homework no. 3 · homework problem 3.1 a double pendulum consists of two bobs of mass m1 and m2,...
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ME 563-Fall 2020
Homework No. 3
Due: September 19, 2020 11:59 pm on Gradescope
ME 563 - Fall 2020 Homework Problem 3.1 A double pendulum consists of two bobs of mass m1 and m2, suspended by inextensible, massless strings of length L1 and L2.
a) Determine the expression for potential energy U and the generalized forces corresponding to F for the generalized coordinates. Use these results to determine the angles θ1 and θ2 corresponding to static equilibrium. Leave these angles in terms of the ratio F/mg. b) Write down an expression for the kinetic energy T in terms of the generalized coordinates θ1 and θ2 and their time derivatives. From this expression, identify the elements mij , where:
c) Determine the mass matrix [M] and the stiffness matrix [K] corresponding small oscillations about the equilibrium state for F/mg = 0.
d) Determine the mass matrix [M] and the stiffness matrix [K] corresponding small oscillations
about the equilibrium state for F/mg = 2. Compare these with those found in part c).
ProbleMD
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2mgs.inQFcosQ2tanQFmg tanQi Fgmg
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Thevelocity vectors
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Stiffness calculation
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22 222 1g mgtacos eenc consider Mmg O and Flgmg 0
Qew zeq tan O OcosQew coscozen
Thelinearmassandstiffness matrices
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m L L2 m O L
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cosQew 11oz
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ME 563 - Fall 2020 Homework Problem 3.2 Consider the system below, whose motion is described by the absolute coordinates shown.
a) Write down the potential energy function U for this four-DOF system and use the following results from lecture to develop the stiffness matrix for the system:
b) Use the method of influence coefficients to develop the flexibility matrix [A] =[K]-1.
c) Check your results in a) and b) above by verifying that [A][K]=[I] where [I] is the identity matrix.
a The potentialenergy can be writtenas
a 42km7 42131445 t4212koxoAGT 42kExcNB
Equilibriumposition maNoAB Nc O
ManaKHA 2 3K4g 2KHoXD 5KX 2KXB
2 2KExpooxo Ktx xD 3koxo 2Koxo kik
2qKKc Xaoh na ar Nogo q
Ki 21,1 K Kia ka 23 0 143 431 221 0 144 212 2502min24s
Kee_222 5k Kas132 2245 2K k24 442 222,125 0
Kss 221 3K Kz4 K z K22ND doxodale
K44 221 L24C
K K O O OO SK 2k OO 2k 3K K
O O K K
b Apply a unit load ANoforce here
m_ff Efx ka 0 a c00attail 921 931 941 0 sincenotconnected
Byreciprocity 9,2 921 0 a3 931 0 ay 9,4 0
Apply a unit load Oa B3k f 2x k 922 923 924 zero strainrun In mn to right
take e e EFX 3Kazz 1 O00_00122 Hazy 922 134 924 134
923 134Bg reciprocity 942 924 1 34 923 932 1310
Apply a unit load a C
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f key 13k2K0 OF c E ktazz Haze
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Apply a unit load D
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Keg 1k 61gk t K
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a Hk O O O
O 43K 43K 43K0 113k 516k 46k0 113k 96k 16K
a K K O O 0 Hk O O O
O SK 2k O O 43k 43K KskO 2k 3K K O 113k k k
O O K K O BK 516k 16K
I O O GO l O 0O O I 0O O O I
ME 563 - Fall 2020 Homework Problem 3.3 A disk of mass m and radius r rolls without slip on a cart of mass 3m. A force F is applied to the cart. The disk has an angular displacement q. Let x, describe the absolute motion of the cart. The springs is unstretched when xA=0. Assume the surface between the cart and ground is smooth.
a) Use a Newton Euler formulation to find the equations of motion. Your final equations should not include any forces of reaction. The final differential equation are in terms of xA and q and should be in matrix form.
b) Use Lagrange’s Equations to find the equations of motion. The generalize coordinates are xA and q. The final differential equation are in terms of xA and q and should be in matrix form.
c) Compare the form of the mass matrix, stiffness matrix and forcing vector of the Newtonian formulation with the Lagrangian formulation. In particular, are they the same? If not, in which formulation are the matrices symmetric ? Do the forcing vectors differ? Explain any similarities or differences.
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