Pxgage 1gage 2x1x2MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.002 MECHANICS AND MATERIALS II HOMEWORK NO. 1 Distributed: Wednesday, February 11, 2004 Due: Wednesday, February 18, 2004 Problem 1 (35 points) A cantilever beam has been constructed from a steel having Young’s modulus E = 208 GP a, Poisson ratio ν = 0.29, and tensile yield strength σy = 410 M P a. The length of the beam, from its base to its tip, is L = 1 m, and the uniform cross-section is rectangular, h = 5 mm thick and b = 30 mm wide. Two axially-mounted 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 � = 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 y-axis, 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: Figure 1: Schematic of tip-loaded cantilever showing positions of strain gages numbers 1 and 2. 1Gage no. axial coordinate strain 1 x1 =? 1200 × 10−6 2 x2 = x1 + 200 mm 900 × 10−6 • (20 points) Evaluate the load P . • (15 points) Using the (now-known) value of P , find the axial locations (x1 and x2) of both gages. 2Problem 2. (25 points) 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 first-mo de natural frequency, ω0) is recorded experimentally. Then the coated cantilever is exposed to a liquid medium. If the sought-for 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 micro-fabricated resonating cantilever beams. (from: Ilic, et al., Applied Physics Letters, 77, #3, 2000, 450-452. Subsequent testing of the added-mass 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 GP a. 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 fre-quency of the cantilever, estimate the first-mode natural frequency, ω0, of the cantilever 3 Image removed due to copyright considerations.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 Δm � m, show that the change in frequency resulting from the added mass, Δω, can be expressed by . �1 Δm �ω0 + Δω = ω0 1 − . 2 m – Hint # 1: You might wish to show that the natural frequency of a uniform beam of total mass ‘m’ can be written as ω ∝ (1/L)2 �EIL/m (neglecting dimensionless factors). – Hint # 2: If the change in total mass (Δm) is very small in comparison to the initial beam mass (m), then the resulting frequency of the perturbed-mass system can be evaluated by taking a Taylor series expansion of the frequency expression about the reference mass. • 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 bacterium-coated cantilever. 4Problem 3. (20 points) Atomic force microscopy resolves the structure of surfaces to near atomic-level resolu-tion. 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 sharp-tipped 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 mea-sured 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 cross-section cantilever from an atomic force microscope (AFM) that has been machined from a single block of silicon using lithographic technology (e.g., precision