EME 5213 Vibrations

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Bacon117
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293360
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EME 5213 Vibrations
Updated:
2015-03-11 22:45:26
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Vibrations Engineering Mechatronics
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Vibrations
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  1. What is the equation for final velocity if you are given distance, gravity, and starting velocity.
    v2 = v02 + 2a(x − x0)
  2. What is the standard equation for oscillatory motion?


    where a and b are constants determined by initial conditions.

    • a=x'(0)/ωn
    • b=x(0)
  3. How do you determine the natural frequency of a simple mass spring oscillation, ignoring damping?
    • ω=√(k/m) = 2 * pi * fn = (2*pi)/Tn
    • ωn is natural freq in rad/s
    • fn  is nat frequency in cycles/s or hz
    • Tn is period of natural oscillation in s
  4. According to Hookes law, what is the force equation of a simple, linear, and massless spring?
    • F=kx
    • k is stiffness in N/m or pound/inch
    • x is deflection distance
  5. What is the standard differential equation of simple oscillatory motion in a simple spring-mass system?
    X″ + ωn2x=0


    Remember ωn2=k/m
  6. In a simple pendulum, what is the restoring force when the pendulum is displaced by a small angle. (Hint, Fnet )
    Fnet = -mg sin(Θ)
  7. What is the differential equation for simple pendulum motion?
    Θ"+(g/L)Θ=0

    • L is pendulum length.
    • This is for small angles since sin(Θ)≈Θ at small theta.
  8. What is the standard equation for simple pendulum oscillation?
    Θ=Θmax sin{√(g/L)t}
  9. What is the moment of inertia about the geometric axis of a disc or cylinder?
    I=1/2 MR2
  10. What is the Inertia of a body from on axis to another (Parallel axis theorem)
    I=Icm + md2


    • Icm is the inertia about the center mass
    • d is the distance between parallel axes.
  11. What is the moment of inertia of a hoop about it's axis of symmetry?
    I=MR2
  12. What is the Moment of inertia of a solid sphere?
    I=2/5 MR2
  13. What is the moment of inertia of a rod about it's center
    I=1/12ML2
  14. What is the equation of motion for a simple mass spring system if the motion starts with zero initial velocity, but with an initial displacement?
    x(t)=x0 cos(ωnt)
  15. What is the equation of motion if initial displacement is 0, but there is an initial velocity (like a hammer impact).
    x(t)=(x0'/ωn)*sin(ωnt)
  16. What is the differential equation of a shaft disk system?
    Θ''+ωn2 Θ=0   (1)

    Derivation I*Θ'' = -ktΘ, so I*Θ''+ktΘ = 0.  This gives Θ''+(kt/I)Θ.  And since ωn2=kt/I, we get the top equation (1).
  17. What is kt of a shaft disk system?
    kt=(G*pi*d4)/(L*32)

    • G= shaft modulus of elasticity
    • d= shaft diameter
    • L= shaft lenght

    Bad mother.. shut yo mouth = Shaft.
  18. For a mass spring system what is the kinetic energy?
    T=1/2m*x'2

    (1/2mv2)
  19. In a mass spring system what is the gravitational potential energy?
    Vg=-mg(Δ+x)
  20. In a mass spring system what is the elastic potential energy?
    Ve=1/2 k(Δ+x)2
  21. In a mass spring system, what is equation for conservation of energy?
    UNonConservative=ΔT + ΔVg + ΔVe
  22. What is the law of conservation of linear momentum?
    M1V01+M2V02=M1Vf1+M2Vf2
  23. What is the effective mass of a spring?
    • 1/3 * ms
    • This is because we take the derivative of y2 to determine the KE of a spring.
  24. What is the natural frequency considering the mass of the spring in a simple mass-spring system.
    ωn=√(k/[m+{ms/3}])
  25. What is the equation for springs in series?
    1/keq = 1/k1 + 1/k2
  26. What is the total spring rate for spring in parallel?
    keq=k1+ k2
  27. What is the equation for simple harmonic motion?
    x(t) = X sin(ωnt)
  28. The amplitude of the oscillation of a simple mass-spring system depends on what?
    The initial conditions (velocity and displacement)
  29. The period of natural frequency of a simple spring-mass system depends on what?
    The parameters of the sytsem (specifically, the mass and spring rate)
  30. What is equation to find the roots of ax2+bx+c=0?
  31. What is the solution to the IDE d2x/dt2+ζωn dx/dt+ωn2=0
    • x(t)=e-2ωnt [Aeωn√(ζ^2-1)+Beωn√(ζ^2-1)]
  32. What is Eulers identity?
    eix=cos(x)+i*sin(x)
  33. What is the critical damping ratio for a simple mass-spring-dashpot system?
    2*√(km)
  34. What is the logarithmic decrement for vibration?
    δ=ln(x1/x2)= (2Πζ)/√(1-ζ2)
  35. What is the damped natural frequency?
    ωdn*√(1-ζ2)=2pi/τd
  36. What is the damping ratio ζ?
    ζ=c/ccr
  37. What is the moment of inertia for a rod, from the end of the rod?
    1/3 mL2
  38. For a given force, how do you convert that to the moment about some point?
    Multiply the force times the distance to the point.
  39. How do you solve this mx''+cx'+kx=0?
    • Assume the solution is in the form on est.
    • Substitute for (ms2+cs+k)est=0.
    • Use quadratic to find roots of ms2+cs+k=0.
    • The solution is
  40. How do you determine ccr from the equation of motion?
    Take equation of motion, and substitute est, this will replace x'' with s2. Take this equation, use the quadratic to find the roots.  Set the portion under the square root to 0.  Solve for c, this is ccr.
  41. What is the general steps to solve a coulomb damping problem?
    • Solve just like an oscillation problem except you have to consider when X' is + or -.
    • Solve for one half of motion, then you need to determine the initial conditions for the second half of the motion.
    • The second half starts when ωn=pi
  42. What is the general equation for sinusoidal excitation?
    • It is a second order ordinary differential non-homogeneous equation in the form:
    • x''+cx'+kx=F0*sin(ωt)
  43. Find zeta from logarithmic decrement.
    ζ=δ/√((2pi)22)

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