Vibration Theory 2.1: Undamped Forced Oscillations

 

At the beginning,

 

In my case, I believe in the results of calculations in physics and mathematics as if they were a creed, through the study of vibrational theory. The calculation results of cosmology, derived from the observation that the current temperature of the universe is 272.5 degrees in absolute temperature, indicate the age of the universe as 13.8 billion years.

 

In the previous section on free oscillations, I explored solutions to problems without external forces. Furthermore, by accurately elucidating the dynamics of forces, I confirmed that the solutions obtained mathematically precisely express physical phenomena.

 

Forced oscillations involve solving problems with external forces for simple sinusoidal waves, such as in the case of a single-degree-of-freedom system. Some may wonder how this can be helpful in solving real problems involving arbitrary external forces, such as seismic waves represented by multi-degree-of-freedom systems. However, it is important to note that understanding the solution for forced oscillations in a single-degree-of-freedom system is not significantly different from finding solutions for arbitrary external forces in multi-degree-of-freedom systems.

 

The great mathematician Fourier established the foundation of modern applied science by expressing arbitrary waveforms as mere summations of sinusoidal waves. Understanding the theory of mode superposition, which states that multi-degree-of-freedom systems are just aggregations of single-degree-of-freedom systems, helps us grasp that all vibrational analyses originate from solutions of single-degree-of-freedom harmonic oscillations.

 

Chapter 4: Forced Vibration(Sinusoidal Loading)

 

The vibrational system depicted in Figure 2.1 undergoes harmonic loading with an amplitude of F0 and a natural frequency of p. In this case, the motion equation for the vibrational system is expressed by equation (4.1).

 

The general solution to the motion equation with an external force term can be expressed as the sum of the fundamental solution (Complementary Solution) xc(t), which represents the solution in the absence of external forces, and the particular solution (Particular Solution) xp(t).

Instead of questioning why this is the case, let's accept it for now and proceed with our understanding.

 

 

 

4.1 Undamped System

 

The vibration equation for an undamped oscillatory system, which is the simplest case without damping, becomes equation (4.3).

 

 

Next, the response due to harmonic vibrations in an inertial pendulum system takes on the form of harmonic vibrations similar to the input. The particular solution can be set as equation (4.5).

 

The assumed solution must satisfy the original vibrational equation, equation (4.3). Substituting equation (4.5) into equation (4.3) yields the following expression.

Separating this into sine and cosine terms and rearranging, the expression becomes as follows.

 

For the above expression to always be 0, both the sine and cosine terms must be 0. Here, a crucial new variable is introduced: β = p/ω.

 

This value, expressed as the ratio of the frequency of external force to the natural frequency of the vibrational system, holds significant physical meaning. In disciplines dealing with the phenomena of moving objects, it is often considered as one of the crucial variables.

Therefore, the general solution represented as the sum of the complementary solution and the particular solution is given by equation (4.6).

 

By substituting the initial condition of the vibrational system, x(0)=0, into equation (4.6), we find A=0. Then, applying x′(0)=0 to the first derivative of equation (4.6), which represents the velocity x′(t), we obtain equation (4.7).

 

Therefore, the final solution is given by equation (4.8).

 

Examining the right-hand side of equation (4.8), F0/k represents the static displacement when a static load is applied, following Hooke's Law.

The second part, 1/(1-β²), signifies the dynamic amplification factor, representing the ratio of dynamic displacement to static displacement when harmonic vibrations are applied.

Throughout the process of solving equation (4.8), we assumed β≠1. However, if β=1, the particular solution takes the form given in equation (4.9).

 

Therefore, differentiating this once and twice, the resulting expressions are as follows.

 

By substituting these into equation (4.3) and solving, we obtain the following expression.

 

As before, rearranging this into sine and cosine terms, the expression becomes as follows.

 

For the above expression to always be 0, the values of the sine and cosine terms must be 0. Therefore, we can obtain a system of equations in terms of G1 and G2 as follows.

 

By solving the system of equations for G1 and G2, we find that G2=0 and G1= -F₀/2mω.

Therefore, the particular solution is given by the following expression.

 

And the general solution x(t) is given by equation (4.10).

 

As before, by substituting the initial condition x(0)=0 into equation (4.10), we find A=0. Then, by substituting the initial condition x′(0)=0 into the first derivative of equation (4.10), we obtain equation (4.11).

 

Therefore, the general solution x(t) is determined as given in equation (4.12).

 

Here, when the value of β is close to 1 and when β is a very large value, the expressions (4.8) for those cases, and the expression (4.12) for the case when β equals 1, are depicted in Figure 4.1.

When β is close to 1, the vibration takes on a beat-like form, including components of the excitation frequency and the natural frequency of the vibrational system.

In the case of β being equal to 1, it represents the phenomenon of resonance, where the vibration progressively increases with time. Figure 4.1 illustrates these scenarios.

 

 

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