Receptance Coupling

Abstract

In Chaps. 1–9 we discussed discrete and continuous beam models that can be used to describe the behavior of vibrating systems. We also detailed experimental techniques that we can use to identify these models. In this chapter, we will introduce an approach to combine models or measurements of individual components in order to predict the assembly’s frequency response function (FRF). This method is referred to as receptance coupling [1]; recall from Sect. 7.1 that a receptance is a type of FRF.

I have made this letter longer than usual,

only because I have not had the time to make it shorter.

—Blaise Pascal

Exercises

• 1. Determine the direct frequency response function, $\frac{X_2}{F_2}$, for the two degree of freedom system shown in Fig. P10.1 using receptance coupling. Express your final result as a function of m, c, k, and the excitation frequency, ω. You may assume a harmonic forcing function, F₂e^iωt, is applied to coordinate X₂.

• 2. Determine the direct frequency response function, $\frac{X_1}{F_1}$, for the two degree of freedom system shown in Fig. P10.2 using receptance coupling. Express your final result as a function of m, c, k, and the excitation frequency, ω. You may assume a harmonic forcing function, F₁e^iωt, is applied to coordinate X₁.

• 3. Use receptance coupling to rigidly join two free-free beams and find the free-free assembly’s displacement-to-force tip receptance. Both steel cylinders are described by the following parameters: 12.7 mm diameter, 100 mm length, 200 GPa elastic modulus, and 7800 kg/m³ density. Assume a solid damping factor of 0.0015. Once you have determined the assembly response, verify your result against the displacement-to-force tip receptance for a 12.7 mm diameter, 200 mm long free-free steel cylinder with the same material properties. Select a frequency range that encompasses the first three bending modes and display your results as the magnitude (in m/N) versus frequency (in Hz) using a semi-logarithmic scale.

• 4. Plot the displacement-to-force tip receptance for a sintered carbide cylinder with free-free boundary conditions. The beam is described by the following parameters: 19 mm diameter, 150 mm length, 550 GPa elastic modulus, and 15,000 kg/m³ density. Assume a solid damping factor of 0.002. Select a frequency range that encompasses the first three bending modes and display your results as magnitude (m/N) vs. frequency (Hz) in a semi-logarithmic format.

• 5. Determine the fixed-free displacement-to-force tip receptance for a sintered carbide cylinder by coupling the free-free receptances to a rigid wall (with zero receptances). The beam is described by the following parameters: 19 mm diameter, 150 mm length, 550 GPa elastic modulus, and 15,000 kg/m³ density. Assume a solid damping factor of 0.002. Select a frequency range that encompasses the first two bending modes and display your results as magnitude (m/N) vs. frequency (Hz) in a semi-logarithmic format. Verify your result by comparing it to the displacement-to-force tip receptance for a fixed-free beam with the same dimensions and material properties.

• 6. For a rigid coupling between two component coordinates x_1a and x_1b, the compatibility condition is _____________.

• 7. For a flexible coupling (spring stiffness k) between two component coordinates x_1a and x_1b, the compatibility condition is _____________. An external force is applied to the assembly at coordinate X_1a.

• 8. For a flexible-damped coupling (spring stiffness k and damping coefficient c) between two component coordinates x_1a and x_1b, the compatibility condition is _____________. An external force is applied to the assembly at coordinate X_1a.

• 9. What are the units for the rotation-to-couple receptance, p_ij, used to describe the transverse vibration of beams?

• 10. What are the (identical) units for the displacement-to-couple, l_ij, and rotation-to-force, n_ij, receptances used to describe the transverse vibration of beams?

Fig. P10.1

Two degree of freedom assembly

Fig. P10.2

Flexible damped coupling of mass (I) to spring-mass-damper (II) to form the two degree of freedom assembly III