Introduction to Mechanical Vibrations. Ronald J. Anderson

Introduction to Mechanical Vibrations

7Figure 7.1 A 2DOF undamped system.Figure 7.2 Free body diagram for the 2DOF undamped system.Figure 7.3 First mode shape for the 2DOF undamped system.Figure 7.4 Second mode shape for the 2DOF undamped system.Figure 7.5 Another sample system.Figure 7.6 A 2DOF undamped, forced, system.Figure 7.7 A model of a machine experiencing large amplitudes.Figure 7.8 The machine with the vibration absorber mounted.Figure 7.9 The cart and pendulum system.Figure 7.10 The cart and pendulum system – FBD.Figure E7.1Figure E7.2Figure E7.3Figure E7.5Figure E7.6Figure E7.7

8 Chapter 8Figure 8.1 A taut string.Figure 8.2 The first five string modes.Figure 8.3 Deflections of a uniform beam.Figure 8.4 A beam element.Figure 8.5 A cantilever beam.Figure 8.6 The first three cantilever beam modes.

9 Chapter 9Figure 9.1 A beam element.Figure 9.2 A non‐uniform beam.Figure 9.3 An elastic rod.Figure 9.4 An elastic rod element.Figure 9.5 An element of the element.Figure 9.6

versus

for the element.Figure 9.7 The rod element with nodal forces applied.Figure 9.8 A statically loaded rod modeled with one element.Figure 9.9 The velocity of the element of the element.Figure 9.10 Two rod elements in an assembly.Figure 9.11 Free body diagrams of the two rod elements.Figure 9.12 The global mass matrix.Figure 9.13 The global stiffness matrix.Figure 9.14 The two‐noded beam element.Figure 9.15

versus

for the element.Figure 9.16 Node and element numbering.Figure 9.17 Node and element numbering.Figure 9.18 Externally applied forces and moments.Figure 9.19 The harmonically varying applied load.Figure E9.3 Figure E9.4

10 Chapter 10Figure 10.1 An inerter implementation.Figure 10.2 A screw.Figure 10.3 The inerter symbol. Figure 10.4 Single degree of freedom system with an inerter.Figure 10.5 Free Body Diagram for the single degree of freedom system with a...Figure 10.6 Two degree of freedom system with inerters.Figure 10.7 Simplified two degree of freedom system with an inerter.Figure 10.8 A single degree of freedom system with harmonic ground motion.Figure 10.9 Free body diagram for the single degree of freedom system with h...Figure 10.10 A single degree of freedom system with an inerter and harmonic ...Figure 10.11 Free body diagram for the single degree of freedom system with ...

11 Chapter 11Figure 11.1 A measured variable

plotted versus time.Figure 11.2 The square of

plotted versus time.Figure 11.3 The example function,

, plotted versus time.Figure 11.4 The DFT amplitudes of the example function,

, plotted versus fr...Figure 11.5 Aliasing.Figure 11.6 The folding frequency.Figure 11.7 Aliased DFT results.Figure 11.8 The DFT for the first example.Figure 11.9 The DFT for the second example.Figure 11.10 CFT approximation to the square wave.Figure 11.11 The Hanning window.Figure 11.12 The data from Equation 11.83.Figure 11.13 The windowed data.Figure 11.14 The DFT for the second example with windowing.Figure 11.15 The DFT before downsampling.Figure 11.16 The DFT after downsampling.Figure 11.17 The time series before and after digital filtering.Figure 11.18 The DFT after decimating.Figure 11.19 A low‐pass filter circuit.Figure 11.20 Low‐pass filter frequency response in dB on a logarithmic scale...Figure 11.21 Low‐pass filter frequency response on a linear scale.Figure 11.22 Exponential moving average smoothed data.Figure 11.23 Exponential moving average DFT.Figure 11.24 Low‐pass digital filter frequency response.Figure 11.25 Low‐pass filter frequency response near the cut‐off frequency.Figure 11.26 Noisy time signal.Figure 11.27 Noisy time signal zoomed.Figure 11.28 DFT – 1 average.Figure 11.29 DFT – 10 averages.Figure 11.30 DFT – 20 averages.Figure E11.5

12 Chapter 12Figure 12.1 A mass on a spring.Figure 12.2 A spring connecting two masses.Figure 12.3 Lift and drag in a wind tunnel.Figure 12.4 Typical lift and drag forces versus angle of attack.Figure 12.5 A suspended airfoil in steady flow.Figure 12.6 Free body diagram of the airfoil.Figure 12.7 Representation of a von Karman vortex street in a wake.Figure 12.8 Helical strakes on tall chimneys.Figure 12.9 A railway truck supported by its two wheelsets.Figure 12.10 Parameters for a railway wheelset.Figure 12.11 Back‐to‐back cones forming a railway wheelset.Figure 12.12 Wheelset degrees of freedom.Figure 12.13 Creep forces.Figure 12.14 Wheelset free body diagram.Figure 12.15 Wheelset instability.Figure 12.16 System with a rigid body mode.Figure 12.17 The three modes.Figure 12.18 A spring/mass system.Figure 12.19 The model of a double wishbone suspension.Figure E12.1 Figure E12.3

13 Appendix AFigure A.1 Three data points and two Least Squares curve fits.

14 Appendix BFigure B.1 Parallel Axis Theorem.