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Mastering Applications of Second Order Differential Equations
Dive into the world of mechanical vibrations and electrical circuits through second order differential equations. Gain practical skills to analyze and solve complex real-world engineering problems.

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Now Playing:Applications of second order differential equations – Example 0a
Intros
  1. What are some applications of second order differential equations?
  2. Mechanical Vibrations and Dampening Forces
Examples

  1. Mechanical Vibrations

    A spring has a weight of 5kg attached to the end of it. The spring has a natural length of 0.3m and a force of 35 newtons is required to stretch the spring to a length 0.8m. If the spring is stretched to 0.5 meters and then released (with zero initial velocity), then what is the position of the mass at time tt?
    Homogeneous linear second order differential equations
    Jump to:NotesConceptFAQsPrerequisites
    Notes

    Mechanical Vibrations:

    Hooke's Law:

    Force=-ky

    Newton's Second Law:

    Force=mass×acceleration=m×d2ydt2Force= mass \times acceleration=m \times \frac{d^2y}{dt^2}

    ⟹\Longrightarrow m×d2ydt2=−kym \times \frac{d^2y}{dt^2}=-ky

    ⟹\Longrightarrow my′′+ky=0my'' + ky =0

    Damping Force:

    Force=−cdydtForce = -c\frac{dy}{dt}

    ⟹\Longrightarrow m×d2ydt2=−cdydt−kym \times \frac{d^2y}{dt^2}=-c\frac{dy}{dt}-ky

    ⟹\Longrightarrow my′′+cy′+ky=0my''+cy'+ky=0

    Electrical Circuits:

    Using second order differential equations we are able to analyze a circuit consisting of a battery, a resistor, an inductor and a capacitor in series. Let us denote Q(t)Q(t) as the charge on the capacitor at time tt. The current is the rate of change of QQ with respect to tt. So the current of the system is equal to I=dQdtI=\frac{dQ}{dt}

    Kirchhoff's Law states that the sum of all voltage drops across a system must equal the supplied charge:

    VL+VR+VC=VbatV_L+V_R+V_C=V_{bat}

    Where VLV_L is the voltage drop across the inductor, VRV_R is the voltage drop across the resistor, VCV_C is the voltage drop across the capacitor, and VbatV_{bat} is the voltage supplied by the battery (or other electrical force).

    Faraday's Law:

    According to Faraday's Law the voltage drop across an inductor is equal to the instantaneous rate of change of current times an inductance constant, denoted by LL (measured in henry's).

    VL=L×dIdtV_L=L \times \frac{dI}{dt}

    Ohm's Law:

    From Ohm's Law the voltage drop across a resistor is equal to the resistance (measured in ohms) times the current:

    VR=IRV_R=IR

    And the voltage drop across a capacitor is proportional to the electrical charge of the capacitor times a constant of capacitance (measured in farads).

    VC=1C×Q(t)V_C=\frac{1}{C} \times Q(t)

    And let us denote the voltage from the battery as some sort of function with respect to time Vbat=E(t)V_{bat}=E(t)

    So inputting all the previously found information into Kirchhoff's Law:

    VL+VR+VC=VbatV_L+V_R+V_C=V_{bat}

    Which will become,

    LdIdt+IR+1CQ(t)=E(t)L \frac{dI}{dt}+IR+\frac{1}{C} Q(t)=E(t)

    And we know that I=dQdtI= \frac{dQ}{dt}. So the equation becomes,

    Ld2Qdt2+RdQdt+1CQ(t)=E(t)L \frac{d^2 Q}{dt^2}+R \frac{dQ}{dt}+\frac{1}{C} Q(t)=E(t)

    Which can also be written as

    LQ′′+RQ′+1CQ=E(t)LQ''+RQ'+\frac{1}{C} Q=E(t)

    Which is a second order, constant coefficient, non-homogeneous differential equation.

    Concept

    Introduction to Applications of Second Order Differential Equations

    Second order differential equations play a crucial role in modeling various real-world phenomena, particularly in the fields of mechanical vibrations and electrical circuits. These equations provide powerful tools for analyzing and predicting the behavior of complex systems. In mechanical vibrations, second order differential equations are used to describe the motion of objects under the influence of forces, such as springs and dampers. They help engineers design everything from suspension systems in vehicles to earthquake-resistant structures. In electrical circuits, these equations model the flow of current and voltage in systems with inductors and capacitors, enabling the design of filters, oscillators, and other essential components. The introduction video accompanying this section offers a visual and intuitive understanding of these applications, demonstrating how second order differential equations bridge the gap between mathematical theory and practical engineering problems. By mastering these concepts, students gain valuable insights into the fundamental principles governing many physical systems, preparing them for advanced studies in engineering and applied mathematics.

    FAQs

    Here are some frequently asked questions about applications of second order differential equations:

    1. What are the main applications of second order differential equations?

      Second order differential equations are widely used in modeling mechanical vibrations and electrical circuits. In mechanical systems, they describe the motion of objects under the influence of forces, such as springs and dampers. In electrical circuits, they model the behavior of current and voltage in systems with inductors and capacitors.

    2. How do second order differential equations relate to mechanical vibrations?

      In mechanical vibrations, these equations describe the motion of objects in terms of displacement, velocity, and acceleration. They incorporate factors like mass, spring constant, and damping coefficient to predict how a system will oscillate or return to equilibrium when disturbed.

    3. What is the significance of damping in mechanical systems?

      Damping is crucial in controlling oscillations in mechanical systems. It determines how quickly the system returns to equilibrium after being disturbed. The damping ratio, derived from the second order differential equation, indicates whether a system is underdamped (oscillatory), critically damped (fastest return to equilibrium), or overdamped (non-oscillatory return).

    4. How are second order differential equations applied in electrical circuits?

      In electrical circuits, particularly RLC circuits, these equations describe the relationship between current, voltage, and time. They help in analyzing circuit behavior, determining resonant frequencies, and designing filters and oscillators. The equation typically involves terms related to resistance (R), inductance (L), and capacitance (C).

    5. What methods are commonly used to solve second order differential equations?

      Common methods include the characteristic equation method, method of undetermined coefficients, variation of parameters, and Laplace transforms. For complex systems, numerical methods like Runge-Kutta are often employed. The choice of method depends on the specific form of the equation and the type of system being analyzed.

    Prerequisites

    Understanding the applications of second order differential equations is crucial in various fields of science and engineering. However, to fully grasp this complex topic, it's essential to have a solid foundation in several prerequisite concepts. These fundamental topics provide the necessary context and tools to tackle more advanced problems in differential equations.

    One of the most important prerequisites is Newton's Second Law of motion. This fundamental principle of physics forms the basis for many applications of second order differential equations, particularly in mechanics and dynamics. By understanding how forces relate to acceleration, students can more easily interpret and solve equations describing real-world systems.

    Another crucial concept is simple harmonic motion, often studied in the context of spring systems. This topic introduces students to oscillatory behavior, which is frequently modeled using second order differential equations. Familiarity with the trigonometric functions involved in simple harmonic motion helps in visualizing and analyzing more complex oscillatory systems.

    When dealing with more advanced applications, knowledge of Laplace transforms becomes invaluable. This mathematical technique allows for the transformation of differential equations into algebraic equations, simplifying the solution process for certain types of problems. Understanding Laplace transforms can greatly enhance a student's ability to solve and analyze complex systems described by second order differential equations.

    The method of undetermined coefficients is another essential tool in the arsenal of techniques for solving second order differential equations. This method is particularly useful for finding particular solutions to non-homogeneous equations, which are common in many practical applications. Mastery of this technique allows students to approach a wide range of problems with confidence.

    By building a strong foundation in these prerequisite topics, students can more effectively navigate the complexities of second order differential equations and their applications. Each concept contributes uniquely to the overall understanding: Newton's Second Law provides physical intuition, simple harmonic motion introduces oscillatory behavior, Laplace transforms offer powerful analytical tools, and the method of undetermined coefficients expands problem-solving capabilities.

    As students progress in their studies, they'll find that these prerequisite topics are not just isolated concepts but interconnected ideas that form a comprehensive framework for understanding and applying second order differential equations. This holistic approach to learning ensures that students are well-prepared to tackle real-world problems in fields such as physics, engineering, and applied mathematics, where second order differential equations play a crucial role in modeling and analysis.