7.4. Solution to the Equation of Motion

The solution to the second-order differential equation is of the form:

x(t)=Acos(ωₙ t)+Bsin(ωₙ t)

The equation presented is a general solution to the equation of motion for an undamped Single Degree of Freedom (SDOF) system. Here, x(t) represents the displacement at any time t, A and B are constants that will affect the shape of the motion, and ωₙ is the natural frequency of the system. Essentially, the displacement can be described using a combination of cosine and sine functions, which means that the motion will be periodic or oscillatory.

Think of a child on a swing. When the swing is set into motion, it moves back and forth in a periodic way. The height of the swing at any given moment can be represented by a cosine or sine function, where the maximum height corresponds to A (the amplitude) and the movement back and forth can be represented by these trigonometric functions.

Where:

• A and B are constants determined by initial conditions (initial displacement x₀ and initial velocity ẋ₀).

The constants A and B in the solution are not arbitrary; they are determined by the initial conditions of the system. For example, if we know how far the system is displaced from its equilibrium position (initial displacement, x₀) and how fast it was moving at that moment (initial velocity, ẋ₀), we can calculate the specific values for A and B. This personalizes the equation to reflect the actual conditions of the system at t=0.

Imagine stretching a rubber band and then letting it go. The initial stretch (displacement) and how fast you let it go (velocity) determine how the rubber band will move afterward. In the same way, the values of A and B define the specific motion of the system based on its start.

Alternatively, using the harmonic form:

x(t)=Xcos(ωₙ t+ϕ)

Another way to express the displacement is in the harmonic form, where X is the amplitude of vibration, and ϕ is the phase angle. The phase angle is crucial because it indicates the position of the wave relative to time, showing how the wave starts at an offset rather than just beginning at zero displacement. This form emphasizes the oscillatory nature of the motion, encompassing both amplitude and phase shifts.

Consider a wave traveling through the ocean. The height of the wave is like the amplitude X, and the phase angle ϕ determines where the wave starts when we begin to observe it. This is akin to tuning into a specific point in the wave's cycle rather than starting when it is at rest.

These parameters can be related back to the initial conditions using:

B X=√A² +B², tan(ϕ)= A

The relationships presented here allow us to connect the constants A and B with the initial conditions of the system. Specifically, we can find the amplitude X as a function of A and B, showing that the total energy of oscillation is related to the quadratic sum of A and B. Furthermore, the tangent of the phase angle (ϕ) is expressed through the ratio of these constants, providing a mathematical means to derive critical parameters of the system's motion from its initial state.

Picture a triangle: if you know two sides, you can find the hypotenuse. Similarly, knowing the coefficients A and B helps you find the amplitude and phase, shaping your understanding of the system's behavior starting from its initial conditions. This is like knowing a car’s speed and direction to figure out where it will be later.

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