Assignment Solution 6
MECH 412- Dynamics of Systems Assignment #6 Due in class on Thursday, March 4, 2010 1. Problem B-7-3. Also draw the equivalent circuit representation of the system. 2. Problem B-7-9. Also draw the electrical circuit representation of the system. 3. Problem B-7-17*. (facultative, not graded) 4. Problem B-7-18. Show that this device may be represented as a transformer in the impedance or the mobility mechanical analog circuit representation. Identify the transformer ratio in terms of the piston areas. 5. Consider a bass-reflex enclosure, i.e. a closed enclosure in which an opening, usually called the port, has been made. Neglecting the acoustic radiation impedance from the speaker diaphragm, Zrad1, and assuming the port has no radiation impedance Zrad2, sketch the analog electro-acoustical circuit for the loudspeaker enclosure with a port. The compliance of the enclosure is CAB. The mass loading on the back side of the diaphragm is MAB. The mass and resistance of the air in the port that penetrates the side of the box, including the inner end correction, are MAP and RAP, respectively. The cone moves back and forth, like a piston, generating a certain flow rate.
CAB, MAB SD
MAP, RAP
Sp
6. Extra problem (facultative, will not be graded). The device shown has circular symmetry about the vertical center line. The base is excited by a simple harmonic force generator F0ejωt. A piston of mass Mp and area Ap entrains fluid of density ρ within a cylinder of mass MT and inner radius a, as shown. The cylinder and the annular cavity, CA1, are completely filled with fluid, which motion may be lumped into three equivalent fluid masses MA1, MA2 , and MA3 (cavity neck) as the piston moves. The cylinder open end has radiation impedance Zrad. C1, C2, and C3 are single coil springs. The cylinder moves vertically with a displacement y. The piston displacement is x, and the friction between piston and cylinder is modeled using a mechanical viscous damper RM2. The piston and cylinder are supported as shown by a support of mass MS, with displacement z, and a base of mass MH with displacement w. Friction between support and base is RM1. Consider small oscillations around the equilibrium position of the system. a) Assuming all mechanical elements to be fixed and rigid, draw the acoustical impedance equivalent circuit for the fluid domain alone, with no mechanical elements. b) Assuming the mechanical system operates in vacuo, with no fluid loading, draw the impedance analog electrical circuit equivalent of the mechanical system. c) draw the impedance analog electrical circuit representation of the complete mechanicalacoustical system in terms of the parameters shown, coupling fluid and mechanical elements through the use of one or more transformers with ratio Ap:1 (mechanical to fluid) or 1:Ap (fluid to mechanical). d) Assuming all mechanical springs to be infinitely stiff and neglecting friction, remove the transformer by passing the mechanical elements through the transformer. Would there be any sound produced? Explain why.
MA2
MA3 y
CA1
MT C3 MA1 RM2
x
Mp, Ap
RM1
C2
z C1 MS MH w
F0 e
iωt
5)
6) See the following page:
DEPARTMENT OF MECHANICAL ENGINEERING MECH 412 – SYSTEM DYNAMICS EQUIVALENT CIRCUITS (15 pts) Name:_________________________ Thursday April 9, 2009 The device shown has circular symmetry about the vertical center line. The base is excited by a simple harmonic force generator F0ejωt. A piston of mass Mp and area Ap entrains fluid of density ρ within a cylinder of mass MT and inner radius a, as shown. The cylinder and the annular cavity, CA1, are completely filled with fluid, which motion may be lumped into three equivalent fluid masses MA1, MA2 , and MA3 (cavity neck) as the piston moves. The cylinder open end has radiation impedance Zrad. C1, C2, and C3 are single coil springs. The cylinder moves vertically with a displacement y. The piston displacement is x, and the friction between piston and cylinder is modeled using a mechanical viscous damper RM2. The piston and cylinder are supported as shown by a support of mass MS, with displacement z, and a base of mass MH with displacement w. Friction between support and base is RM1. Consider small oscillations around the equilibrium position of the system. a) Assuming all mechanical elements to be fixed and rigid, draw the acoustical impedance equivalent circuit for the fluid domain alone, with no mechanical elements. b) Assuming the mechanical system operates in vacuo, with no fluid loading, draw the impedance analog electrical circuit equivalent of the mechanical system. c) draw the impedance analog electrical circuit representation of the complete mechanicalacoustical system in terms of the parameters shown, coupling fluid and mechanical elements through the use of one or more transformers with ratio Ap:1 (mechanical to fluid) or 1:Ap (fluid to mechanical). d) Assuming all mechanical springs to be infinitely stiff and neglecting friction, remove the transformer by passing the mechanical elements through the transformer. Would there be any sound produced? Explain why.
y
CA1 MT C3 MA1 RM2
x
Mp, Ap
C2
RM1
z C1
MH w iωt
F0 e
System
potential flux quantity quantity a b
resistance element a R= b 2H Rl = Q
liquid level
h [m]
q 3 [m /s]
pneumatic
p [Pa]
qm [kg/s]
⎧ d ( Δp ) ⎫ Rpn = ⎨ ⎬ ⎩ dq ⎭eq
acoustic
p [Pa]
U [m3/s]
⎧ d ( Δp ) ⎫ RA = ⎨ ⎬ ⎩ dU ⎭eq
hydraulic
p [Pa]
qm [kg/s]
⎧ d ( Δp ) ⎫ Rh = ⎨ ⎬ ⎩ dqm ⎭eq
thermal
θ [K]
qh [Watt]
Rt =
mechanical impedance mechanical mobility electrical
f [N] u [m/s] V [V]
u [m/s] f [N] i [A]
d ( Δθ ) 1 = dqh K
capacitance element 1 a = ∫ b dt C Cl = Atank
Vres nRgasT
Cpn =
CA =
inductance element db a=L dt L M l = pipe gApipe
ρ Lneck
Vcav V = cav 2 ρ c0 ργ RT
MA =
ρVres β
Mh =
Ch =
Lhose Ahose
M pn =
Aneck Lhose Ahose
Ct = mc p,V
RM
CM=1/k
MM
rM=1/RM
MM
CM=1/k
R
C
L
CAB, MAB SD
MAP, RAP
Sp
6. Extra problem (facultative, will not be graded). The device shown has circular symmetry about the vertical center line. The base is excited by a simple harmonic force generator F0ejωt. A piston of mass Mp and area Ap entrains fluid of density ρ within a cylinder of mass MT and inner radius a, as shown. The cylinder and the annular cavity, CA1, are completely filled with fluid, which motion may be lumped into three equivalent fluid masses MA1, MA2 , and MA3 (cavity neck) as the piston moves. The cylinder open end has radiation impedance Zrad. C1, C2, and C3 are single coil springs. The cylinder moves vertically with a displacement y. The piston displacement is x, and the friction between piston and cylinder is modeled using a mechanical viscous damper RM2. The piston and cylinder are supported as shown by a support of mass MS, with displacement z, and a base of mass MH with displacement w. Friction between support and base is RM1. Consider small oscillations around the equilibrium position of the system. a) Assuming all mechanical elements to be fixed and rigid, draw the acoustical impedance equivalent circuit for the fluid domain alone, with no mechanical elements. b) Assuming the mechanical system operates in vacuo, with no fluid loading, draw the impedance analog electrical circuit equivalent of the mechanical system. c) draw the impedance analog electrical circuit representation of the complete mechanicalacoustical system in terms of the parameters shown, coupling fluid and mechanical elements through the use of one or more transformers with ratio Ap:1 (mechanical to fluid) or 1:Ap (fluid to mechanical). d) Assuming all mechanical springs to be infinitely stiff and neglecting friction, remove the transformer by passing the mechanical elements through the transformer. Would there be any sound produced? Explain why.
MA2
MA3 y
CA1
MT C3 MA1 RM2
x
Mp, Ap
RM1
C2
z C1 MS MH w
F0 e
iωt
5)
6) See the following page:
DEPARTMENT OF MECHANICAL ENGINEERING MECH 412 – SYSTEM DYNAMICS EQUIVALENT CIRCUITS (15 pts) Name:_________________________ Thursday April 9, 2009 The device shown has circular symmetry about the vertical center line. The base is excited by a simple harmonic force generator F0ejωt. A piston of mass Mp and area Ap entrains fluid of density ρ within a cylinder of mass MT and inner radius a, as shown. The cylinder and the annular cavity, CA1, are completely filled with fluid, which motion may be lumped into three equivalent fluid masses MA1, MA2 , and MA3 (cavity neck) as the piston moves. The cylinder open end has radiation impedance Zrad. C1, C2, and C3 are single coil springs. The cylinder moves vertically with a displacement y. The piston displacement is x, and the friction between piston and cylinder is modeled using a mechanical viscous damper RM2. The piston and cylinder are supported as shown by a support of mass MS, with displacement z, and a base of mass MH with displacement w. Friction between support and base is RM1. Consider small oscillations around the equilibrium position of the system. a) Assuming all mechanical elements to be fixed and rigid, draw the acoustical impedance equivalent circuit for the fluid domain alone, with no mechanical elements. b) Assuming the mechanical system operates in vacuo, with no fluid loading, draw the impedance analog electrical circuit equivalent of the mechanical system. c) draw the impedance analog electrical circuit representation of the complete mechanicalacoustical system in terms of the parameters shown, coupling fluid and mechanical elements through the use of one or more transformers with ratio Ap:1 (mechanical to fluid) or 1:Ap (fluid to mechanical). d) Assuming all mechanical springs to be infinitely stiff and neglecting friction, remove the transformer by passing the mechanical elements through the transformer. Would there be any sound produced? Explain why.
y
CA1 MT C3 MA1 RM2
x
Mp, Ap
C2
RM1
z C1
MH w iωt
F0 e
System
potential flux quantity quantity a b
resistance element a R= b 2H Rl = Q
liquid level
h [m]
q 3 [m /s]
pneumatic
p [Pa]
qm [kg/s]
⎧ d ( Δp ) ⎫ Rpn = ⎨ ⎬ ⎩ dq ⎭eq
acoustic
p [Pa]
U [m3/s]
⎧ d ( Δp ) ⎫ RA = ⎨ ⎬ ⎩ dU ⎭eq
hydraulic
p [Pa]
qm [kg/s]
⎧ d ( Δp ) ⎫ Rh = ⎨ ⎬ ⎩ dqm ⎭eq
thermal
θ [K]
qh [Watt]
Rt =
mechanical impedance mechanical mobility electrical
f [N] u [m/s] V [V]
u [m/s] f [N] i [A]
d ( Δθ ) 1 = dqh K
capacitance element 1 a = ∫ b dt C Cl = Atank
Vres nRgasT
Cpn =
CA =
inductance element db a=L dt L M l = pipe gApipe
ρ Lneck
Vcav V = cav 2 ρ c0 ργ RT
MA =
ρVres β
Mh =
Ch =
Lhose Ahose
M pn =
Aneck Lhose Ahose
Ct = mc p,V
RM
CM=1/k
MM
rM=1/RM
MM
CM=1/k
R
C
L
