1. ME321 Kinematics and Dynamics of Machines
Steve Lambert
Mechanical Engineering,
U of Waterloo
1/18/2019

2. Vibrations Vibration: repeated or cyclic motion.
usually small scale (small amplitude),
usually undesirable, and
can lead to noise, fatigue failures, etc.
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3. Natural Frequencies Preferred frequencies of vibration for a system.
Excitation (loading) at these frequencies can lead to resonance.
The excitation frequency is the frequency of the applied loading.
Examples include shock loading, wave loading, out-of-balance shafts, etc.
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4. Damping Energy ‘absorption’ (actually transfer to heat energy)
Reduces the amplitude of vibration with time and can result from
‘natural damping’ of a material,
friction (Coulomb),
air resistance, or
fluid viscosity, for example.
A shock absorber is designed to produce damping.
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5. Mass Spring Damper Systemk c x
F(t)
F(t): excitation force (N)
m: mass (kg)
k: spring stiffness (N/m)
c: viscous damping coefficient (Ns/m)
To continue, we draw the free body diagram for the mass
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6. Free Body Diagram From force equilibrium: or:
This is the governing differential equation for forced vibration
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7. Forced Displacement
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8. Types of Loading
Transient - represented by non-zero initial conditions for displacement and velocity
Steady-state - represented by sinusoidal loading, or a combination of sinusoidal loading using Fourier Series analysis
Random - represented using Fourier Series, so solution technique similar to steady-state
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9. Types of Solution
All types of loading can be handled by just two solutions
Unforced Vibration: F(t) = 0
Forced Vibration: F(t) = F0 sin  t
Forced vibration solution requires the unforced (natural) solution to obtain the general solution, and we then add the particular solution
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