ME260W Final Exam Review A. Strain gage and strain gage circuit: * Hooke’s law in 2D form, (pp431) 1,,yyxxx y xy xymm mmEE EE Gσσσσενε νγ τ=− =− = * General Wheatstone bridge (quarter-bridge & half-bridge configuration) (pp.204~206 & pp.438~445) R1R2UoR4R3Ui * “Neighbors work against each other.” (pp.442~443) ()012434iUGFUδεεεε=−+− * Output voltage of general Wheatstone bridge (p.205) 3112 34oiRRUURRRR⎛⎞=−⎜⎟++⎝⎠ * Gage factor (p.436) RRGFδε= * For a quarter bridge, if R1=R2=R3=R4 initially, and dR/R << 1 0/44iURR GFUδδε∗== * Principle of temperature compensation (pp.447~448) ()()yaxybx′+=, ydydx′=: B. General solution of the linear first order ODE: ()()()()Ax Axye C bxedx−=+∫ () ()Ax axdx=∫C. Response of a first-order system (pp.76~87) * Governing equation, ()yyKFtτ+=∗& * Time response of first order system to unit step function input (pp.76~79)ME260W Final Exam Review ()0()exptyt y y yτ∞∞⎛⎞=+ − −⎜⎟⎝⎠ * Methods to determine time constants: 63.2% rise time, log-t plot (pp.79~83) * Sinusoidal response of first-orders system (pp.84~85): magnitude ratio ()Mω, dynamic error ()δω If the solution to ()yyKASintτω+=&is () () ()()()() ()1/22sin111tyt Ce B tBMKAMτωωφωωωτδω ω−=+ +⎡⎤⎣⎦==⎡⎤+⎣⎦=− D. General solution of the second order ODE:2nd Order Linear Homogeneous ODE with Constant Coefficients: Characteristic Equation: Solutions of Characteristic Equation , General Solution 1 2 3 E. Second-order system: (p87~94) * General concept and terminologies: * 2nd-order SDOF vibration, equation of motion, 0vmx c x kx++=&& & * Natural frequencynω, nkmω= * Critical damping, cc22cnkcm mmω==ME260W Final Exam Review * Damping ratioζ, (underdamped1ζ<; critically damped1ζ=; overdamped, 1ζ>) vcccζ= * Damped vibration frequency dω, 21dnωζω=− * Period of the damped vibration 2ddTπω= * Logarithmic decrement δ, 122ln1kndkATAπζδζωζ+⎛⎞−==−=⎜⎟−⎝⎠ * Equation of motion with Coulomb (friction) damping, ()0cmx c sign x kx++=&& & * Amplitude decrement, ∆, due to Coulomb damping, 14ckkcAAk+∆= − = * Step-function response (pp.88~90) * Concept of rise time & settling time * Ringing frequency 21dnωωζ=− * Optimum damping ratio, 0.7 * Sin-function response (pp.92~94) * Magnitude ratio, ()Mω (equation 3.12 on p.92) * Resonance frequency (different from ringing frequency!) 212rnωωζ=− F. Transverse vibration of Euler beam: (see handout) * Equation of motion, 424221, vvaEIxat Aρ∂∂=− =∂∂ * General solution of equation of motion, X(x) is mode shape, ω is the associated frequency ()()()()cos sinvXx A t B tωω=∗ +ME260W Final Exam Review Define kaω=, then () ( )()()(1234cos cosh cos cosh sin sinh sin sinhX x C kx kx C kx kx C kx kx C kx kx=++−+++−) * Natural frequency of i-th mode of simple-simple supported beam, 222, 1,2,3,...iiEIilAπωρ== * Modes of vibration of simple-simple supported beam, sin , 1,2,3...iiixXD ilπ== G. Probability and statistics: (pp.109~143) * Concept of histogram and frequency distribution. (pp.109~113) * Normal distribution, x’ is called true mean value, 2σis called true variance ()()2211exp22xxpxσπσ⎛⎞′−=−⎜⎟⎜⎟⎝⎠ * Transformation to normal error function xxzσ′−= * P% interval for measured value with normal distribution (z1 from Table 4.3, p118) ()1 %xxz Pσ′=± * Finite statistics, Sample mean value 11NiixxN==∑ Sample variance ()22111NxiiSxN==−−∑x Sample standard deviation xS * Determine precision interval of a random variable using t-estimator, t from table 4.4, p122. (), %iPxxxtSPν=± * Standard deviation of mean (p.123), and precision interval of the mean of a random variable, t from table 4.4 (),%xiPSxxt PNν=± * Least-square regression (pp132~134,) and the concept of R square (pp135~136) 2221yxySrS=−ME260W Final Exam Review * Number of measurements required to estimate the true mean value with acceptable precision (),95,95 95%2xxtSSCIdt NdNνν⎛⎞==±