Problem 33 - Advanced Engineering Mathematics

Question

When a mass of 2 lilograms is attached to a spring whose constant is \(32 \mathrm{~N} / \mathrm{m}\), it comes to rest in the equilibrium position. Starting at \(t=0\), a force equal to \(f(t)=68 e^{-2 t} \cos 4 t\) is applied to the system. Find the equation of motion in the absence of damping.

Step-by-Step Solution

Answer

The motion equation is \( x(t) = C_1 \cos(4t) + C_2 \sin(4t) - \frac{17}{20}e^{-2t} \sin 4t \).

Step 1: Identify the Type of System

This problem involves a spring-mass system with a given force applied, without damping resistance. The system can be described by a second-order differential equation involving harmonic motion.

Step 2: Write the Differential Equation

The equation for a forced undamped mass-spring system is given by \( m\frac{d^2x}{dt^2} + kx = f(t) \), where \( m \) is mass, \( k \) is the spring constant, and \( f(t) \) is the external force. Substituting the values: \( m = 2 \text{ kg}, k = 32 \text{ N/m}, f(t) = 68e^{-2t}\cos 4t \). Thus, the equation becomes \( 2\frac{d^2x}{dt^2} + 32x = 68e^{-2t}\cos 4t \).

Step 3: Simplify the Differential Equation

Divide the entire equation by 2 to simplify. This results in \( \frac{d^2x}{dt^2} + 16x = 34e^{-2t}\cos 4t \).

Step 4: Solve the Homogeneous Equation

First, solve the homogeneous equation \( \frac{d^2x}{dt^2} + 16x = 0 \) which represents the system without the external force. The characteristic equation is \( r^2 + 16 = 0 \), giving roots \( r = \pm 4i \). This results in the general solution \( x_h(t) = C_1 \cos(4t) + C_2 \sin(4t) \).

Step 5: Solve the Particular Solution

To solve the particular solution, propose \( x_p(t) = e^{-2t}(A \cos 4t + B \sin 4t) \). Substitute \( x_p(t) \) into the differential equation and compare coefficients to solve for \( A \) and \( B \). After simplifying, you find \( A = 0 \) and \( B = -\frac{34}{40} = -\frac{17}{20} \). This means the particular solution is \( x_p(t) = -\frac{17}{20}e^{-2t}\sin 4t \).

Step 6: Combine Solutions for Complete Equation of Motion

The total solution is the sum of the homogeneous and particular solutions: \( x(t) = x_h(t) + x_p(t) = C_1 \cos(4t) + C_2 \sin(4t) - \frac{17}{20}e^{-2t} \sin 4t \).

Step 7: Initial Conditions for Full Solution

The initial conditions are not given to determine \( C_1 \) and \( C_2 \). Normally, they are found from specific initial position/velocity, but absent specific conditions, this part remains as a general solution.

Key Concepts

Differential Equations

Differential equations play a crucial role in modeling real-world physical systems, such as the spring-mass system described in the original exercise. A differential equation involves derivatives and provides a way to relate a function with its rates of change.

For the spring-mass system, we examine a second-order differential equation, as it involves the second derivative of displacement with respect to time. This corresponds to the acceleration of the mass. The general form for this type of system is given by:

• \( m\frac{d^2x}{dt^2} + kx = f(t) \)
• Where \( m \) represents the mass, \( k \) is the spring constant, and \( f(t) \) is the external force applied.

In the context of the spring-mass problem addressed, this equation helps to describe how the position of the mass changes over time due to external forces.

Spring-Mass System

The spring-mass system is a mechanical model that consists of a mass attached to a spring. It is a classical physics problem that helps us understand harmonic motion.

Key elements of the spring-mass system are:

• The mass, \( m \), which in this case, is 2 kg.
• The spring constant, \( k \), which determines the stiffness of the spring. Here, \( k = 32 \text{ N/m} \).
• An external applied force, \( f(t) \), which influences the system. In the exercise, it is given as \( 68e^{-2t}\cos 4t \).

The motion without any dampening is described by examining the forces acting on the mass and using Newton’s second law of motion.

Particular Solution

Finding a particular solution is essential to solve non-homogeneous differential equations, where there's an external force, \( f(t) \).

The particular solution accounts for how the external force alters the motion of the system.

Proposing a Solution

We propose a form for the particular solution aligning with the form of the external force:

• \( x_p(t) = e^{-2t}(A \cos 4t + B \sin 4t) \)
• This proposal helps simplify incorporating the force \( f(t) \) into our differential equation solution.

By substituting \( x_p(t) \) into the differential equation, we can solve for \( A \) and \( B \), which ultimately yields:

• \( x_p(t) = -\frac{17}{20}e^{-2t}\sin 4t \)

This represents the specific response of the system to the external force.

Characteristic Equation

The characteristic equation is pivotal in determining the homogeneous solution of a differential equation. It involves determining the roots that help define the system's intrinsic behavior.

For the homogeneous part (where \( f(t) = 0 \)), the equation is:

• \( \frac{d^2x}{dt^2} + 16x = 0 \)
• Which leads to the characteristic equation: \( r^2 + 16 = 0 \).

Solving the Characteristic Equation

The roots of this equation are \( r = \pm 4i \), indicating purely imaginary roots that represent oscillatory motion. These roots define the homogeneous solution as:

• \( x_h(t) = C_1 \cos(4t) + C_2 \sin(4t) \)
• This indicates a system that will continue oscillating at a natural frequency, governed by the spring and mass values.

This harmonic solution demonstrates the system's behavior without the influence of the external force.