Two Degree of Freedom Forced Vibration

Abstract

Let’s extend the two degree of freedom free vibration analysis from Chap. 4 to include externally applied forces so that we can analyze two degree of freedom forced vibration. The general case is that a separate harmonic force is applied at each coordinate; see Fig. 5.1. However, we are considering only linear systems, so we can apply superposition. This means that we can determine the system response due to each force separately and then sum the results to find the combined effect.

Our achievements of today are but the sum total of our thoughts of yesterday.

—Blaise Pascal

Exercises

• 1. A two degree of freedom spring-mass-damper system is shown Fig. P5.1. For harmonic forced vibration (due to the external force at coordinate x₂), complete the following if k₁ = 2 × 10⁵ N/m, c₁ = 60 N s/m, m₁ = 2.5 kg, k₂ = 5.5 × 10⁴ N/m, c₂ = 16.5 N s/m, and m₂ = 1.2 kg.

  • (a) Verify that proportional damping exists.

  • (b) Define the modal matrix and determine the modal mass, stiffness, and damping matrices. Note that the mode shapes should be normalized to the force location.

  • (c) Write expressions for the uncoupled single degree of freedom FRFs in modal coordinates, $\frac{Q_1}{R_1}$ and $\frac{Q_2}{R_2}$, and the direct FRF in local coordinates, $\frac{X_2}{F_2}$.

  • (d) Plot the real and imaginary parts of the direct FRF, $\frac{X_2}{F_2}$. Units should be m/N for the vertical axis and rad/s for the horizontal (frequency) axis. Use a frequency range of omega = 0:0.01:500; (rad/s).

• 2. A two degree of freedom spring-mass-damper system is shown in Fig. P5.2. For harmonic forced vibration (due to the external force applied at coordinate x₂), complete the following if k₁ = 8 × 10⁷ N/m, c₁ = 1000 N s/m, m₁ = 50 kg, k₂ = 5 × 10⁷ N/m, c₂ = 500 N s/m, and m₂ = 12 kg.

  • (a) Show that proportional damping does not exist.

  • (b) Write a symbolic expression for the direct FRF $\frac{X_2}{F_2}$ as a function of the frequency, ω, and mass, stiffness, and damping values, m_1,2, k_1,2, and c_1,2. Use the complex matrix inversion approach.

  • (c) Write a symbolic expression for the cross FRF $\frac{X_1}{F_2}$ as a function of the frequency, ω, and mass, stiffness, and damping values, m_1,2, k_1,2, and c_1,2. Use the complex matrix inversion approach.

  • (d) Plot the real and imaginary parts of the cross FRF, $\frac{X_1}{F_2}$. Units should be m/N for the vertical axis and rad/s for the horizontal (frequency) axis. Use a frequency range of omega = 0:0.01:3500; (rad/s).

• 3. Consider the single degree of freedom freedom spring-mass system shown in Fig. P5.3, where k = 4 × 10⁵ N/m and m = 8 kg. It is being excited by a harmonic forcing function, ${F}_1{e}^{i{\omega}_ft}$, at a frequency, ω_f.

  • (a) If the excitation frequency is 200 rad/s, design a dynamic absorber to eliminate the vibration at coordinate x₁. The only available spring for use in the absorber is identical to the one already used in the system.

  • (b) If the 4 × 10⁵ N/m absorber spring is used in conjunction with a 2 kg absorber mass, at what forced excitation frequency (in rad/s) will the steady-state vibration of coordinate x₁ be eliminated?

• 4. A two degree of freedom spring-mass-damper system is shown in Fig. P5.4. For harmonic forced vibration (due to the external force at coordinate x₁), complete the following if k₁ = 2 × 10⁵ N/m, c₁ = 60 N s/m, m₁ = 2.5 kg, k₂ = 5.5 × 10⁴ N/m, c₂ = 16.5 N s/m, and m₂ = 1.2 kg.

  • (a) Verify that proportional damping exists.

  • (b) Define the modal matrix and determine the modal mass, stiffness, and damping matrices. Note that the mode shapes should be normalized to the force location.

  • (c) Write expressions for the uncoupled single degree of freedom FRFs in modal coordinates, $\frac{Q_1}{R_1}$ and $\frac{Q_2}{R_2}$, and the direct FRF in local coordinates, $\frac{X_1}{F_1}$.

  • (d) Plot the real and imaginary parts of the direct FRF, $\frac{X_1}{F_1}$. Units should be m/N for the vertical axis and rad/s for the horizontal (frequency) axis. Use a frequency range of omega = 0:0.1:500; (rad/s).

• 5. For the two degree of freedom spring-mass-damper system shown in Fig. P5.5, complete the following if k_a = 2 × 10⁵ N/m, k_b = 5.5 × 10⁴ N/m, c_a = 60 N s/m, c_b = 16.5 N s/m, m_a = 2.5 kg, and m_b = 1.2 kg.

  • (a) Obtain the equations of motion in matrix form and transform them into modal coordinates q₁ and q₂. Normalize your eigenvectors to the force location, coordinate x₂. Verify that proportional damping exists.

  • (b) Determine the FRFs $\frac{Q_1}{R_1}$, $\frac{Q_2}{R_2}$, and $\frac{X_2}{F_2}$. Express them in equation form and then plot the real and imaginary parts (in m/N) vs. frequency (in rad/s). Use a frequency range of 0:0.1:600; (rad/s).

• 6. A dynamic absorber is to be designed to eliminate the vibration at coordinate x₁ for the system shown in Fig. P5.6, where the excitation frequency is 400 rad/s and the force magnitude is 100 N. For the given system constants, determine the values of the mass and spring constant for the dynamic absorber if the magnitude of vibration for the absorber mass is 5 mm.

• 7. Given the modal mass matrix, ${m}_q=\left[\begin{array}{cc}2& 0\\ {}0& 2\end{array}\right]$ kg, the modal stiffness matrix, ${k}_q=\left[\begin{array}{cc}5.858\times 1{0}^6& 0\\ {}0& 3.414\times 1{0}^7\end{array}\right]$ N/m, the modal matrix, $\left[P\right]=\left[\begin{array}{cc}0.707& -0.707\\ {}1& 1\end{array}\right]$, and the modal damping ratios, ζ_q1 = 0.04 and ζ_q2 = 0.02, complete the following.

  • (a) Plot the imaginary part (m/N) of the direct FRF $\frac{X_2}{F_2}$. Use a frequency range of 0:0.1:5000; (rad/s).

  • (b) Plot the imaginary part (in m/N) of the cross FRF $\frac{X_1}{F_2}$. Use a frequency range of 0:0.1:5000; (rad/s).

• 8. After installation, it was found that a particular machine exhibited excessive vibration due to a harmonic excitation force with a frequency of 100 Hz. A dynamic absorber was designed and added to the original system to attenuate this vibration. If the resulting vibration magnitude of the absorber mass was 2 mm at 100 Hz and the excitation force magnitude was 25 N, determine the stiffness of the spring (N/m) and mass (kg) used to construct the absorber. You may neglect damping in your analysis.

• 9. Given the eigenvalues and eigenvectors for the two degree of freedom system shown in Fig. P5.9, complete the following.

  $$ {s_1}^2=-1\times {10}^6\mathrm{rad}/{\mathrm{s}}^2\kern1em {s_2}^2=-7\times {10}^6\mathrm{rad}/{\mathrm{s}}^2 $$
  $$ {\psi}_1=\left\{\begin{array}{c}0.5\\ {}1\end{array}\right\}\kern1em {\psi}_2=\left\{\begin{array}{c}-2.5\\ {}1\end{array}\right\} $$

  • (a) Determine the modal matrices m_q (kg), c_q (N s/m), and k_q (N/m).

  • (b) Plot the imaginary part (in m/N) of the cross frequency response function, $\frac{X_1}{F_2}$. Use a frequency range of 0:0.1:3500; (rad/s).

• 10. Given the eigenvalues and eigenvectors for the two degree of freedom system shown in Fig. P5.9, determine the DC (zero frequency) compliance for the real part of the direct FRF $\frac{X_2}{F_2}$.

  $$ {s_1}^2=-1\times {10}^6\mathrm{rad}/{\mathrm{s}}^2\kern1em {s_2}^2=-7\times {10}^6\mathrm{rad}/{\mathrm{s}}^2 $$
  $$ {\psi}_1=\left\{\begin{array}{c}0.5\\ {}1\end{array}\right\}\kern1em {\psi}_2=\left\{\begin{array}{c}-2.5\\ {}1\end{array}\right\} $$

Fig. P5.1

Two degree of freedom spring-mass-damper system under forced vibration

Fig. P5.2

Two degree of freedom spring-mass-damper system under forced vibration

Fig. P5.3

Single degree of freedom system excited by the harmonic forcing function ${F}_1{e}^{i{\omega}_ft}$

Fig. P5.4

Two degree of freedom spring-mass-damper system under forced vibration

Fig. P5.5

Two degree of freedom spring-mass-damper system under forced vibration

Fig. P5.6

Single degree of freedom system excited by the harmonic forcing function 100e^i400t

Fig. P5.9

Two degree of freedom spring-mass-damper system under forced vibration