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Truss Structures - Natural Frequency Manipulation via SDP Brian W. Anthony Robert M. Freund 1 7 Background : Mechanical Systems 8 Simple Mechanical System k1 m1 u1 9 10 Natural Frequencies – How a Mechanical System wants to vibrate when forced. 1 DOF System 3 2.5 Amplitude 2 1.5 1 0.5 (k/m).5 0 0 2 4 6 8 10 12 14 Frequency (Hz or Cycles/Sec) 16 18 20 Examples • Ball on a string • Beam(s) • For a simple mechanical systems it is relatively easy to systematically effect a change in the natural frequency. 11 Beam Vibration Narrow Beam How would we change the frequency of vibration? Beam Vibration Narrow Beam Narrow Beam Narrow and Short Beam Wide Beam - Natural Frequency = 5.6 Hz Natural Frequency = 11.7 Hz Natural Frequency = 7.8 Hz 12 Another Mechanical System k1 k2 m1 k3 m3 m2 F(t) u1 u2 u3 Write Newton’s Law for Each Mass F = ma Another Mechanical System k1 k2 m1 k3 m3 m2 F(t) u1 u2 u3 The Dynamics Model (The Equations of motion). m1(d2u1 /dt2) + k1u1+ k2(u1-u2) =0 2 2 k2(u2-u1) + k3(u2-u3) = 0 m2(d u2 /dt ) + 2 2 m3(d u3 /dt ) + k3(u3-u2) = F(t) In Matrix Form M(d2U /dt2) + KU = F(t) 13 Equations of Motion And Eigenvalue Analysis The Equations of motion M(d2U /dt 2) + KU = F(t) M d 2U + KU = F (t ) dt 2 • The Eigenvalues of M-1K are the natural frequencies of vibration (squared). • The Eigenvectors of M-1K are the mode shapes (the relative displacement of each degree of freedom) Natural Frequencies – The Rate a Mechanical System wants to vibrate 3 DOF System 3 2.5 Amplitude 2 1.5 1 0.5 0 1 2 3 4 5 6 7 Frequency (Hz or Cycles/Sec) 8 9 10 14 Natural Modes – (The Eigen-Modes) Are the shape of the Vibration 3 DOF System 3 2.5 Mode 1 m2 m3 Amplitude 2 m1 1.5 1 Mode 2 0.5 0 m1 m2 m3 m2 m3 1 2 3 4 5 6 7 Frequency (Hz or Cycles/Sec) 8 9 10 Mode 3 m1 Movies… Modification? 3 DOF System 3 2.5 Mode 1 m2 m3 Amplitude 2 m1 1.5 1 Mode 2 0.5 0 m1 m2 m3 Mode 3 m1 m2 m3 1 2 3 4 5 6 7 Frequency (Hz or Cycles/Sec) 8 9 10 It is no longer obvious what we have to do in order to change the lowest natural frequency. 15 Truss Structure Truss Structures • Rigid beams – Axial forces only • Pin-connected – Concentric joints – Welded or bolted • Bridges, towers, exoskeletons 16 A truss has large number of natural frequencies and complex motion. Truss : Model as Lumped Masses with Connecting Springs Link Stiffness is a function of Length, Area, density Half of the mass of Each Link is assumed to sit at a node. (A function of Areas) 17 Truss : Modeled as Lumped Masses with Connecting Springs 8 4 9 6 7 1 2 3 5 8 4 9 6 7 y 1 2 3 5 x Step 6. Determine System of Equations by applying Newton’s Law at each node (on each ball of mass) M(d2U /dt 2) + KU = F(t) 18 22 20 18 16 14 12 10 8 6 4 2 0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 19 Each Node (Mass) has, in general, 3 Degrees of Freedom 20 Equations of Motion And Eigenvalue Analysis The Equations of motion M(d2U /dt 2) + KU = F(t) • The Eigenvalues of M-1K are the natural frequencies of vibration (squared). • The Eigenvectors of M-1K are the mode shapes (the relative displacement of each degree of freedom) 21 22 23 24 25 A proposed design of a Cell Phone Tower Planar Approximation: All beams have an Area of 1 square centimeter 50 Bars 40 Degrees of Freedom 22 20 18 16 14 12 10 8 6 4 2 0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 26 A proposed design of a Cell Phone Tower Natural Frequency = 159 Hz ( smallest eigenvalue of M-1K ) 22 20 18 16 14 12 10 8 6 4 2 0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Natural Frequencies – The Rate a Mechanical System wants to vibrate N DOF System Amplitude 10 10 10 0 -5 -10 0 0.5 1 1.5 Frequency (Hz or Cycles/Sec) 2 2.5 x 10 4 27 Natural Modes (Eigenvectors or Eigen-modes) - The Shape of the Vibration N DOF System 0 Amplitude 10 -5 10 -10 10 0 159.7156 Hz 0.5 1 1.5 Frequency (Hz or Cycles/Sec) 920.3246 Hz 2 Movies… 2.5 x 10 4 1618.3 Hz 2333.4 Hz Cell Phone Tower • The initial design of the cell tower has a natural frequency of 159 Hz. • We expect ground vibration induced by a nearby railroad near 100 Hz and 159 Hz. 22 20 18 16 14 12 10 8 6 4 Movies… 2 0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 28 Optimized Tower Natural Frequency = 250 Hz towershrt250 22 20 New Areas: 18 16 1.0353 1.5332 2.0785 14 12 10 8 6 2.6420 3.2065 4 2 0 3.7667 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Cell Phone Tower • The optimized design of the cell tower has a natural frequency of 250 Hz. • This vibration of 100 Hz and 159 Hz will now have minimal effect. towershrt250 22 20 18 16 14 12 10 8 6 4 2 Movies… 0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 29 Optimized Tower Natural Frequency = 500 Hz towershrt500 22 New Areas: 2.0561 4.0845 20 18 16 14 12 7.3261 11.8377 10 8 6 17.2199 22.8216 4 2 0 28.2383 -1 -0.5 0 0.5 1 1.5 2 2.5 3 30 Continuous Mechanical System – Truss Approximation Step 0. Complex (Continuous) Mechanical Structure 31 Step 1. Apply a Mesh (Truss) to the Mechanical Structure Step 2. Form Local Regions around ‘Nodes’ of the Mesh 32 All of this mass is applied to the contained node. Step 3. Use Local Regions around ‘Nodes’ to approximate Node Mass (mi) and Link Stiffness (kij) mj kij mi Step 4. The Finite Element (Truss) Approximation 33 8 4 9 6 7 1 Step 5. Label the Nodes 2 3 5 8 4 9 6 7 y 1 2 3 5 x Step 6. Determine System of Equations by applying Newton’s Law … m6(d2 x6 /dt2) + k61x(x6-x1) + k62x(x6-x2) + … = F6x(t) m6(d2 y6 /dt2) + k61y(y6-y1) + k62y(y6-y2) + … = F6y(t) … 34 8 4 9 6 7 y 1 2 3 5 x Step 6. Determine System of Equations by applying Newton’s Law M(d2U /dt 2) + KU = F(t) 35 If Time Remains 36 37 38 39 40