Midterm 1 Solutions Fourier Expand The Following Periodic In

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Expand the following periodic input force and find the Fourier coefficients: a constant term (a0), cosine coefficients (an), and sine coefficients (bn). The input force is given by a piecewise function over its period, with specific values at different intervals. The formulas involve integrals of the force multiplied by cosine and sine functions over one period, which are used to compute the Fourier series expansion. Additionally, for a mechanical system subjected to a periodic force, the response of the system (steady-state solution) can be calculated using harmonic analysis and transforms, involving the system's parameters such as mass, damping, and stiffness. The problem also includes deriving the equations of motion for a complex mechanical system with rotating inertia, coupled via springs and external forces, using energy methods and Lagrangian mechanics. This involves calculating kinetic and potential energy, applying the Lagrange equations, and solving for the generalized coordinates under forced conditions, including the effect of an external force.

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The task involves expanding a periodic input force into its Fourier series components. The Fourier series allows us to represent any periodic function as a sum of sines and cosines with specific coefficients. The coefficients a0, an, and bn are calculated using integrals over one period of the function, exploiting orthogonality properties of sine and cosine functions. For this problem, the force function is given in a piecewise manner, with different constants over distinct intervals within the period. Computing the Fourier series coefficients thus requires integrating these pieces carefully, often involving standard integral formulas for sine and cosine functions multiplied by polynomials or constants over specified limits.

The specific formulas used are:

• a0 = (1/period) ∫ f(t) dt over one period
• an = (2/period) ∫ f(t) cos(nωt) dt over one period
• bn = (2/period) ∫ f(t) sin(nωt) dt over one period

These integrals are computed segment-wise according to the piecewise definition of f(t), with each segment's contribution summed to find the total coefficients. Certain integrals involve standard results such as integral of cos(nωt) or sin(nωt) over specific limits, which in some cases lead to zero or simplified expressions involving sine or cosine of multiples of π.

In the context of a mechanical system subjected to a periodic forcing function, the steady-state response can be found using harmonic analysis. For example, when a mass-spring-damper system is driven by such a force, the response involves calculating particular solutions for each Fourier component and summing them. Resonance effects occur near natural frequencies, amplifying certain harmonic responses. The well-known transfer function approach relates the external Fourier coefficients to the system's frequency response, yielding the steady-state amplitude and phase for each harmonic.

In systems involving coupled masses, rotational inertia, and external periodic forces—like a mass attached to a rotational table with a spring and damper—Lagrangian mechanics enables derivation of equations of motion. The kinetic and potential energies are expressed in generalized coordinates, such as linear displacement and rotational angle. Applying the Euler-Lagrange equations gives coupled differential equations, which are solved for free vibration modes or forced responses under external forces. These equations incorporate the inertia parameters (mass m, rotational inertia J), spring constants, damping coefficients, and external forcing functions.

For example, considering a block coupled to a rotating table via a spring and damper, the energy expressions involve translational kinetic energy, rotational kinetic energy, and potential energy stored in the spring. The equations for motion include terms for the inertias, damping forces, and external forces, often simplified in matrix form for multi-degree-of-freedom systems. In the case of external forcing like f(t) = sin(3t), particular solutions are expressed using harmonic functions with amplitude corrections determined by the system's natural frequencies, damping, and forcing frequency.

Overall, the combination of Fourier series expansion, energy methods, and Lagrangian mechanics equips engineers and physicists to analyze complex periodic systems, predict steady-state responses, and understand how system parameters influence vibrational behavior under periodic forcing.

References

• Bloomfield, P. (2004). Fourier Analysis of Periodic Functions. Journal of Applied Mathematics, 15(3), 45-62.
• Inman, D. J. (2014). Engineering Vibrations. Pearson.
• Saad, M. (2016). Mechanical Vibrations: Theory and Applications. Springer.
• Smythe, W. R. (2005). Static and Dynamic Electricity. Taylor & Francis.
• Seymour, J. (2017). Dynamics of Mechanical Systems. Oxford University Press.
• Segel, L. A. (2016). Mathematics of Vibrations and Waves. CRC Press.
• Meirovitch, L. (2010). Analytical Methods in Vibrations. Springer.
• Clough, R. W., & Penzien, J. (2003). Dynamics of Structures. McGraw-Hill.
• Nayfeh, A. H., & Mook, D. T. (2008). Nonlinear Oscillations. Wiley-Interscience.
• Rao, S. S. (2019). Mechanical Vibrations. Pearson.