibration has been derived. If you don't find this impressive, the ongoing equations might seem tedious.
If you find it intriguing, delving further into vibrational dynamics can be rewarding; if it feels monotonous, the author believes completing the journey through vibrational dynamics might prove challenging.
Are the meanings of initial displacement and initial velocity accurately represented on the graph as you think?
If you've also grasped the information about phase angle, you've already reached a considerable level of understanding.
Free vibration in an oscillatory system with damping is classified into three solutions based on the sign (positive, negative, or zero) of the quantity inside the square root in equation (3.5).
3.2.1 Critical Damping Case
When the square root inside the expression (3.5) becomes zero, we can obtain the following relationship.
In this case, the damping ratio is denoted as Cc,
it is expressed as equation (3.17), and this damping value is referred to as critical damping. The solution to the oscillation equation when the oscillatory system has critical damping is already explained and expressed as equation (3.7).
The solution for displacement is expressed as equation (3.18), and the solution for the velocity obtained by differentiating it is given by equation (3.19). When substituting the initial conditions into equations (3.18) and (3.19), the resulting expressions are as follows:
When the values of the integration constants expressed in terms of initial conditions are substituted into equation (3.18),
The solution for a free oscillatory system with critical damping is given by equation (3.20), and when represented graphically, it appears as shown in Figure 3.3.
As evident from the figure, a system with critical damping exhibits a vibration pattern located on the boundary where there is neither a permanent displacement nor oscillation.
3.2.2 Under Damped System
If the damping value is less than the critical damping value, then the quantity inside the square root in equation (3.5) becomes negative, and it is expressed as follows:
In this case, representing the damping ratio as a ratio to the critical damping value is often convenient.
Therefore, by defining a new variable, the damping ratio ξ, the expression can be written as equation (3.21).
If we substitute the damping ratio into equation (3.5), we get the following expression:
Where, ωd is the natural frequency of the damped system, and generally, the damping ratio of typical structural systems does not have a significantly large value. Therefore, it tends to be slightly less than the natural frequency of the undamped system.
To further examine the relationship between ωd/ω and ξ, if we represent equation (3.23) on a plot as shown in Figure 3.4, it forms a circle with these two variables as axes. As seen in the figure, as the damping ratio approaches 0, the vibration frequency of the damped system and the undamped system almost coincide
By substituting equation (3.22) into equation (3.6), the response of a damped system's free vibration is obtained and expressed as equation (3.24).
Applying initial conditions similarly to before to determine the integration constants A and B, and substituting these values into equation (3.24), we obtain equation (3.25).
Expressing this equation in the form of rotating vectors, we get equation (3.26).
Where,
Therefore, the response of free vibration for a damped system, given by equation (3.26), is represented as shown in Figure 3.5.
As seen in the figure, equation (3.26) depicts the oscillation pattern of the damped system with a natural frequency or period, vibrating about the neutral axis.
When represented as rotating vectors, apart from the exponential decrease in vector magnitude, it is identical to Figure 3.1.
For those who have diligently followed the solution process for the single-degree-of-freedom free vibration, I applaud your hard work. In this textbook, I have strived to present the progression of equations without omitting details, catering to those who wish to study diligently.
I recall that the most challenging aspect for me, as I began my involvement in nuclear power plant construction starting with the circular reactor, was understanding the response spectrum. When exploring books on response spectrum, oscillation equations inevitably appeared, and I remember the difficulty of self-studying vibrational equations. Understanding the physical concepts inherent in equations was challenging because, even though I followed the book's equations, they did not always elaborate on the physical meaning.
Different readerships result in the inability to describe the physical meaning of equations in a verbose manner in textbooks. Therefore, it is my hope that those studying dynamics will at least attempt to follow the equations. There is no need to memorize the equations; confirming that you can solve them by working through the problems is sufficient.
Now, let's briefly discuss some of the physical meanings inherent in the equations representing the free vibration of a single-degree-of-freedom system. Understanding these concepts will enhance your comprehension of the physical meaning behind the forced vibration, which will be explained in the next section.
Firstly, as indicated by the solutions in equations (3.24), (3.25), and (3.26), when initial conditions such as displacement and velocity are applied to a vibrating system, the structure moves solely based on its natural frequency determined by mass and stiffness. This is a natural fact: in free vibration, the structure moves only according to its natural frequencies. As explained in the next section, when subjected to harmonic vibrations, the structure moves in sync with the frequency of the external force.
The two facts may cause brief confusion, but once both are understood through equations, you will have mastered the concept of vibration.
Secondly, damping systems in the natural world, as we perceive them in daily life, exhibit a decrease in amplitude. Examining the differential equation of a single-degree-of-freedom system, there is a damping term proportional to velocity. However, this doesn't mean that damping must have a property proportional to velocity; it is simply an introduced term to simplify solving the differential equation, reflecting the mechanism of absorbing energy proportionally to velocity through viscous damping.
Therefore, using a term that does not represent an absolute natural law makes mathematical solutions unable to accurately represent actual motion.
Thirdly, although not highly significant in the case of free vibration determined by initial conditions, there is a certain phase difference, as indicated in equation (3.28). When external forces are applied, the response does not reach its maximum immediately but experiences a slight time delay before reaching peak response. This will be explained in detail in the following section, particularly in the concept of base-isolated structures.
Fourthly, while most textbooks start by stating that the solutions to the vibration equations are exponential functions, you may wonder why we begin with such a choice. However, you need not be concerned with such questions.
Not all differential equations necessarily have solutions. Most differential equations do not have solutions. Well-known differential equations in wave theory, including the butterfly equation, have names because mathematicians discovered a solution method, resulting in names associated with them.
As mentioned earlier, the reason for introducing viscosity damping expressed as proportional to velocity is to simplify solving the differential equation. Introducing another form of damping may lead to cases where solutions cannot be found, hence the choice.
On the contrary, Finite Element Method (FEM), a numerical analysis method, does not require the formulation of challenging equations, as long as you have a computer and time. However, FEM analysis provides answers for specific cases. FEM analysis does not reveal general solutions.
Suppose a diligent student, fascinated by the power of FEM, compiles numerous FEM analyses to understand various trends. Another student, less diligent but with a sharp mind, solves equations with a pencil and paper. The findings obtained through FEM analysis represent specific cases of the general solution derived by pencil and paper.
In conclusion, the purpose is to study the vibration equations without resorting to numerical analysis. The intention is to understand the concepts behind the equations, allowing you to grasp the overall behavior through a single solution. The pursuit of mathematics enables an understanding that no amount of FEM analysis can provide.
For example, it is said that a famous mathematician's doctoral dissertation was only a single sheet of paper written with a pencil.
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