Solution Manual for Applied Engineering Mathematics, 1st Edition By Brian Vick Contains Chapter 3-11

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Chapter 3

Modeling of Physical Processes Problem 3.1: Ground Level Determine the solutions to the following problems. a

b

c

d

𝑑𝑑𝑑𝑑 1 = − 𝜃𝜃 𝑑𝑑𝑑𝑑 𝜏𝜏 𝜃𝜃 = 0 , 𝑡𝑡 = 0 2 𝑑𝑑 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑚𝑚 2 + 𝑐𝑐 + 𝑘𝑘 ∙ 𝑥𝑥 = 0 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑥𝑥 = 0 , = 0 , 𝑡𝑡 = 0 𝑑𝑑𝑑𝑑 𝜕𝜕 2 𝑇𝑇 𝜕𝜕 2 𝑇𝑇 + =0 𝜕𝜕𝜕𝜕 2 𝜕𝜕𝜕𝜕 2 𝑇𝑇 = 0 on all boundaries 𝜕𝜕𝜕𝜕 𝜕𝜕 2 𝑇𝑇 = 𝛼𝛼 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 = 0 , 𝑥𝑥 = 0 𝜕𝜕𝜕𝜕 𝑇𝑇 = 0 , 𝑥𝑥 = 𝐿𝐿 𝑇𝑇 = 0 , 𝑡𝑡 = 0

Solution All these cases have zero forcing functions, thus they all remain in equilibrium. a) 𝜃𝜃 = 0 b) 𝑥𝑥 = 0 c) 𝑇𝑇 = 0 d) 𝑇𝑇 = 0

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Problem 3.2: Mathematical Models Consider the following systems of equations. In each case identify if the mathematical model is formulated properly in order to allow for the possibility of a unique solution. If the system is not properly formulated, what additions or changes are necessary? a b

c

d

e

f

Solution a b c d e f

𝑑𝑑𝑑𝑑 1 = − 𝜃𝜃 + 𝑆𝑆 𝑑𝑑𝑑𝑑 𝜏𝜏 𝑑𝑑𝑑𝑑 1 = − 𝜃𝜃 + 𝑆𝑆 𝑑𝑑𝑑𝑑 𝜏𝜏 𝜃𝜃 = 𝜃𝜃0 , 𝑡𝑡 = 0 𝜃𝜃 = 𝜃𝜃1 , 𝑡𝑡 = 𝑡𝑡1 2 𝑑𝑑 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑚𝑚 2 + 𝑐𝑐 + 𝑘𝑘 ∙ 𝑥𝑥 = 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑥𝑥 = 𝑥𝑥0 , 𝑡𝑡 = 0 2 𝑑𝑑 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑚𝑚 2 + 𝑐𝑐 + 𝑘𝑘 ∙ 𝑥𝑥 = 𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑥𝑥 = 𝑥𝑥0 , 𝑡𝑡 = 0 𝑥𝑥 = 𝑥𝑥1 , 𝑡𝑡 = 𝑡𝑡1 2 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑇𝑇 𝑔𝑔(𝑥𝑥, 𝑡𝑡) = 𝛼𝛼 2 + , 0 < 𝑥𝑥 < 𝐿𝐿 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 −𝑘𝑘 = 𝑞𝑞0" (𝑡𝑡) , 𝑥𝑥 = 0 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 2 𝑇𝑇 𝑔𝑔(𝑥𝑥, 𝑡𝑡) = 𝛼𝛼 2 + , 0 < 𝑥𝑥 < 𝐿𝐿 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 −𝑘𝑘 = 𝑞𝑞0" (𝑡𝑡) , 𝑥𝑥 = 0 𝜕𝜕𝜕𝜕 𝑇𝑇 = 𝑇𝑇0 (𝑡𝑡) , 𝑥𝑥 = 0 𝑇𝑇 = 𝑇𝑇0 , 𝑡𝑡 = 0

Requires an initial condition 𝜃𝜃 = 𝜃𝜃0 , 𝑡𝑡 = 0 Too many initial conditions. Remove 𝜃𝜃 = 𝜃𝜃1 , 𝑡𝑡 = 𝑡𝑡1 𝑑𝑑𝑑𝑑

Requires a second initial condition 𝑑𝑑𝑑𝑑 = 𝑣𝑣0 , 𝑡𝑡 = 0 Remove 𝑥𝑥 = 𝑥𝑥1 , 𝑡𝑡 = 𝑡𝑡1

𝑑𝑑𝑑𝑑

Requires a second initial condition 𝑑𝑑𝑑𝑑 = 𝑣𝑣0 , 𝑡𝑡 = 0

Requires a boundary condition at x=L. Requires an initial condition.

Requires a boundary condition at x=L. Over specified at x+L. Remove one of the conditions.

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Problem 3.3: Coupled Mass-Spring-Dampers Consider the following system with two sets of mass-spring-dampers. A force f1 is applied directly to mass-1 while a force f2 is applied to mass-2. The spring constants are k1 and k2, and the damping coefficients are b1 and b2. The equilibrium positions with f1=f2=0 correspond to x1=x2=0.

x2

x1 b1

b2 m1

f1

k1

m2

f2

k2

Derive the complete mathematical model of this system.

Solution Draw free body diagrams of the two masses and apply Newton’s 2nd law.

Equation of Motion for Mass-1 𝑚𝑚1 𝑥𝑥̈ 1 = � 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓

𝑚𝑚1 𝑥𝑥̈ 1 = −𝑏𝑏1 𝑥𝑥̇ 1 − 𝑏𝑏2 (𝑥𝑥̇ 1 − 𝑥𝑥̇ 2 ) − 𝑘𝑘1 𝑥𝑥1 − 𝑘𝑘2 (𝑥𝑥1 − 𝑥𝑥2 ) + 𝑓𝑓1 𝑚𝑚1 𝑥𝑥̈ 1 + 𝑏𝑏1 𝑥𝑥̇ 1 + 𝑏𝑏2 (𝑥𝑥̇ 1 − 𝑥𝑥̇ 2 ) + 𝑘𝑘1 𝑥𝑥1 + 𝑘𝑘2 (𝑥𝑥1 − 𝑥𝑥2 ) = 𝑓𝑓1 Equation of Motion for Mass-2 𝑚𝑚2 𝑥𝑥̈ 2 = � 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓

Initial conditions at t=0

𝑚𝑚2 𝑥𝑥̈ 2 = −𝑏𝑏2 (𝑥𝑥̇ 2 − 𝑥𝑥̇ 1 ) − 𝑘𝑘2 (𝑥𝑥2 − 𝑥𝑥1 ) + 𝑓𝑓2 𝑚𝑚2 𝑥𝑥̈ 2 + 𝑏𝑏2 (𝑥𝑥̇ 2 − 𝑥𝑥̇ 1 ) + 𝑘𝑘2 (𝑥𝑥2 − 𝑥𝑥1 ) = 𝑓𝑓2 𝑥𝑥1 = 𝑥𝑥1,0 𝑥𝑥2 = 𝑥𝑥2,0 𝑥𝑥̇ 1 = 𝑣𝑣1,0 𝑥𝑥̇ 2 = 𝑣𝑣2,0

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Problem 3.4: A Series of Spring-Mass-Dampers A series of coupled mass-spring-dampers consisting of n-masses in series are connected by springs and dampers. The masses m (kg) are identical, the springs are identical with spring constant k (N/m), and the dampers are identical with damping coefficient b (N m/s). x2

x1

f1

The initial conditions are

xn

fn

f2

•

𝑥𝑥1,0 𝑣𝑣1,0 𝑥𝑥1 (0) ⎡ 𝑥𝑥• 1 (0) ⎤ 𝑥𝑥2,0 𝑣𝑣2,0 𝑥𝑥 (0) ⎢ ⎥ � 2 � = � ⋮ � and ⎢ 𝑥𝑥2 (0) ⎥ = � ⋮ � ⋮ ⎢• ⋮ ⎥ 𝑥𝑥𝑛𝑛,0 𝑣𝑣𝑛𝑛,0 𝑥𝑥𝑛𝑛 (0) ⎣𝑥𝑥𝑛𝑛 (0)⎦ Each mass has an individual external force applied, 𝑓𝑓𝑖𝑖 (𝑡𝑡). The displacements are measured from static equilibrium with no applied forces. Derive the mathematical model of the system for the displacements xi, for i =1 to n.

Solution Using free body diagrams for mass-1, general mass-i, and mass–n, we get the following coupled set of ODEs. Mass-1: 𝑚𝑚𝑥𝑥̈ 1 + 𝑏𝑏𝑥𝑥̇ 1 + 𝑏𝑏(𝑥𝑥̇ 1 − 𝑥𝑥̇ 2 ) + 𝑘𝑘𝑥𝑥1 + 𝑘𝑘(𝑥𝑥1 − 𝑥𝑥2 ) = 𝑓𝑓1

Mass-i, i=2,…n-1 𝑚𝑚𝑥𝑥̈ 𝑖𝑖 + 𝑏𝑏(𝑥𝑥̇ 𝑖𝑖 − 𝑥𝑥̇ 𝑖𝑖 −1 ) + 𝑏𝑏(𝑥𝑥̇ 𝑖𝑖 − 𝑥𝑥̇ 𝑖𝑖+1 ) + 𝑘𝑘(𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖−1 ) + 𝑘𝑘(𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖+1 ) = 𝑓𝑓𝑖𝑖 Mass-n: 𝑚𝑚𝑥𝑥̈ 𝑛𝑛 + 𝑏𝑏𝑥𝑥̇ 𝑛𝑛 + 𝑏𝑏(𝑥𝑥̇ 𝑛𝑛 − 𝑥𝑥̇ 𝑛𝑛−1 ) + 𝑘𝑘𝑥𝑥𝑛𝑛 + 𝑘𝑘(𝑥𝑥𝑛𝑛 − 𝑥𝑥𝑛𝑛−1 ) = 𝑓𝑓𝑛𝑛

4

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