Post on 29-Jan-2021
Q1: Establish the equation of motion of each system about x. Regarding (1), answer the critical damping coefficient and damping ratio.
m
mc
x
m๐ ๐
Exercise of mechanical vibration (Damped vibration)
cx
(2) (3)
(1) x
๐ ๐๐
Basic
Q3: The top view of a door is shown in the figure. The door has a mass of 40 kg, 2.1 m height, 1.2 m width, and 0.05 m thickness. The door has a torsional spring and damper of coefficient ๐ 13.6 Nm/rad and ๐ Nms/rad, respectively. Determine the critical value of ๐ when the damping ratio is ๐ 1.
๐๐๐Torsional springand damper
Basic
Q5: Highway crash barriers are designed to absorb a vehicleโs kineticenergy without bringing the vehicle to such an abrupt stop that theoccupants are injured. The barrierโs materials and thickness arechosen to accomplish this. It can be modeled as the springโmassโdamper system. For this application, t = 0 denotes the time ofcollision at x(0) = 0. The speed of the vehicle at this moment is๐ฅ 0 22 m/s. For a particular barrier, k = 18,000 N/m and c =20,000 Nใปs/m. Determine how long it takes to for the vehicle ofmass 1,800 kg to stop, and how far it compresses the barrier.
Basic
Q6: Obtain the response of the following equation of motion
for each of the following three cases:a) ๐ฅ 0 2, ๐ฅ 0 5;b) ๐ฅ 0 2, ๐ฅ 0 5;c) ๐ฅ 0 2, ๐ฅ 0 0.And, draw x(t) for the three cases using a graph drawing software such as MS Excel.
Basic
Q8: Obtain the response of the following equation of motion
for the following case:๐ฅ 0 3, ๐ฅ 0 5.And, draw x(t) for the three cases using a graph drawing software such as MS Excel.
Basic
Rk
ฮฑ
x
ฮธ c
Q10: Consider a cylinder that rolls without slipping. Let x = 0 denote therest position of the cylinder. Neglect the mass of the spring anddamper. The mass moment of inertia of the cylinder is I.(1) Obtain the equation of motion in terms of x.(2) Determine the damped natural angular frequency of the
cylinder.
Basic
Q11: Derive the equation of motion for ฮธ. When ฮธ = 0, the spring is at itsnatural length. Assume that ฮธ is small (small angle assumption).Mass of the lever is negligible. Gravitational acceleration is g.Answer the damped natural angular frequency.
m
l1
l2
l3
ฮธ
c
k
Basic
Q12: Consider a lever system shown in the figure. Obtain its equation ofmotion in terms of x, and determine the damped natural angularfrequency. x = 0, when the lever is at its rest (statically equilibrium)position.
xma mb
ab
kc
Basic
Q14: In the following system, the spring coefficient and mass are k=10000 N/m and m=20 kg, respectively. Find the critical damping coefficient.
ใใญในใ P51โ2.9a
m
k
c
2k
Basic
Q15: ๅทฆ็ซฏใๆฏ็นใจใใฆๅ่ปขใงใใๅไฝๆฃใฎๅ ็ซฏใซ่ณช้ใฎ็ฉไฝใใคใใฆใใใใฐใญๅฎๆฐ k ,็ฒๆงๆธ่กฐไฟๆฐ c ๏ผ(1) ๐ใซ้ขใใฆใฎ้ๅๆน็จๅผใๆฑใใ๏ผ(2) ่จ็ๆธ่กฐไฟๆฐใๆฑใใ๏ผ
ใใญในใ P51โ2.10a
m
l
l
l
Basic
Q16: Consider a pendulum supported by a springโdamper system. A mass is attached at the end of a weightless rod. The rotation angle ฮธ is zero at the equilibrium position. Let the gravity acceleration be g.(1) Answer the equation of motion about ๐.(2) Answer c such that the pendulum is critically damped (๐ is
small enough).
m
k
c
๐
๐๐
Basic
Q18: If applicable, compute ฮถ, ฯn, and ฯd for the following characteristic roots of 1โd.o.f damped systems.(1) ๐ 2 6๐(2) ๐ 10(3) ๐ , (or 1 )
Basic
Q20: ไธ่จ(1)โ(3)ใฏ๏ผๆธ่กฐใฎใใ1่ช็ฑๅบฆๆฏๅ็ณปใฎ้ๅๆน็จๅผใงใใ.x is the displacement of the system. ๅบๆๆน็จๅผใฎๆ นใๅบใซ๏ผใใใใใ้ๆธ่กฐ็ณป๏ผไธ่ถณๆธ่กฐ็ณป๏ผ่จ็ๆธ่กฐ็ณปใฎใใใใงใใใใๅคๆญใใ๏ผ
(1) 3๐ฅ 2๐ฅ ๐ฅ 0(2) 2๐ฅ 5๐ฅ 2๐ฅ 0(3) ๐ฅ 4๐ฅ 4๐ฅ 0
Basic
Q21: Consider a cart of mass m being supported by a spring of coefficient k and a damper of coefficient c. x is the displacement of the cart. At the equilibrium position, x = 0. The three curves in the figure depict the displacement of the cart after being released with the initial conditions being ๐ฅ ๐ก 0 ๐ฅ and ๐ฅ ๐ก 0 ๐ฃ . Answer the correct response of the system from X, Y, and Z when (1) ๐ 2 ๐๐(2) ๐ 2 ๐๐(3) ๐ 2 ๐๐.
0
x0
Time [s]
x: Cart d
isplacemen
t
X
YZ
Basic
Q22: ๆง้ ็ฉใซๅ ใใๅf ใจใใฎใใใฟxใฎๆฏf/xใ่ค็ด ๆฐใ็จใใฆ่กจใใจ๏ผๅพฎๅใ็จใใใใใ็ฐกไพฟใจใชใ๏ผ็่งฃใไฟ้ฒใใใใจใใใ๏ผไธๅณใฎ็ฒๆงๆตๆใฎใฟใใใชใๆง้ ็ฉใไพใจใใ๏ผ็ฒๆงๆตๆใฎๅ ็ซฏใซๅf ใๅ ใ๏ผใใฎใใใฟใxใจใใ๏ผๆง้ ็ฉใ้็ๅนณ่กกใซใใใจใ๏ผx = 0ใงใใ๏ผใใฎ็ณปใฎ้ๅๆน็จๅผใฏ๐ ๐๐ฅใงใใ๏ผใใใง๏ผ๐ฅ ๐ดsin๐๐ก ใจใใใจ๏ผ๐ ๐๐ด๐cos๐๐ก๐๐๐๐ฅใจใชใ๏ผๅใจใใใฟใฎๆฏใฏ๏ผ๐๐ฅ ๐๐๐ใจ่กจใใใ๏ผใใฎใจใใ๏ผ็ฒๆง่ฆ็ด ใฏ๏ผ
๐ฅ ๐ก๐ใใใฟใฎๅจๆณขๆฐ๐ใซไพๅญใ๏ผใใใฟใใใ ๐/2 rad ไฝ็ธใ้ฒใใ ๏ผใใใฟใฎ้
ๅบฆใซๅฏพใใๆๅใ็บ็ใใใ๏ผ
Basic
Answer the f/x ratios of the following systems using a complex unit.
๐๐ฅ ๐ก
๐ ๐(1)
๐๐ฅ ๐ก๐ ๐
(2)
๐๐ฅ ๐ก
๐๐(3)
๐๐๐ฅ ๐ก๐ ๐
(4)
๐
Q23 Consider a lever system on a horizontal plane, for which we do not need to consider the effect of the gravity. As shown in the figure, two point masses are fixed on the lever, one of which end is fixed to a frictionless pin. The lever rotates around this pin, and its rotation angle is small enough such that sin๐~๐. At its static equilibrium position, ๐ 0. Answer the following problems.(1) Find the mass moment of inertia of the lever with the two
masses about the pin.(2) Find the equation of motion of the system about ๐. Use I as the
mass moment of inertia of the system.(3) Solve the equation acquired in (2) about ๐ with the initial
conditions ๐ 0 ๐ and ๐ 0 0 when the system is underdamped.
Basic
mm
c
Q24 The behaviors of 1โDOF systems can be understood from their characteristic roots. Complete the following roots locus by finding the correct combinations of the loci of roots (ใโใป) and the motions of 1โDOF springโmassโdamper systems (AโE).
Im
Re
ใ
ใฏ
ใปใซ
ใ
Roots locus of 1โDOF springโmassโdamper systems
Basic
Motions (displacement) after being released from rest
Time
(A)
(C)
(B)
(D)
(E)
Time
Time
Displacemen
t
Displacemen
t
Q26 When the equation of motion of 1โd.o.f vibration system is given asbelow, answer the following problems.๐ฅ 4๐ฅ 2๐ฅ 0(1) Choose one among ใ, ใ, and ใฏ, to express the nature of the
system.ใ: Overdampedใ: Critically dampedใฏ: Underdamped
(2) Find the solution of ๐ฅ with the initial conditions being ๐ฅ 0 1,๐ฅ 0 0.
Basic
Q28 Consider a pendulum supported by a springโdamper system. A mass is attached at the end of a weightless rod. The rotation angle ฮธ is zero at the equilibrium position. Let the gravity acceleration be g.(1) Answer the equation of motion about ๐.(2) Answer c such that the pendulum is
critically damped (๐ is small enough). (3) When the system is underdamped,
find the frequency of free vibration ๐ . (4) Find the solution of free vibration ๐ ๐ก with the initial condition ๐ 00 and ๐ 0 ๐ . m
k
c
๐ ๐๐
Basic
Q2: An inverted pendulum is supported by two springs as shown in the figure. Assume small angles of vibration and neglect the rod mass. Answer the following problems.
a) Derive the equation of motion of the system about ๐.
b) What is the relation among ๐ , ๐ , ๐, and ๐ for the characteristic roots to include at least one positive real value?
c) Describe the behavior of the system when the characteristic roots include positive real values.
๐๐ ๐๐๐ ๐
๐
๐๐
Intermediate
Q7: Consider a disk that rotates on the floor without slippage. Its centeris connected to walls by way of springโdamper systems withoutfriction of which spring and damping coefficients are k and c,respectively. The diskโs radium and mass are r and m, respectively.(1) Obtain the equation of motion of the disk in terms of x.(2) When the system is critically damped, answer the following two
problems.(1) Solve the motion of the disk with initial conditions being๐ฅ 0 0 and ๐ฅ 0 ๐ฃ .(2) Find tโ at which the displacement of the disk becomes
greatest.(3) When the system is underdamped (ฮถ < 1), solve the response of
the system with the initial conditions being ๐ฅ 0 0 and๐ฅ 0 ๐ฃ .Intermediate
(4) Find the logarithmic decrement (ๅฏพๆฐๆธ่กฐ็) ฮด, which is thelogarithmic ratio of two successive amplitudes.
๐ฟ ln ๐๐x(t)
t
a1 a2 a3r
Q13: A homogeneous stick of mass m and length L revolves around O.The stick is supported by a spring and two dampers. Gravityacceleration g acts on the stick.
(1) Answer the mass moment of inertial of the stick around O.
(2) Answer the damped natural angular frequency of the stick. You may use a small angle assumption.
ฮธ
L/4
L/2
L
O
Intermediate
Q19: A hammer strikes a metal plate at the initial speed of v0. Thehammer mass is ๐ and the plate mass is ๐ . The metal plate issupported by the stiffness ๐ and damping ๐. Answer the expressionof the plate about x, of which equilibrium position is ๐ฅ 0. Do thisfor two values of the coefficient of restitution (1) e = 1 and (2) e = 0.Note, when e = 0, mp and mh moves as a mass of mp+mh. When e =1, mh rebounds after an extremely short contact period. Note that๐ 2 ๐ ๐ and ๐ 2 ๐ ๐ ๐.
Intermediate
Q25 Consider a cart system with Coulombโs friction. It is supported by a spring of coefficient k. The position of the cart is denoted by x and its origin is the equilibrium position of the system with no friction. Answer the following problems.(1) Find the equation of motion of the system when the friction
force is given by ๐ ๐ฅ 0๐ ๐ฅ 0 .(2) Solve the above equation about x at ๐ก 0, with initial
position ๐ฅ 0 0 and velocity ๐ฅ 0 ๐ฃ 0 .(3) Find the maximum position of the cart at ๐ก 0, .
Intermediate
Q17: Consider an inverted pendulum supported by a rotational spring of constant k. The rotational spring produces the restoring torque proportional to the deflection angle. A mass of m is on the tip of the rod of length l. The rotational angle of the pendulum is denoted by ฮธ, and ฮธ = 0 when the pendulum stands straight. The spring is at its natural length when ฮธ = ฮธ0. The gravitational acceleration g acts on the mass. Answer the following problems. (1) Answer the mass moment of inertia of the pendulum about the
pivot.(2) Establish the equation of motion about ฮธ.(3) Consider a case where ฮธ0 = 0. The spring is at its natural length
when the pendulum stands straight.3โ1) Establish the equation of motion for a small angle (sinฮธ ~
ฮธ).3โ2) Answer the characteristic root of the system.
ๅคงๅญฆ้ขๅ ฅ่ฉฆๅ้กใ่งฃใใฆใฟใใ! 2017ๅนด ๅๅคๅฑๅคงๅญฆ ๅคงๅญฆ้ขๅ ฅ่ฉฆๅ้ก
3โ3) Depending on the k value, the behavior of the system can be categorized into three cases. Describe all of them.
3โ4) In the case where the pendulum vibrates around ฮธ = 0, answer the natural angular frequency.
(4) Consider a case where the pendulum is statically equivalent at ๐ . We discuss the behavior at ๐ .4โ1) Answer the equation of motion about ฮธ.4โ2) Answer the natural angular frequency of the system.
mฮธ
kl
m
k
๐ฅ ๐กQ25 (Figure) Q27 Consider a 1โdof springโmassโdamper system in the figure. The
mass is supported by two springs, a beam, and dashpot. The massmoves only horizontally and its displacement is small. The length,Youngโs modulus, and second moment of are L, E, and I,respectively. Answer the following problems.
ๅคงๅญฆ้ขๅ ฅ่ฉฆๅ้กใ่งฃใใฆใฟใใ! 2019ๅนด ๆฑๅๅคงๅญฆ ๅคงๅญฆ้ขๅ ฅ่ฉฆๅ้ก
(1) Find the equivalent spring coefficient of the beam: ๐ .(2) Find the collective spring coefficient of the entire system, K using๐ .(3) Find the critical damping coefficient of the system, Cc, using K.(4) When the system is underdamped, find the period of damped
natural vibration T.(5) Find the ratio of vibratory amplitudes at two time points: ๐ฅ ๐ก๐ to ๐ฅ ๐ก .
A: Translate the following paragraph into Japanese.
The free response of damped vibration system has equation
Its characteristic roots are
There are three cases depending on the sign of the expressionunder the square root.i ๐ 4๐๐ (this will be underdamping, c is small relative to m
and k)ii ๐ 4๐๐ (this will be overdamping, c is large relative to m andk)iii ๐ 4๐๐ (this will be critical damping, c is just between overand under damping).
๐๐ฅ ๐๐ฅ ๐๐ฅ 0..
i) UnderdampingDamping force generates heat and dissipates energy. When thedamping constant is small, the system oscillates, but withdecreasing amplitude. Over time, it comes to rest at equilibrium.
ii) OverdampingWhen the damping is large the damping force is so great that thesystem cannot oscillate.
iii) Critical dampingAs in the overdamped case, this does not oscillate. For fixed m andk, choosing c to be the critical damping constant gives the fastestreturn of the system to its equilibrium position. In engineeringdesign, this is often a desirable property.
B: Translate the following paragraph about springโmassโdamper systems.
Consider a springโmassโdamper system described by๐๐ฅ ๐๐ฅ ๐๐ฅ 0where m, k, and c are the mass, spring coefficient, and dampingcoefficient, respectively. This equation is rewritten as๐ฅ 2๐๐๐ฅ ๐ ๐ฅ 0where ๐ ๐/๐ and ๐ ๐/2 ๐๐. ๐ is a dimensionless valuecalled the damping ratio and defined by the ratio of c and thecritical damping coefficient. The characteristic roots are๐ ๐๐ ๐๐ 1 ๐ .Its imaginary part is the frequency of the oscillation. This frequencyis called the damped natural angular frequency and is smaller thanundamped natural frequency.
COLUMN: ๆฆ็ฅ็ๅญฆ็ฟ
ๆฆ็ฅ็ๅญฆ็ฟใจใฏ๏ผ่ฏใๆ็ธพ๏ผAใใใใฏB๏ผใงๅไฝใๅๅพใใฆใใใใใฎๅน็็ใชๅญฆ็ฟใฎใใจใงใ๏ผไพใใฐ๏ผ้ๅปใฎ่ฉฆ้จๅ้กใไธญๅฟใซๅๅผทใใใ๏ผใใฎๆผ็ฟๅ้กใ่งฃใใจใใ๏ผ่งฃๆณใใใฎๆ้ ใใฎใใฎใ่จๆถใใใใจใซๅชใใใใใๅญฆ็ฟๆณใงใ๏ผๆฆ็ฅ็ๅญฆ็ฟใฎๅ้กใฏ๏ผใชใใใฎใใใช่งฃๆณ/ๅ็ใซใชใใฎใใ็ใซ็่งฃใใใใจใใๅงฟๅข๏ผใใชใใกๆ้ทใฎๆฉไผ๏ผใๅคฑใฃใฆใใพใใใจใงใ๏ผๅทฅๅญฆ่ ใใใใฏๆ่ก่ ใจใใฆๆฐใใๅ้กใซ้ขใใใจใ๏ผ็ใใใๅพๆผใใใฆใใใใฎใฏๅ็ใฎๆทฑใ็่งฃใงใ๏ผ
๐๐ฅ ๐ ๐ ๐ฅ ๐๐ฅ 0๐ 2 ๐๐๐ ๐๐๐ ๐ ๐2 ๐๐๐๐ฅ ๐ถ ๐ถ๐ถ ๐ถ ๐ฅ 0
AnswerQ1(1) ๐๐ฅ 2๐๐ฅ 0(2)
(3)
Q2 ๐๐ ๐ 2๐๐ ๐๐๐ ๐ 0a) 2๐๐ ๐๐๐ 0b)c) The pendulum falls down or does not remain around ๐ 0.The general solution is ๐ ๐ด exp ๐ ๐ก ๐ด exp ๐ ๐ก.If ๐ or ๐ includes a positive real value, ๐ diverges over time.
Q3 ๐ 401.2 0.05 2003๐ผ ๐ ๐ ๐๐ฅ๐๐ฆ. 19.23 kgm.๐ผ ๐ ๐ ๐ ๐๐ 0ฮถ 1 ๐2 ๐๐พEquation of Motion
๐ 2 ๐ผ๐ 32.3 Nms/rad
Q4: Translate the following paragraph into Japanese.
ๆธ่กฐๆฏๅ็ณปใฎ่ช็ฑๅฟ็ญใฏ๏ผไธ่จใฎๅผใง่กจ็พใใใ:
ใใฎ็นๆงๆน็จๅผใฎๆ นใฏ๏ผ
ใซใผใใฎไธญใฎ็ฌฆๅทใซใใฃใฆ๏ผ3ใคใฎๅ ดๅใใใ๏ผi ๐ 4๐๐ (ใใใฏไธ่ถณๆธ่กฐใงใใ๏ผc ใฏ m ใจ k ใซๆฏในใฆๅฐใใ)ii ๐ 4๐๐ (ใใใฏ้ๆธ่กฐใงใใ๏ผc ใฏ m ใจ k ใซๆฏในใฆๅคงใใ)iii ๐ 4๐๐ (ใใใฏ่จ็ๆธ่กฐใงใใ๏ผc ใฏไธ่ถณๆธ่กฐใจ้ๆธ่กฐใฎ่ชฟๅบฆ้ใงใใ).
๐๐ฅ ๐๐ฅ ๐๐ฅ 0..
i) ไธ่ถณๆธ่กฐๆธ่กฐๅใฏ็ฑใ็บ็ใใ๏ผใจใใซใฎใผใๆฃ้ธใใใ๏ผๆธ่กฐไฟๆฐใๅฐใใใจใ๏ผใทในใใ ใฏๆฏๅใใใ๏ผใใฎๆฏๅน ใฏๆธๅฐใใฆใใ๏ผๆ้ใ็ตใคใจ๏ผๅนณ่กก็ถๆ ใง้ๆญขใใ๏ผ
ii) ้ๆธ่กฐๆธ่กฐใๅคงใใๆ๏ผๆธ่กฐๅใฏๅคงใใใชใ๏ผใทในใใ ใฏๆฏๅใงใใชใ๏ผ
iii) ่จ็ๆธ่กฐ้ๆธ่กฐใฎๅ ดๅใจๅๆงใซ๏ผใใใฏๆฏๅใใชใ๏ผm ใจ k ใๅบๅฎใใใฆใใใจใ, c ใ่จ็ๆธ่กฐไฟๆฐใจใชใใใใซ้ธๆใใใจ๏ผใทในใใ ใฏๆใๆฉใ๏ผๅนณ่กกไฝ็ฝฎใซๆปใ๏ผๅทฅๆฅญใใถใคใณใงใฏ๏ผใใใฏใใฐใใฐๆใพใใๆง่ณชใงใใ๏ผ
Q5 1800๐ฅ 20000๐ฅ 18000๐ฅ 0By solving the characteristic roots of the above equation of motion1800๐ 20000๐ 18000 0๐ 0.99๐ 10.12๐ฅ ๐ก ๐ด๐ ๐ต๐ ๐ฅ 0 ๐ด ๐ต 0๐ฅ 0 ๐ด๐ ๐ต๐ 22๐ฅ ๐ก 2.41 ๐ . ๐ .
๐ฅ ๐ก 2.41 0.99๐ . 10.12๐ . 0When the car stops ๐ฅ ๐ก 0. ๐ก 0.254 s๐ฅ ๐ก 0.254 2.41 ๐ . ๐ . 1.69 m
๐ด 133 , ๐ด 73 ๐ด 1, ๐ด 1๐ด 83 , ๐ด 23(a) (b)
(c)
Q6๐ฅ ๐ก ๐ด ๐ ๐ด ๐
0
1
2
3
0 1 2 3 4 5
x(t)
t
a
b c
Q7
(1) 32 ๐๐ฅ ๐๐ฅ 2๐๐ฅ 0(2โ1)๐ฅ ๐ฃ ๐กexp ๐3๐ ๐ก (2โ2)๐กโฒ 1๐(3) ๐ฅ ๐ฃ๐ 1 ๐ exp ๐ ๐๐ก sin ๐ 1 ๐ ๐ก ๐ 2
๐3๐(4) ๐ฟ 2๐๐1 ๐
๐ ๐2 3๐๐๐ฅ ๐ฃ ๐กexp 2 ๐3๐ ๐ก
or
Q8 ๐ฅ ๐ก ๐ด ๐ก๐ด ๐ ๐ด 3, ๐ด 11
0
1
2
3
4
0 1 2 3 4 5
x(t)
t
Q9: Translate the following paragraph about springโmassโdampersystems.
ๆฌกใฎๅผใง่กจใใใใฐใญโใในโใใณใ็ณปใ่ใใ.๐๐ฅ ๐๐ฅ ๐๐ฅ 0.ใใใง๏ผ m, k, ใใใณ c ใฏใใใใ๏ผ่ณช้๏ผใฐใญไฟๆฐ๏ผๆธ่กฐไฟๆฐใงใใ๏ผใใฎๅผใฏ๐ฅ 2๐๐๐ฅ ๐ ๐ฅ 0ใจๆธใๆใใใ๏ผ๐ ๐/๐ ใใใณ ๐ ๐/2 ๐๐ ใงใใ๏ผ๐ ใฏๆธ่กฐๆฏใจๅผใฐใใ็กๆฌกๅ ้ใงใใ๏ผc ใจ่จ็ๆธ่กฐไฟๆฐใฎๆฏใงๅฎ็พฉใใใ๏ผ็นๆงๆน็จๅผใฎๆ นใฏ๐ ๐๐ ๐๐ 1 ๐ใงใใ๏ผใใฎ่ๆฐ้จใๆฏๅใฎๅจๆณขๆฐใงใใ๏ผใใฎๅจๆณขๆฐใฏ๏ผๆธ่กฐๅบๆ่งๅจๆณขๆฐใจๅผใฐใ๏ผๆธ่กฐใฎ็กใๅบๆๅจๆณขๆฐใใใๅฐใใใชใ๏ผ
(1)
(2)
๐ ๐ฅ ๐๐ฅ ๐๐ฅ 0๐
Q10
๐ ๐ 1 ๐ ๐ ๐2 ๐ ๐ผ๐ ๐Q11: Derive the equation of motion for ฮธ. When ฮธ = 0, the spring is at
its natural length. Assume that ฮธ is small. Mass of the lever isnegligible. Gravitational acceleration is g. Answer the dampednatural angular frequency.
m
l1
l2
l3
ฮธ
c
k
๐ ๐ 1 ๐๐ ๐๐๐ ๐๐๐๐๐ ๐๐2 ๐๐ ๐๐๐ ๐๐
๐๐ ๐ ๐๐ ๐ ๐๐๐ ๐๐ ๐ 0
Q12: Consider a lever system shown in the figure. Obtain its equationof motion in terms of x, and determine the damped naturalangular frequency. x = 0, when the lever is at its rest (staticallyequilibrium) position.
xma mb
ab
kc
๐ ๐ ๐๐ ๐ฅ ๐๐ฅ ๐ ๐๐ ๐ฅ 0๐ ๐ 1 ๐ ๐ ๐พ๐ ๐ ๐2 ๐๐พ๐ ๐ ๐ ๐๐ ๐พ ๐ ๐๐
ใใณใใฎๅๆๆธ่กฐไฟๆฐใฏ๏ผ้ๅบฆใฎ้ขไฟใใ ใฎ้ขไฟ
ใๆใ็ซใคใฎใง๏ผ ๐ ๐ ๐๐ ๐ใจใชใ๏ผๅพใ๐ใซใคใใฆ๏ผ่ฟไผผใชใใง่กจใใ้ๅๆน็จๅผใฏ๏ผใจใชใ๏ผๅพฎๅฐๆฏๅใงใใใฎใง๏ผไปฅไธใฎใใใซๆด็ใงใใ๏ผ
ใใใซ๏ผๆธ่กฐๅบๆ่งๆฏๅๆฐใฏ
(1) ๅ็นOใพใใใฎๆ ฃๆงใขใผใกใณใ๐ผ ใฏ๐ผ ๐๐ฟ ๐ ๐๐ 13 ๐๐ฟ
(2)
016223
1 22
21
212
LkLmgL
ccccmL
Q13
2cos
sin2
4cossin
4sin
231
21
212
22 L
dt
Ld
ccccLLkLmg
dtdmL
2
21
21
)(43
41
823
ccmcckLmg
mLd
Q14
ใใญในใ P51โ2.9a
m
k
c
2k
๐ถ 2 ๐๐พโด ๐ 2 20 3 10000400 15 Ns/m
๐ ๐ฅ ๐๐ฅ 3๐๐ฅ 0 ๐ ๐๏ผ ๐ถ ๐๏ผ ๐พ 3๐ใจใใใจ่จ็ๆธ่กฐใจใชใๆ๏ผ ๏ผ
Q15
ml
ll
(1) ๐ใซ้ขใใฆใฎ้ๅๆน็จๅผใๆฑใใ
(2)
๐ผ ๐๐๐ผ ๐ ๐ ๐ ๐๐๐ ๐ ๐ 23 ๐๐ 23 ๐ 49 ๐๐ ๐๐ ๐ 13 ๐๐ 13 ๐ 19 ๐๐ ๐9๐๐ ๐๐ 4๐๐ 0๐ถ 2 9๐ 4๐12 ๐๐
Q16(1) Answer the equation of motion about ๐.(2) Answer c such that the pendulum is critically damped.
(1)๐๐ ๐ ๐ถ๐ ๐ ๐๐ ๐๐๐ ๐ 0(2)๐ 2 ๐๐พ๐๐ 2 ๐๐ ๐๐ ๐๐๐โด ๐ 2๐ ๐๐ ๐๐ ๐๐๐
Q17.(1) ๐ผ ๐๐ (2) ๐ผ๐ ๐ ๐ ๐ ๐๐๐sin๐(3โ1)๐ผ๐ ๐๐ ๐๐๐๐ (3โ2) ๐ ๐๐๐ ๐๐๐(3โ3)
Case 1: ๐๐๐ ๐ 0็นๆงๆน็จๅผใฎๆ นใๆญฃใฎๅฎ้จใๆใ๏ผ่้จใๆใใชใใฎใง๏ผ่งฃใฏ็บๆฃใใ๏ผใใชใใก๏ผใฐใญๅฎๆฐใๅฐใใ๏ผๆฏๅญใๆฏใใใใจใใงใใชใใฎใง๏ผๆฏๅญใ่ปขๅใใ๏ผ
Case 2: k = mgl็นๆงๆน็จๅผใฎๆ นใ0ใซใชใใฎใง๏ผ่งๅบฆใฏๅคๅใใชใ๏ผใคใพใ๏ผๆฏๅญใฏ็ไธใๅใใใพใพ้ๆญขใใ๏ผ
Case 3: ๐๐๐ ๐ 0็นๆงๆน็จๅผใฎๆ นใ่้จใฎใฟใซใชใใฎใง๏ผฮธ = 0 ใไธญๅฟใซๆฏๅใใ๏ผใฐใญๅฎๆฐใ้ๅใซๅฏพใใฆๅๅใซๅคงใใ๏ผ
(3โ4)
๐ ๐ ๐๐๐๐ผ ใใ ใ๏ผk โ mgl > 0(4โ1)๐ผ๐ ๐ ๐ ๐2 0 ้ๅใฎๅฝฑ้ฟใ่ๆ ฎใใชใใฆใ่ฏใๆกไปถใงใใ๏ผ
๐ ๐๐ผ(4โ2)
Q18.Hint: Use ๐ ๐ ๐ ๐ ๐ 1 when ๐ 1๐ ๐ ๐ ๐๐ 1 ๐ when ๐ 1(1) ๐ , ๐ 2 10, ๐ 6(2) ๐ 10, ๐ 1(3) ๐ , ๐
Q19.ใทในใใ ใฏไธ่ถณๆธ่กฐ๏ผ๐ 2 ๐๐๏ผใงใใใฎใง๏ผไธ่ฌ่งฃใฏ๐ฅ ๐ก A๐ cos ๐ ๐ก ๐ใจใชใ๏ผ
(1) e = 1 ใฎๅ ดๅ่ก็ชๅๅพใฎ้ๅ้ใไฟๅญใใ๏ผๅ็บไฟๆฐใฎๆกไปถใๆบใใใใใใจใใ๏ผใใณใใจๆฟใฎ่ก็ชๅใฎ้ๅบฆใใใใใ๏ผvh, vp ใจใ๏ผ่ก็ชๅพใฏvโh, vโp ใจใใใจ๏ผ1 ๐ฃ ๐ฃโฒ๐ฃ ๐ฃใใใณ ๐ฃ ๐ ๐ฃ ๐ ๐ฃโฒ ๐ ๐ฃโฒ ๐ใๆบใใใใชใใใฐใชใใชใ๏ผ๐ฃ 0ใใค๐ฃ ๐ฃ ใงใใใฎใง๏ผใใใใๆบใใ่งฃใจใใฆ๏ผ ๐ฃโฒ 2๐ฃ ๐๐ ๐๐ฃโฒ ๐ฃ ๐ ๐๐ ๐ใๅพใใใ๏ผ
Q19ๅๆๅคไฝใฏ ๐ฅ 0ใงใใ๏ผใใใใฃใฆ๏ผ้ๅฑๆฟใฎๅฟ็ญใฏไธ่จใฎใใใซใชใ:๐ฅ ๐ก 2๐ฃ ๐๐ ๐ ๐ ๐ sin ๐ ๐ก๐ ๐๐ , ๐ ๐2 ๐ ๐ , ๐ ๐ 1 ๐(2) e = 0 ใฎๅ ดๅ
ใใณใใจ้ๅฑๆฟใฏ่ก็ชๅพใซใฏไธไฝใจใชใฃใฆๆฏๅใใ๏ผใใณใใฎ่ก็ชๅๅพใง้ๅ้ใไฟๅญใใใชใใใฐใชใใชใใฎใง๏ผ้ๅฑๆฟใจใใณใใฎ่ก็ช็ดๅพใฎ้ๅบฆใvใจใใใจ๏ผ ๐ ๐ฃ ๐ฃ ๐ ๐ใๆ็ซใใ๏ผ่ก็ชๅพใฏ(1)ใจๅๆงใซ่ช็ฑๆฏๅใซใชใใใ๏ผ้ๅฑๆฟใฎๅฟ็ญใฏไธ่จใฎใใใซใชใ:๐ฅ ๐ก ๐ ๐ฃ๐ ๐ ๐ ๐ sin ๐ ๐ก๐ ๐๐ ๐ , ๐ ๐2 ๐ ๐ ๐ , ๐ ๐ 1 ๐
Q20(1) ไธ่ถณๆธ่กฐ็ณป(2) ้ๆธ่กฐ็ณป(3) ่จ็ๆธ่กฐ็ณป
Q211) Z. ๆใๆฉใๅนณ่กก็นใซๅๆใใใฎใ๏ผ่จ็ๆธ่กฐใฎ็นๅพดใงใใ๏ผ2) X. ใใฎ3ๆกไปถใฎไธญใง๏ผๆฏๅใ็ใใใฎใฏใใฎไธ่ถณๆธ่กฐใฎใฟใงใ
ใ๏ผ3) Y. ็ฒๆงใๅคงใใ๏ผๅๆใซๆ้ใ่ฆใใใฎใ้ๆธ่กฐใงใใ๏ผ
(1)
Q22 ๐๐ฅ ๐ ๐๐๐(2) ๐๐ฅ 1๐ 1๐๐๐ ๐๐๐๐๐ ๐๐๐(3) ๐๐ฅ 1๐ 1๐ ๐๐๐(4) ๐๐ฅ ๐ 1๐ 1๐๐๐
(2)ใฎ็ดๅๆฅ็ถใฏ๏ผๆฏๅใฎๅจๆณขๆฐใๅฐใใใจใใฏ๏ผๆใใใ๏ผๆตๆใๅฐใใ๏ผ๏ผๅจๆณขๆฐใๅคงใใใชใใซใคใใฆ๏ผๆฌๆฅใฎใฐใญใฎๅๆงใๆฉ่ฝใใใจใใ็นๆงใๆใใ๏ผ
๐
๐๐/๐ฅ๐0
0
๐2
Q23(1) ๐ผ ๐๐ ๐ 2๐ 5๐๐(2) ๐ผ๐ 4๐๐ ๐ 4๐๐ ๐ 0(3)
๐ ๐ก exp ๐ ๐๐ก ๐ด cos๐ ๐ก ๐ด sin๐ ๐ก้ๅๆน็จๅผใฎไธ่ฌ่งฃใฏไธ่จใฎ้ใใงใใ๏ผๆชๅฎๆฐA1, A2ใฏๅๆๆกไปถใใๆฑบๅฎใใใ๏ผ
๐ 0 ๐ด ๐๐ 0 ๐ด ๐ ๐ ๐ด ๐ 0๐ 4๐๐๐ผ๐ ๐๐๐ผ๐๐ ๐ 1 ๐๐ด ๐ , ๐ด
Im
Re
A
B
CD
E
Roots locus of 1โDOF springโmassโdamper systems
Motions (displacement) after being released from rest
Time
(A)
(C)
(B)
(D)
(E)
Q24 The behaviors of 1โDOF systems can be understood from their characteristic roots. Complete the following roots locus by finding the correct combinations of the loci of roots (ใโใป) and the motions of 1โDOF springโmassโdamper systems (AโE). (4 pt).
Q25(1) ๐๐ฅ ๐๐ฅ ๐ (๐ฅ 0)๐๐ฅ ๐๐ฅ ๐ (๐ฅ 0)(2) ๐ฅ ๐ก ๐ฃ๐ sin๐ ๐ก ๐๐ cos๐ ๐ก ๐๐ ๐ ๐๐(3)
(2)ใฎๅผใๅคๅฝขใ๏ผ๐ฅ ๐ก ๐ฃ๐ ๐๐ sin ๐ ๐ก ๐ ๐๐ใๅพใใจ๏ผx(t)ใฎๆๅคงๅคใฏ๏ผไธ่จใจใชใใใจใๅใใ๏ผ๐ฅ ๐ก ๐ฃ๐ ๐๐ ๐๐
Q26(1) ใ: Overdamped
(2)
๐ฅ 2 12 exp 2 2 ๐ก 2 12 exp 2 2 ๐ก
๐ฅ ๐ด exp 2 2 ๐ก ๐ด exp 2 2 ๐ก๐ฅ 0 ๐ด ๐ด 1๐ฅ 0 ๐ด 2 2 ๐ด 2 2 0General solution:
Q27(1) ๐ 3๐ธ๐ผ๐ฟ(2) ๐พ ๐ ๐๐ ๐ ๐(3) ๐ถ 2 ๐๐พ(4) ๐ 2๐๐ 2๐๐ 1 ๐๐ ๐๐พ ๐ ๐2 ๐๐พ
(5) ๐ฅ ๐ก ๐๐ cos ๐ ๐ก ๐ใจใใฆ๏ผ๐ฅ ๐ก ๐๐ฅ ๐ก exp ๐2๐ ๐exp ๐2๐ 2๐๐ 1 ๐exp 2๐๐4๐๐ ๐
Q28 (1) ๐ผ๐ ๐ๅ่ปข้ๅใงใใใใ๏ผใขใผใกใณใใฎ้ๅๆน็จๅผใใใง๏ผ ๐ผใฏๆฏใๅญใฎๅบๅฎ่ปธๅจใใฎๆ ฃๆงใขใผใกใณใใงใใ๏ผ๐ใฏๆฏใๅญใซไฝ็จใใใขใผใกใณใใงใใ๏ผๆฏใๅญใฎๅบๅฎ่ปธๅจใใฎๆ ฃๆงใขใผใกใณใ ๐ผ ๐๐ ใจ่กจใใ๏ผๆฏใๅญใซไฝ็จใใใขใผใกใณใ๐ ใฏ
ใปใฐใญใฎๅพฉๅ ๅใซใใใขใผใกใณใ๏ผ๐ ๐ ยท ๐๐ ยท ๐ใปใใณใใฎๆธ่กฐๅใซใใใขใผใกใณใ๏ผ๐ ๐ ยท ๐๐ ยท ๐ใปใในใซใใใขใผใกใณใ๏ผ๐ ๐๐ sin ๐ ยท ๐ ๐๐๐๐
ใฎ3ใคใใ่กจใใใ๏ผไปฅไธใโ ใซไปฃๅ ฅใ๏ผๆฑใใ้ๅๆน็จๅผ๐๐ ๐ ๐๐ ๐ ๐๐ ๐ ๐๐๐๐
โ
๐๐ ๐ ๐๐ ๐ ๐๐ ๐๐๐ ๐ 0 โกm
k
c
๐ ๐๐
๐ ๐
๐
Q28 (2) โกใฎ้ๅๆน็จๅผใๅคๅฝขใ๏ผโข๐๐ ๐๐ ๐ ๐๐ ๐โ ๐ 0
่งฃใฎๅฝขใ๐ ฮ๐ ใจไปฎๅฎใ๏ผโขใซไปฃๅ ฅใใฆๆด็ใใใจ๏ผ็นๆงๆน็จๅผ๐๐ ๐๐ ๐ ๐๐ ๐โ 0ใๅพใใใ๏ผใใใใ๏ผๆ น๐ ๐ ๐ 4๐ ๐ ๐๐ ๐โ2๐ใๅพใ๏ผๆฑใใ่จ็ๆธ่กฐไฟๆฐ๐ ใฏ๏ผใใฎ๐ใ้ๆ นใจใชใใจใใฎ๐ใใ๏ผ๐ 4๐ ๐ ๐๐ ๐โ
๐ 2 ๐ ๐๐ ๐๐๐
โฃ
โค
Q28 (3) ๐ ๐ ๐ 4๐ ๐ ๐๐ ๐โ2๐ ๐2๐ ๐ 4๐ ๐ ๐๐ ๐โ ๐2๐ ๐๐ ๐๐ใๅพใ๏ผใใใง๏ผ ๐ ใฏ็ณปใฎๅบๆ่งๆฏๅๆฐ๏ผ๐ใฏๆธ่กฐๆฏใงใใ๏ผ
โฅ
โง
็ณปใไธ่ถณๆธ่กฐใงใใๅ ดๅ๏ผ ๐ ๐ ใใฎๆ๏ผ
โฆ
๐ 4๐ ๐ ๐๐ ๐โ ๐2๐ 4๐๐ ๐๐ ๐๐ ๐ ๐2๐๐
๐ , ๐ , ๐๐ ใงใใ๏ผโฅ๏ผโฆใใ๏ผๆฑใใ่ช็ฑๆฏๅใฎๅจๆณขๆฐ(ๆธ่กฐๅบๆ่งๆฏๅๆฐ) ๐ ใฏ
Q28 (4)
๐ ๐ก ๐๐ ๐ sin ๐ ๐ก2๐๐๐4๐๐ ๐๐ ๐๐ ๐ ๐ ๐ sin 4๐๐ ๐๐ ๐๐ ๐ ๐2๐๐ ๐ก
(2)ใงไปฎๅฎใใ่งฃใฎๅฝข๐ ฮ๐ ใซ๏ผ๐ ๐๐ ๐๐ ใไปฃๅ ฅใใ๏ผ๐ ฮ๐ ฮ๐ ฮ ๐ cos ๐ ๐ก ๐โจใ๐กใงๅพฎๅใ๏ผ โจ๐ ๐๐ ฮ ๐ cos ๐ ๐ก ๐ ๐ ฮ ๐ sin ๐ ๐ก ๐
โฉๅๆๆกไปถ๐ 0 0 and ๐ 0 ๐ ใโจ๏ผโฉใซไปฃๅ ฅใ๏ผ๐ 0 ฮ cos ๐ 0 ๐ ๐2๐ 0 ๐๐ ฮ cos ๐2 ๐ ฮ sin ๐2 ๐ฮ ๐๐ใใใใโจใซไปฃๅ ฅใ๏ผ