Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 23

Develop the appropriate dimensionless paremeters for the period \(\tau\) of transverse vibration of a turbine rotor of mass \(m\) connected to a shaft of stiffness \(k \doteq F / L\) and length \(\ell \doteq L\). Other relevant dimensional parameters are the eccentricity \(\varepsilon \doteq L\) of the center of mass of the rotor, the speed of rotation \(N \doteq 1 / T\) of the shaft, and the amplitude \(A \doteq L\) of the vibration.

Short Answer

Expert verified
The dimensionless parameters formed for the problem are \(\tau N\), \(m k / \ell^2\), and \(A \varepsilon / \ell^2\).

Step by step solution

01

Identify physical quantities with dimensions

The physical quantities given in the problem are period of vibration \(\tau\), mass of the rotor \(m\), stiffness of the shaft \(k\), length of the shaft \(\ell\), eccentricity of the center of mass \(\varepsilon\), speed of rotation \(N\) and amplitude of vibration \(A\). All these quantities have dimensions except for \(N\) which is frequency and has unit \(1 / T\).
02

Express quantities in basic units

The basic units involved are mass \(M\), length \(L\) and time \(T\). The mass \(m\) is already in basic units. The stiffness \(k = F / L\) can be expressed in basic units as \(ML / T^2\), where \(F\) is force with unit \(ML/T^2\). The length \(\ell\), eccentricity \(\varepsilon\) and amplitude \(A\) all have unit length \(L\). The period \(\tau\) has unit time \(T\). The speed of rotation \(N\) is inverse of time \(1 / T\).
03

Combine quantities to form dimensionless parameters

The objective of this step is to combine the physical quantities in such a way as to eliminate the units. Let's form a dimensionless parameter using the period of vibration \(\tau\) and the speed of rotation \(N\) as \(\tau N\), which are dimensionless since they cancel out each other's units \((T * 1/T)\). We can also form dimensionless parameter using the mass \(m\), stiffness \(k\), and the length \(\ell\) as \(m k / \ell^2 = m (ML / T^2) / L^2 = T^{-2}\), which is also dimensionless. Another possible dimensionless parameter is formed by \(A \varepsilon / \ell^2 = A * \varepsilon / L^2 = L^{2} / L^2 = 1\), which is also dimensionless.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The dimensionless parameters for a ball released and falling from rest in a fluid are $$C_{D}, \quad \frac{g t^{2}}{D}, \quad \frac{\rho}{\rho_{b}}, \quad \text { and } \quad \frac{V_{t}}{D}$$ where \(\left.C_{\mathbf{D}} \text { is a drag coefficient (assumed to be constant at } 0,4\right)\) \(g\) is the acceleration of gravity, \(D\) is the ball diameter, \(t\) is the time after it is released, \(\rho\) is the density of the fluid in which it is dropped, and \(\rho_{b}\) is the density of the ball. Ball 1 , an aluminum ball \(\left(\rho_{b_{1}}=2710 \mathrm{kg} / \mathrm{m}^{3}\right)\) having a diameter \(D_{1}=1.0 \mathrm{cm},\) is dropped in water \(\left(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}\right) .\) The ball velocity \(V_{\mathrm{t}}\) is \(0.733 \mathrm{m} / \mathrm{s}\) at \(t_{1}=\) 0.10 s. Find the corresponding velocity \(V_{2}\) and time \(t_{2}\) for ball 2 having \(D_{2}=2.0 \mathrm{cm}\) and \(\rho_{b_{2}}=\rho_{b_{1}} .\) Next, use the computed value of \(V_{2}\) and the equation of motion,$$\rho_{b} V_{g}-\frac{\mathrm{C}_{\mathrm{D}}}{2} \rho A V^{2}=\rho_{b} V \frac{d V}{d t}$$ where \(Y\) is the volume of the ball and \(A\) is its cross-sectional area. to verify the value of \(t_{2}\). Should the two values of \(t_{2}\) agree?

You are to conduct wind tunnel testing of a new football design that has a smaller lace height than previous designs (see Videos \(V 6.1\) and \(V 6.2\) ). It is known that you will need to maintain Re and St similarity for the testing. Based on standard college quarterbacks, the prototype parameters are set at \(V=40 \mathrm{mph}\) and \(\alpha=300 \mathrm{rpm},\) where \(V\) and \(\omega\) are the speed and angular velocity of the football. The prototype football has a 7 -in. diameter. Due to instrumentation required to measure pressure and shear stress on the surface of the football, the model will require a length scale of 2: 1 (the model will be larger than the prototype). Determine the required model free stream velocity and model angular velocity.

Consider a typical situation involving the flow of a fluid that you encounter almost every day. List what you think are the important physical variables involved in this flow and determine an appropriate set of pi terms for this situation.

As winds blow past buildings, complex flow patterns can develop due to various factors such as flow separation and interactions between adjacent buildings. (See Video \(\vee 7.13\).) Assume that the local gage pressure, \(p\), at a particular location on a building is a function of the air density, \(\rho,\) the wind speed, \(V\), some characteristic length, \(\ell,\) and all other pertinent lengths, \(\ell_{i},\) needed to characterize the geometry of the building or building complex. (a) Determine a suitable set of dimensionless parameters that can be used to study the pressure distribution. (b) An eight-story building that is \(100 \mathrm{ft}\) tall is to be modeled in a wind tunnel. If a length scale of 1: 300 is to be used, how tall should the model building be? (c) How will a measured pressure in the model be related to the corresponding prototype pressure? Assume the same air density in model and prototype. Based on the assumed variables, does the model wind speed have to be equal to the prototype wind speed? Explain.

A vapor bubble rises in a liquid. The relevant dimensional parameters are the liquid specific weight, \(\gamma_{\ell},\) the vapor specific weight, \(\gamma_{\nu},\) bubble velocity, \(V,\) bubble diameter, \(d,\) surface tension, \(\sigma,\) and liquid viscosity, \(\mu .\) Find appropriate dimensionless parameters.
See all solutions

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Read Explanation

Astrophysics

Read Explanation

Fields in Physics

Read Explanation

Rotational Dynamics

Read Explanation

Kinematics Physics

Read Explanation

Electromagnetism

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free