/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 The period \(T\) of vibration of... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 23

The period \(T\) of vibration of a beam is a function of its length \(L,\) area moment of inertia \(I,\) modulus of elasticity \(E,\) density \(\rho,\) and Poisson's ratio \(\sigma .\) Rewrite this relation in dimensionless form. What further reduction can we make if \(E\) and \(I\) can occur only in the product form EI? Hint: Take \(L, \rho,\) and \(E\) as repeating variables.

Short Answer

Expert verified
The dimensionless groups are \(\Pi_1 = \frac{T}{\sqrt{\frac{L^3}{E}}}\) and \(\Pi_2 = \frac{L \rho}{E}\).

Step by step solution

01

Identify Variables and Their Dimensions

Determine the dimensions of the given variables.- Length \(L\): \([L]\)- Area moment of inertia \(I\): \([L^4]\)- Modulus of elasticity \(E\): \([ML^{-1}T^{-2}]\)- Density \(\rho\): \([ML^{-3}]\)- Poisson's ratio \(\sigma\): dimensionless- Period \(T\): \([T]\)
02

Select Repeating Variables

Use the hint to choose the repeating variables as \(L, \rho, E\). These variables will help us form the dimensionless groups.- \(L\): \([L]\)- \(\rho\): \([ML^{-3}]\)- \(E\): \([ML^{-1}T^{-2}]\)
03

Write Pi Terms

Formulate Pi terms using these repeating variables and the other variables \(I\) and \(T\):For the period \(T\), \[ \Pi_1 = T L^a \rho^b E^c \]For moment of inertia \(I\),\[ \Pi_2 = I L^d \rho^e E^f \]
04

Solve for Exponents in Pi Terms

Balance the dimensions for each \(\Pi_i\) such that the overall dimension is dimensionless, i.e., \([\Pi] = [1]\).For \(\Pi_1\):\[[T]^1 [L]^a [M^b L^{-3b}] [M^c L^{-c} T^{-2c}] = [1]\]Equate the powers of \(M, L, T\) to zero:- M: \(b + c = 0\)- L: \(a - 3b - c = 0\)- T: \(1 - 2c = 0\)From \(1 - 2c = 0\), we get \(c = \frac{1}{2}\).From \(b + c = 0\), \(b = -\frac{1}{2}\).Substitute \(b, c\) into \(a - 3b - c = 0\):\(a + \frac{3}{2} - \frac{1}{2} = 0 \Rightarrow a = -1\).
05

Construct Pi Term for I and Rewrite Dimensionless Equation

Now construct \(\Pi_2\) for \(I\):\[ \Pi_2 = I L^d \rho^e E^f \]Follow similar steps to solve for \(d, e, f\):\[[L^4] [L^d] [M^e L^{-3e}] [M^f L^{-f}] = [1]\] Equate the powers of \(M, L, T\) to zero:- M: \(e + f = 0\)- L: \(4 + d - 3e - f = 0\)Substituting \(f = -e\) into the equation for \(L\):\[4 + d - 3e + e = 0 \Rightarrow 4 + d - 2e = 0\]Choose convenient values; solve \(d = -4, e = 2\), then \(f = -2\).The dimensionless groups are then: \(\Pi_1 = \frac{T}{\sqrt{\frac{L^3}{E}}}\), \(\Pi_2 = \frac{I \rho^2}{E^2 L^4}\).
06

Further Simplification Using EI Only

Given that \(E\) and \(I\) only occur together in the form of \(EI\), simplify \(\Pi_2\):If \(EI\) occurs together, replace it in \(\Pi_2\):\[ \Pi_2 = \frac{L \rho}{E} \]And our dimensionless relation becomes: \(\Pi_1 = \phi(\Pi_2)\), where \(\Pi_1 = \frac{T}{\sqrt{\frac{L^3}{E}}}\) and \(\Pi_2 = \frac{L \rho}{E}\).

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Key Concepts

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

Area Moment of Inertia
The area moment of inertia, often simply called "moment of inertia," is a critical concept in structural engineering and mechanics. It refers to the geometric distribution of a cross-sectional area and its orientation relative to a given axis.
This distribution indicates how the area is spread out and affects the flexural capacity of a structure. The larger the moment of inertia, the greater the beam or column's ability to resist bending and deflection.
  • Units and Dimensions: The area moment of inertia is expressed in units of \([L^4]\). Examples include cubic inches (in鈦�) or meters to the fourth power (m鈦�), widely used in engineering calculations.
  • Calculation: For a simple rectangular section, the moment of inertia about an axis through its base is calculated using the formula \( I = \frac{1}{3} bh^3 \), where \(b\) is the base width and \(h\) is the height.
  • Significance: Higher values of the moment of inertia indicate a stiffer structure that is better at resisting bending forces, essential for designing beams, bridges, and more to ensure they can bear loads without excessive deformation.
Modulus of Elasticity
The modulus of elasticity, often referred to as Young's modulus, ties into how materials deform under stress. It reflects the stiffness of a material and its ability to return to its original shape after deformation.
In essence, it quantifies the material's elasticity by illustrating the relationship between stress and strain.
  • Units and Dimensions: Modulus of elasticity is measured in Pascals (Pa) in the SI system, often expressed in gigapascals (GPa). Its dimensions are \( [ML^{-1}T^{-2}] \), derived from force per unit area.
  • Relevance in Structures: Materials with a high modulus of elasticity are stiffer, meaning they will not deform easily under load. Metals like steel have a high modulus, making them ideal for structures requiring strength and rigidity.
  • Understanding the Impact: In design, engineers use the modulus of elasticity to determine how much a material will stretch or compress under given loads, aiding in decisions about material suitability for specific engineering applications.
Density in Mechanics
Density plays a crucial role in mechanics, linking to various properties of materials and their behavior in structural applications. It represents the mass per unit volume of a material, affecting factors like weight, buoyancy, and stability.
Knowing the density of materials helps predict how they will perform under different conditions.
  • Units and Dimensions: Measured in kilograms per cubic meter (kg/m鲁) in the metric system, density has the dimensional formula \( [ML^{-3}] \).
  • Importance in Engineering: Understanding density is crucial for material selection, especially for structures where weight and balance are essential considerations, such as in aerospace or marine engineering.
  • Applications: Density assists in calculating the overall weight of structures and in finding the distribution of mass, which is vital for understanding structural vibrations, stability under loads, and thermal properties.

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Most popular questions from this chapter

In turbulent flow near a flat wall, the local velocity \(u\) varies only with distance \(y\) from the wall, wall shear stress \(\tau_{w},\) and fluid properties \(\rho\) and \(\mu .\) The following data were taken in the University of Rhode Island wind tunnel for airflow, \(\rho=0.0023\) slug \(/ \mathrm{ft}^{3}, \mu=3.81 \mathrm{E}-7 \mathrm{slug} /(\mathrm{ft} \cdot \mathrm{s})\) and \(\tau_{w}=0.029 \mathrm{lbf} / \mathrm{ft}^{2}\) $$\begin{array}{l|l|l|l|l|l|l} y, \text { in } & 0.021 & 0.035 & 0.055 & 0.080 & 0.12 & 0.16 \\ \hline u, \mathrm{ft} / \mathrm{s} & 50.6 & 54.2 & 57.6 & 59.7 & 63.5 & 65.9 \end{array}$$ (a) Plot these data in the form of dimensionless \(u\) versus dimensionless \(y,\) and suggest a suitable power-law curve fit. (b) Suppose that the tunnel speed is increased until \(u=90\) ft/s at \(y=0.11\) in. Estimate the new wall shear stress, in \(1 \mathrm{bf} / \mathrm{ft}^{2}\)

A long, slender, smooth 3 -cm-diameter flagpole bends alarmingly in \(20 \mathrm{mi} / \mathrm{h}\) sea-level winds, causing patriotic citizens to gasp. An engineer claims that the pole will bend less if its surface is deliberately roughened. Is she correct, at least qualitatively?

An East Coast estuary has a tidal period of \(12.42 \mathrm{h}\) (the semidiurnal lunar tide) and tidal currents of approximately \(80 \mathrm{cm} / \mathrm{s} .\) If a one-five-hundredth-scale model is constructed with tides driven by a pump and storage apparatus, what should the period of the model tides be and what model current speeds are expected?

The pressure drop per unit length \(\Delta p / L\) in smooth pipe flow is known to be a function only of the average velocity \(V,\) diameter \(D,\) and fluid properties \(\rho\) and \(\mu .\) The following data were obtained for \(\mathrm{Aw}\) of water at \(20^{\circ} \mathrm{C}\) in an 8 -cm-diameter pipe \(50 \mathrm{m}\) long: $$\begin{array}{l|l|l|l|l} Q, \mathrm{m}^{3} / \mathrm{s} & 0.005 & 0.01 & 0.015 & 0.020 \\ \hline \Delta p, \mathrm{Pa} & 5800 & 20,300 & 42,100 & 70,800 \end{array}$$ Verify that these data are slightly outside the range of Fig. \(5.10 .\) What is a suitable power-law curve fit for the present data? Use these data to estimate the pressure drop for Aw of kerosene at \(20^{\circ} \mathrm{C}\) in a smooth pipe of diameter \(5 \mathrm{cm}\) and length \(200 \mathrm{m}\) if the flow rate is \(50 \mathrm{m}^{3} / \mathrm{h}\)

The rate of heat loss \(\dot{Q}_{\text {loss }}\) through a window or wall is a function of the temperature difference between inside and outside \(\Delta T,\) the window surface area \(A,\) and the \(R\) value of the window, which has units of \(\left(\mathrm{ft}^{2} \cdot \mathrm{h} \cdot^{\circ} \mathrm{F}\right) /\) Btu. (a) Using the Buckingham Pi Theorem, find an expression for rate of heat loss as a function of the other three parameters in the problem. \((b)\) If the temperature difference \(\Delta T\) doubles, by what factor does the rate of heat loss increase?
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