What to bring and what to study

What to bring and what to study One 8.5 X 11 formula sheet, one side only, no examples. Save the other side for test 2. Put your name on it and turn it in with the test. If you number the formulas. Suggest you use the same numbers as the text, you will be free to refer to them on the test. That is: “ from equation (1.37): Mechanical Engineering at Virginia Tech

Material (sections) Covered • 1.1,1.2, 1.3, 1.4, 1.5, 1.7 • 2.1, 2.2 • Log decrement from 1.6 Mechanical Engineering at Virginia Tech

Also bring • Paper, pencil, calculator • No other resources allowed • Your honor, but no anxiety • Knowledge of all examples worked in class or presented in the text • All assigned homework Mechanical Engineering at Virginia Tech

Expect 4 to 5 problems • An example covered in class • A homework problem • An example from the book, not covered in class • A problem involving combining parts of any of the above in “two steps” and/or • A derivation • 25% (or 20%) each • the last problem(4 and/or 5) intended to • sort out the A’s and B’s Mechanical Engineering at Virginia Tech

Free-body diagram and equations of motion • Newton’s Law: Mechanical Engineering at Virginia Tech

2nd Order Ordinary Differential Equation with Constant Coefficients Mechanical Engineering at Virginia Tech

Periodic Motion amplitude x(0) Displacement Maximum Velocity Phase Time usually sec Mechanical Engineering at Virginia Tech

Frequency We often speak of frequency in Hertz, but we need rad/s in the arguments of the trigonometric functions. Mechanical Engineering at Virginia Tech

Amplitude & Phase from the ICs Mechanical Engineering at Virginia Tech

Other forms of the solution: See window 1.4, page 12 for relationships among these. Mechanical Engineering at Virginia Tech

Peak Values Mechanical Engineering at Virginia Tech

Spring-mass-damper systems • From Newton’s law: Mechanical Engineering at Virginia Tech

Solution: Given m, c, k, x0, v0 find x(t) Mechanical Engineering at Virginia Tech

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Three possibilities: Mechanical Engineering at Virginia Tech

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Underdamped 0 < z < 1 Reduces to undamped formulas for z = 0 Mechanical Engineering at Virginia Tech

Potential and Kinetic Energy The potential energy of mechanical systems U is often stored in “springs” (remember that for a spring F=kx) x=0 x0 k M Spring Mass The potential energy of mechanical systems U is also gravitational: The kinetic energy of mechanical systems T is due to the motion of the “mass” in the system Mechanical Engineering at Virginia Tech

Conservation of Energy For a simply, conservative (i.e. no damper), mass spring system the energy must be conserved: Equation of motion At two different times t1 and t2 the increase in potential energy must be equal to a decrease in kinetic energy (or visa-versa). An expression for the natural frequency Mechanical Engineering at Virginia Tech

Deriving equation of motion x=0 x k M Spring Mass Mechanical Engineering at Virginia Tech

Natural frequency If the solution is given by Asin(wt+f) then the maximum potential and kinetic energies can be used to calculate the natural frequency of the system Mechanical Engineering at Virginia Tech

Static Deflection Mechanical Engineering at Virginia Tech

Combining Springs • Equivalent Spring Mechanical Engineering at Virginia Tech

Harmonically Excited SystemsEquations of motion (c =0): Mechanical Engineering at Virginia Tech

Linear nonhomogenous ode: • Solution is sum of homogenous and particular solution • The particular solution assumes form of forcing function (physically the input wins) To be determined Driving frequency Mechanical Engineering at Virginia Tech

Substitute into the equation of motion: Thus the particular solution has the form: Mechanical Engineering at Virginia Tech

Add particular and homogeneous solutions to get general solution: Mechanical Engineering at Virginia Tech

Apply the initial conditions to evaluate the constants Solving for the constants and substituting into x yields Mechanical Engineering at Virginia Tech

2.2 Harmonic excitation of damped systems Mechanical Engineering at Virginia Tech

Substitute the values of As and Bs into xp: Add homogeneous and particular to get total solution: Note: that A and f will not have the same values as in Ch 1, as t gets large, transient dies out Mechanical Engineering at Virginia Tech

Magnitude: Non dimensional Form: Phase: Frequency ratio: Mechanical Engineering at Virginia Tech

Magnitude plot Mechanical Engineering at Virginia Tech

Phase plot p/2 Mechanical Engineering at Virginia Tech

What to bring and what to study