Solid mechanics Homework Help

Solid mechanics assignments help Problem 1 A cantilever beam has been constructed from a steel having Young’s modulus E = 208 GPa, Poisson ratio ν = 0.29, and tensile yield strength σy = 410 MPa. The length of the beam, from its base to its tip, is L = 1 m, and the uniform cross­section is rectangular, h = 5 mm thick and b = 30 mm wide.

Two axially­mounted strain gages have been mounted on the top surface (y = h/2).

The precise absolute axial position “x” is not known for either gage; however, it is known that the two gages are spaced an axial distance of l = 200 mm apart along the length of the beam.

A tip load of magnitude ‘P’ is applied to the cantilever, acting in a direction parallel to the y­axis, causing the two gages (#1, which is nearer the base of the cantilever, and #2, which is nearer its tip) to register the following strain values:

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Figure 1: Schematic of tip­loaded cantilever showing positions of strain gages numbers 1 and 2.

• Evaluate the load P. • Using the (now­known) value of P, find the axial locations (x1 and x2) of both gages.

Problem 2 There is current interest in the use of microfabricated cantilever beams in detecting the presence of bacteria in a liquid solution. Following fabrication of the cantilever, its surface is coated with an antibody that is specific to the presence of the bacterium of interest, and the free vibration characteristics of the cantilever (its first­mo de natural frequency, ω0) is recorded experimentally.

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Then the coated cantilever is exposed to a liquid medium. If the sought­for bacteria are present in solution, they will preferentially attach themselves to the antibody coating on the surface of the cantilever, in the process increasing the vibrating mass of the cantilever by an amount Δm = nb mb, where nb is the number of bacterium cells that attach, and mb is the mass of the bacterium cell.

Figure 2: Scanning electron microscope images of E. coli bacteria attached to various microfabricated resonating cantilever beams. (from: Ilic, et al., Applied Physics Letters, 77, #3, 2000, 450­452.

Subsequent testing of the added­mass cantilever should reveal a progressively decreasing natural frequency as more bacterium cells are added.

The cantilever of interest has been fabricated of silicon nitride, having a mass density of ρ = 3.1 × 103 kg/m3 and Young’s modulus of E = 100 GPa. It has a uniform rectangular cross section, of thickness h = 320 nm, width b = 15 µm, and total length L = 100 µm.

• Assuming that the antibody coating itself does not appreciably affect the natural frequency of the cantilever, estimate the first­mode natural frequency, ω0, of the cantilever

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in the absence of adhered bacteria. (Note: please refer to the Lab 1 Handout Notes on vibration of cantilever beams for relevant analysis.)

• Assume that the added mass Δm = nbmb is uniformly distributed along the length of the beam, and further, assume that the presence of the adhered bacteria does not affect the stiffness of the beam. Let the total mass of the coated beam be m = ρbhL; assuming that show that the change in frequency resulting from the added mass, Δω, can be expressed by

– Hint # 1: You might wish to show that the natural frequency of a uniform beam of total mass ‘m’ can be written as (neglecting dimensionless factors).

– Hint # 2: If the change in total maresulting frequency of the perturbed­mass system can be evaluated by taking a Taylss (Δm) is very small in comparison to the initial beam mass (m), then the or series expansion of the frequency expression about the reference mass.

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• Assume that the cantilever has been covered by nb = 100 bacterium cells.

From the figures, it appears that the diameter, Db, of each bacterium cell is near Db = 1µm; assuming each cell to be spherical and to have a mass density, ρb, equal to that of water, . justify an estimate of each cell’s mass as mb = 5.24 × 10−16 kg. Using this estimate, and assuming that the cell “added mass” is uniformly distributed over the surface of the beam, evaluate the change in natural frequency that you can expect to see for the bacterium­coated cantilever.

Problem 3. Atomic force microscopy resolves the structure of surfaces to near atomic­level resolution. A key structural element of an atomic force microscope (AFM) is a small cantilevered beam fabricated from a material such as silicon using lithographic technology (e.g., precision etching). The beam has a sharp­tipped diamond stylus at its tip, the stylus is tracked across the specimen surface, and very local surface interactions between the tip and the specimen surface cause the tip of the cantilever to deflect. The magnitude of the tip deflection is measured by reflecting a laser beam off its back side and imaging the location of the reflected light. The tip/surface forces are computed from the deflections using beam theory, and they are used to map the structure of the surface.

Figure 3 shows a small rectangular cross­section cantilever from an atomic force microscope (AFM) that has been machined from a single block of silicon using lithographic technology (e.g., precision etching). The nominal dimensions of this cantilever are length l = 460 µm, width w = 50 µm, and thickness t = 2.0 µm. Elastic constants of silicon are Young’s modulus E = 107 GPa, and Poisson ratio ν = 0.2.

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The stiffness k = P/δ of the AFM cantilever is an important design parameter.

• Compute the stiffness of this AFM cantilever.

• Unfortunately, dimensional tolerances in the etching technique inevitably lead to a range of dimensions over a population of the AFM cantilever products. The manufacturer quotes the following tolerances for this cantilever: l = 460 µm ± 5µm; w = 50 µm ± 3µm; and tmin = 1.5 µm ≤ t ≤ tmax = 2.5 µm. Accordingly, the stiffness of any given cantilever can vary from its nominal value, depending on its actual dimensions. Provide estimates of the minimum and maximum stiffnesses, kmin and kmax , that can be expected is this cantilever, corresponding to extreme ranges of the dimensions.

Problem 4 CDL Problem 8.15 Hint/suggestion: Problem 8.15a is the continuation of the “self­weight” cantilever problem we looked at in the lecture. You should be well on the way to solving for its moment distribution, and then you should be able to follow the procedures adopted for the tiploaded cantilever to calculate the corresponding lateral displacement distribution, v(x), and anything else needed.

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Problem 8.15(b) is statically indeterminant, meaning that there is no direct way to evaluate its bending moment distribution, M(x), without simultaneously considering the displacement distribution, v(x). There are two straightforward ways to approach such statically indeterminant beam problems.

• Combine CDL equations 3.11 and 3.12 to obtain

From here on, there are 4 integrations, 4 constants of integration, and 4 boundary conditions (on v(0), φ(0), M(L), and v(L)) to determine the constants. It will work, but the algebra is deadly dull.

• On the other hand, you can use the principle of superposition. In this case, observe that the stated problem is equivalent to a cantilever, subject to (a) the distributed loading q(x) = −w, AND to a concentrated upwards­directed tip load, of [unknown] magnitude RL. The total vertical tip displacement of the cantilever is vtotal (L) = 0 and by superposition,

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where it is understood that there is a downward tip displacement, vq(L) < 0 (from problem 8.15a), as well as an opposite­direction tip displacement, vRL > 0, from the unknown positive­upward tip reaction force, RL. This leads to a simple linear equation to solve for RL; once RL is obtained, all parts of the solution to 8.15b are known by superposition... Try whichever method suits you.

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Problem 1

This beam is under bending and shear. Strain gauges are located at the top surface of the beam. On the top surface, there is zero transverse shear. Thus the beam is only in a stress state due to bending at the location of the strain gauges. We must know how to compute the stress due to a bending moment and relate that stress to the strain in order to solve this problem.

First, the axial stress due to bending is

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where I and y on the top of surface of the beam are the following

M can be found via taking a section cut in the beam and balancing the moments. Be sure to use proper sign convention

Re-write (1) using (2), (3), (4)

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This is in the Linear-Elastic regime so,

Re-write (5) with (6) to get

We have two values of strains σ1 and σ2 at positions x1 and x2 so we now have two equations (7), (8), with three unknowns x1 , x2 , and P

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The third equation needed comes from the geometry condition given in the problem statement

Now we just have 3 unknowns and 3 equations so it can be solved anyway you like. One was is shown below To solve for P, combine equations (8), (9), (10)

Now solve (11) for P, and plug in values. We know

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Knowing P we can get the positions of the strain gauges. Solve (8) for x1 .

Now get x2

Problem 2 The natural frequency of a simple harmonic oscillator depends on both the stiffness of the restoring (elastic) member in the system and the mass which is being accelerated/decelerated. For a rigid mass m connected to a massless spring of linear stiffness k (dimensions: force/length), having one end grounded while the other is attached to the moving mass, the natural frequency is simply

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For the first-mode natural frequency of continuous uniform beam, δ0 , we have

where ρ = 0.24. To see where it comes from, please refer to section 8 of Lab No.1 handout.

1). Estimate the first-mode natural frequency, δ0

We are assuming that the antibody coating itself does not appreciably affect the natural frequency of the cantilever. So, based on the geometry of the beam, we have the beam stiffness,

where I is the area moment of inertia of the cross-section, which, for rectangular crosssections of this orientation, is equal to:

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with b the width and h the thickness of the beam.

The mass of this cantilever is equal to its volume times its density, which is,

Substitute eqn 18 and eqn 19 into eqn 17, we have the first mode natural frequency of this microfabricated cantilever,

2). Frequency change from the added mass, According to eqn 18 and eqn 21, the natural frequency of a cantilever is

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The sought-for bacteria will preferentially attach themselves to the antibody coating on the surface of the cantilever, in the process increasing the vibrating mass of the cantilever by an amount where nb is the number of bacterium cells that attach, and mb is the mass of the bacterium cell.

We are assuming the added mass is uniformly distributed along the length of the beam, and further, assume that the presence of the adhered bacteria does not affect the stiffness of the beam. From Eqn 22, the change in frequency resulting from the add mass can be expressed by

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In our case,

Then, the change in frequency resulting from the adding mass can be expressed as the following

3). Evaluate the change in natural frequency for the bacterium-coated cantilever

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The total mass of all the bacterium cells, which is the ”added mass”, is Figure 2: Scanning electron microscope images of E. coli bacteria attached to various microfabricated resonating cantilever beams. (from: Ilic, et al., Applied Physics Letters, 77, #3, 2000, 450-452.

Using eqn 26, and assuming the cell ”added mass” is uniformly distributed over the surface of the beam, the change in natural frequency is

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Problem 3

In this problem, the CNT(Carbon nanotube) can be assumed to be a cantilever beam. For elastic response, the lateral displacement of the cantilever v(x) is related to the bending moment by

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Remember that k , the stiffness, is a function of structure parameters such as geometry. That is, for the same material, two beams with different widths for example will have different stiffnesses.

Young’s modulus, on the other hand, depends ONLY on the material and thus is a material property. Finding a way to correlate these two is a reasonable approach to solving this problem.

From the definition of stiffness,

Furthermore, as we learned in class and in the lab, the following is true:

Setting (1) equal to (2) we get:

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From Equation 3, it can be concluded that k is proportional to the width and 3 3 ( thickness ) of the beam and inversely proportional to the ( length ) . Therefore, to get the minimum stiffness, we need to use the minimum width and thickness and the maximum length for our calculations using Equation 3 and vice versa for the maximum stiffness.

More specifically:

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Problem 4

Let’s derive the necessary results instead of using Table 8.1 in CDL. For beam (a), using the sign convention found in CDL, we have:

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Converting it to an equation with dimensionless parameters for graphing purposes:

From Equation 3, we can conclude that M ( x ) is a strictly increasing function

From the moment-curvature relation and Equation 4:

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The following figure summarizes the deflection and moment profile along the beam

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Let’s use the principle of superposition for part (b) This problem can be broken into two parts: a beam with a uniformly distributed load and an identical beam with an unknown tip load RL .

We know the deflection d1 and bending moment M for the first part from part (a) (Equations 4 & 8) and for the second part ( d 1 2 and M2 accordingly) from the lab handout:

Make sure you use consistent sign convention. The assumption here is that this is an upwards-directed tip load.

By superposition:

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Similarly for the bending moment:

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