Dynamics of Machines - Part II - IFS.pdf
Dynamics of Machines - Part II - IFS.pdf Dynamics of Machines - Part II - IFS.pdf
By understanding the topics related to mechanical vibration in low frequency ranges (high flexibility and slow motions detectable by human senses), it gets easier to understand mechanical vibration in high frequency ranges where one has faster motions with small amplitudes, just detectable by sensors and electronic devices. Figure 1: Experimental investigation of a mechanical continuous system with concentrated masses modelled as equivalent spring-mass systems with 1, 2 and 3 degrees of freedom (D.O.F.). Figure 2: Signal analyzer and shaker used for inducing and measuring mechanical vibrations while analyzing the behavior of the spring-mass systems with 1 D.O.F., 2 D.O.F. and 3 D.O.F. 4
1.3 Data of the Mechanical System ρ material density of the beam 7, 800 Kg/m 3 E elasticity modulus 2 × 10 11 N/m 2 L total length of the beam 0.600 m b width of the beam 0.030 m h thickness 0.0012 m mi concentrated mass (i = 1, ...,6) 0.191 Kg Table 1: Data of the mass-spring system ”A”. ρ material density of the beam 7, 800 Kg/m 3 E elasticity modulus 2 × 10 11 N/m 2 L total length of the beam 0.300 m b width of the beam 0.025 m h thickness 0.0010 m ρ material density (steel) 7, 800 Kg/m 3 mi concentrated mass (i = 1, ...,6) 0.191 Kg Table 2: Data of the mass-spring system ”B”. 1.4 Calculating Equivalent Stiffness Coefficients – Beam Theory (a) (b) Figure 3: (a) Flexible beam – clamped-free boundary condition case with force applied to the end L; (b) clamped-free boundary condition case with force applied to a general position L ∗ ; By applying a vertical force F at the end of the beam as shown in figure 3(a) and using Beam Theory, one can write: EI d4y(x) = 0 (1) dx4 Integrating in X once, one has: EI d3 y(x) dx 3 = F(x) = C1 (2) 5
- Page 1 and 2: DYNAMICS OF MACHINES 41614 PART I -
- Page 3: 1 Introduction to Dynamical Modelli
- Page 7 and 8: 1.5 Calculating Stiffness Matrices
- Page 9 and 10: 1.6 Mechanical Systems with 1 D.O.F
- Page 11 and 12: Demanding (λ 2 + 2ξωnλ + ω 2 n
- Page 13 and 14: 1 yini − A det λ1 vini − A C2
- Page 15 and 16: 1.6.4 Analytical and Numerical Solu
- Page 17 and 18: (a) y(t) [m] (b) y(t) [m] (c) y(t)
- Page 19 and 20: 1.6.6 Homogeneous Solution or Free-
- Page 21 and 22: (a) Amplitude [m/s 2 ] x Signal 10
- Page 23 and 24: (a) Amplitude [m/s 2 ] (b) Amplitud
- Page 25 and 26: Imag(A(ω)) [m/N] 0 −1 −2 −3
- Page 27 and 28: 1.6.11 Superposition of Transient a
- Page 29 and 30: (a) Amplitude [m/s 2 ] (b) Amplitud
- Page 31 and 32: 1.7 Mechanical Systems with 2 D.O.F
- Page 33 and 34: ⎧ ⎫ ⎪⎨ ˙y1(t) ⎪⎬ ˙y
- Page 35 and 36: zini = U c + A ⇒ c = U −1 {(zin
- Page 37 and 38: 1.7.4 Modal Analysis using Matlab e
- Page 39 and 40: 0.7 0.6 0.5 0.4 0.3 0.2 0.1 First M
- Page 41 and 42: %__________________________________
- Page 43 and 44: ||y 1 (ω)|| [m/N] Phase [ o] Excit
- Page 45 and 46: ||y i (ω)|| [m/N] Phase [ o] 0.8 0
- Page 47 and 48: Imag(y i (ω)/f 1 (ω)) (i=1,2) [m/
- Page 49 and 50: 1.8 Mechanical Systems with 3 D.O.F
- Page 51 and 52: which could be verified using Modal
- Page 53 and 54: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
By understanding the topics related to mechanical vibration in low frequency ranges (high<br />
flexibility and slow motions detectable by human senses), it gets easier to understand mechanical<br />
vibration in high frequency ranges where one has faster motions with small amplitudes, just<br />
detectable by sensors and electronic devices.<br />
Figure 1: Experimental investigation <strong>of</strong> a mechanical continuous system with concentrated<br />
masses modelled as equivalent spring-mass systems with 1, 2 and 3 degrees <strong>of</strong> freedom (D.O.F.).<br />
Figure 2: Signal analyzer and shaker used for inducing and measuring mechanical vibrations<br />
while analyzing the behavior <strong>of</strong> the spring-mass systems with 1 D.O.F., 2 D.O.F. and 3 D.O.F.<br />
4
Change language