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Chapter 5: Problem 4

Problems deal with the mass-and-spring system shown in Fig. 5.3.11 with stiffness matrix $$ \mathbf{K}=\left[\begin{array}{cc} -\left(k_{1}+k_{2}\right) & k_{2} \\ k_{2} & -\left(k_{2}+k_{3}\right) \end{array}\right] $$ and with the given mks values for the masses and spring constants. Find the two natural frequencies of the system and describe its two natural modes of oscillation. $$ m_{1}=m_{2}=1 ; k_{1}=1, k_{2}=2, k_{3}=1 $$

Short Answer

Expert verified
The natural frequencies are 1 and \(\sqrt{5}\). The modes are in-phase and out-of-phase oscillations.

Step by step solution

01

Stiffness Matrix Substitution

First, substitute the given values of the spring constants \(k_1\), \(k_2\), and \(k_3\) into the stiffness matrix \(\mathbf{K}\). We have:\[k_1 = 1, \ k_2 = 2, \ k_3 = 1\]Substitute into:\[\mathbf{K} = \begin{bmatrix} -(k_1 + k_2) & k_2 \ k_2 & -(k_2 + k_3) \end{bmatrix} = \begin{bmatrix} -(1 + 2) & 2 \ 2 & -(2 + 1) \end{bmatrix} = \begin{bmatrix} -3 & 2 \ 2 & -3 \end{bmatrix}\]
02

Mass Matrix

For simplicity, because both masses \(m_1\) and \(m_2\) are equal to 1, the mass matrix \(\mathbf{M}\) is:\[\mathbf{M} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]
03

Solve the Eigenvalue Problem

The problem requires solving the eigenvalue problem \( \mathbf{K} \mathbf{x} = \lambda \mathbf{M} \mathbf{x} \) to find the system's frequencies:First solve for eigenvalues \( \lambda \) of the matrix \( \mathbf{K} - \lambda \mathbf{M} = 0 \):\[\begin{vmatrix} -3 - \lambda & 2 \ 2 & -3 - \lambda \end{vmatrix} = 0\]
04

Eigenvalue Calculation

Calculate the determinant to find \( \lambda \):\[(-3 - \lambda)(-3 - \lambda) - (2)(2) = 0\]\[\lambda^2 + 6\lambda + 9 - 4 = 0 \]\[\lambda^2 + 6\lambda + 5 = 0\]Solving this quadratic equation gives:\[\lambda = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}\]\[\lambda = \frac{-6 \pm \sqrt{36 - 20}}{2} = \frac{-6 \pm \sqrt{16}}{2} = \frac{-6 \pm 4}{2}\]\[\lambda_1 = -1, \quad \lambda_2 = -5\]
05

Natural Frequencies

Natural frequencies are the square roots of the absolute values of the eigenvalues \( \lambda \):\[\omega_1 = \sqrt{|\lambda_1|} = \sqrt{1} = 1, \quad \omega_2 = \sqrt{|\lambda_2|} = \sqrt{5} \]
06

Natural Modes of Oscillation

Determine the eigenvectors corresponding to each eigenvalue to describe the natural modes of oscillation. For \( \lambda_1 = -1 \), solve:\[\begin{bmatrix} -2 & 2 \ 2 & -2 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = 0\]One solution is \( x_1 = 1, x_2 = 1 \) (both masses moving in-phase).For \( \lambda_2 = -5 \), solve:\[\begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = 0\]One solution is \( x_1 = 1, x_2 = -1 \) (masses moving out-of-phase).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass-and-Spring System
The mass-and-spring system is a fundamental model used to describe phenomena in mechanics and engineering. It consists of two masses connected by springs, which can compress or extend. In mechanics, this setup is often used to study oscillations and vibrations.
  • Components: Two masses (\(m_1 = m_2 = 1\)) are connected in a series via three springs with spring constants \(k_1 = 1\), \(k_2 = 2\), and \(k_3 = 1\).
  • Stiffness Matrix: The system's stiffness matrix, \(\mathbf{K}\), expresses how the masses resist deformation under force. Based on the given spring constants, the matrix becomes:\[\mathbf{K} = \begin{bmatrix} -3 & 2 \ 2 & -3 \end{bmatrix}\]
Each element in \(\mathbf{K}\) corresponds to the influence the system's springs have on the masses' motion. Understanding this relationship is crucial for calculating the natural frequencies and modes of oscillation.
Eigenvalue Problem
The eigenvalue problem is at the heart of analyzing oscillations in the mass-and-spring system. It involves finding characteristic values (eigenvalues) that reveal the system鈥檚 natural properties.
  • Basic Concept: To solve the eigenvalue problem, you need to solve the equation \(\mathbf{K} \mathbf{x} = \lambda \mathbf{M} \mathbf{x}\), where \(\mathbf{M}\) is the mass matrix. With both masses as 1, \(\mathbf{M}\) simplifies to the identity matrix:\[\mathbf{M} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]
  • Finding Eigenvalues: Solve for \(\lambda\) such that \(\det(\mathbf{K} - \lambda \mathbf{M}) = 0\). This leads to the characteristic equation:\[\lambda^2 + 6\lambda + 5 = 0\]
  • Solving the Equation: The roots of this equation, \(\lambda_1 = -1\) and \(\lambda_2 = -5\), are the eigenvalues.
These eigenvalues help determine the system's natural frequencies, as they are directly related to how the system responds to external inputs.
Natural Frequencies
Natural frequencies are the rates at which a system oscillates in the absence of external forces. They are derived from the eigenvalues of the system's matrix equation.
  • Calculation: The natural frequencies, denoted \(\omega\), are the square roots of the absolute values of the eigenvalues:\[\omega_1 = \sqrt{|\lambda_1|} = 1, \quad \omega_2 = \sqrt{|\lambda_2|} = \sqrt{5}\]
  • Physical Meaning: These frequencies determine how fast the system oscillates once it鈥檚 set in motion. \(\omega_1\) corresponds to a slower mode and \(\omega_2\) to a faster oscillation.
Understanding the natural frequencies is essential for predicting the behavior of the system under vibration and is critical for designing systems to avoid resonance.
Oscillation Modes
Oscillation modes, also known as normal modes, describe the specific patterns of motion that the system can naturally adopt while oscillating.
  • In-Phase Mode: For the eigenvalue \(\lambda_1 = -1\), the corresponding eigenvector indicates that both masses move in the same direction, or in-phase:\[\begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 1 \ 1 \end{bmatrix}\]
    This describes both masses rising and falling together.
  • Out-of-Phase Mode: For \(\lambda_2 = -5\), the solution indicates an out-of-phase movement:\[\begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 1 \ -1 \end{bmatrix}\]
    This pattern reflects one mass moving up while the other moves down.
These modes are fundamental because they depict all possible independent movements of the system, illustrating how complex vibrations can be broken down into simpler, understandable parts.

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Most popular questions from this chapter

Apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. $$ x_{1}^{T}=2 x_{1}+3 x_{2}, \quad x_{2}^{\prime}=2 x_{1}+x_{2} $$

The characteristic equation of the coefficient matrix \(\mathbf{A}\) of the system $$ \mathbf{x}^{\prime}=\left[\begin{array}{rrrr} 3 & -4 & 1 & 0 \\ 4 & 3 & 0 & 1 \\ 0 & 0 & 3 & -4 \\ 0 & 0 & 4 & 3 \end{array}\right] \mathbf{x} $$ is $$ \phi(\lambda)=\left(\lambda^{2}-6 \lambda+25\right)^{2}=0 $$ Therefore, \(\mathrm{A}\) has the repeated complex conjugate pair \(3 \pm 4 i\) of eigenvalues. First show that the complex vectors \(\mathbf{v}_{1}=\left[\begin{array}{llll}1 & i & 0 & 0\end{array}\right]^{T} \quad\) and \(\quad \mathbf{v}_{2}=\left[\begin{array}{llll}9 & 0 & 1 & i\end{array}\right]^{T}\) form a length 2 chain \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\) associated with the eigenvalue \(\lambda=3-4 i\). Then calculate the real and imaginary parts of the complex-valued solutions $$ \mathbf{v}_{1} e^{\lambda, t} \quad \text { and } \quad\left(\mathbf{v}_{1} t+\mathbf{v}_{2}\right) e^{\lambda t} $$ to find four independent real-valued solutions of \(\mathbf{x}^{\prime}=\mathrm{Ax}\).

Use a calculator or computer system to calculate the eigenvalues and eigenvectors (as illustrated in the \(5.2\) Application below) in order to find a general solution of the linear system \(\mathbf{x}^{\prime}=\mathbf{A x}\) with the given coefficient \(\operatorname{matrix} \mathbf{A} .\) $$ \mathbf{A}=\left[\begin{array}{rrr} -20 & 11 & 13 \\ 12 & -1 & -7 \\ -48 & 21 & 31 \end{array}\right] $$

Apply the method of undetermined coefficients to find a par ticular solution of each of the systems in Problems. If initial conditions are given, find the particular solution that satisfies these conditions. Primes denote derivatives with respect to \(t\). $$ x^{\prime}=x-5 y+2 \sin t, y^{\prime}=x-y-3 \cos t $$

Apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. $$ x_{1}^{\prime}=x_{1}-5 x_{2}, \quad x_{2}^{\prime}=x_{1}-x_{2} $$
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