Q1: Establish the equation of motion of each system about x. Regarding (1), answer the critical damping coefficient and damping ratio. m m c x m ଵ ଶ Exercise of mechanical vibration 3 c x (2) (3) (1) x ଵ ଶ 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 . b) What is the relation among ଵ , ଶ , , and 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. ଶ ଵ ଵ Intermediate 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 ൌ13.6 Nm/rad and Nms/rad, respectively. Determine the critical value of when the damping ratio is ൌ1. Torsional spring and damper Basic Q4: 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 expression under the square root. iሻ ଶ ൏ 4 (this will be underdamping, c is small relative to m and k) iiሻ ଶ 4 (this will be overdamping, c is large relative to m and k) iiiሻ ଶ ൌ 4 (this will be critical damping, c is just between over and under damping). ሷ ሶ ൌ 0. േ మ ସ ଶ .
13
Embed
Exercise of mechanical vibration 3 Q2: An inverted … Establish the equation of motion of each system about x. Regarding (1), answer the critical damping coefficient and damping ratio.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Q1: Establish the equation of motion of each system about x. Regarding (1), answer the critical damping coefficient and damping ratio.
m
mc
x
m𝑐 𝑐
Exercise of mechanical vibration 3
cx
(2) (3)
(1) x
𝑐 𝑐𝑘
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 𝜃.
b) What is the relation among 𝑙 , 𝑙 , 𝑚, and 𝑘 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.
𝑚𝑘 𝑘𝑙𝑙 𝑙
𝜃
𝜃𝑔
Intermediate
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 𝑘 13.6 Nm/rad and 𝑐 Nms/rad, respectively. Determine the critical value of 𝑐 when the damping ratio is 𝜁 1.
𝜃𝑐𝑘Torsional springand damper
Basic
Q4: 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 𝑐 4𝑚𝑘 (this will be underdamping, c is small relative to m
and k)ii 𝑐 4𝑚𝑘 (this will be overdamping, c is large relative to m andk)iii 𝑐 4𝑚𝑘 (this will be critical damping, c is just between overand under damping).
𝑚𝑥 𝑐𝑥 𝑘𝑥 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.
Q5: Highway crash barriers are designed to absorb a vehicle’s kineticenergy without bringing the vehicle to such an abrupt stop that theoccupants are injured. The barrier’s materials and thickness arechosen to accomplish this. It can be modeled as the spring‐mass‐damper system. For this application, t = 0 denotes the time ofcollision at x(0) = 0. The speed of the vehicle at this moment is𝑥 0 22 m/s. For a particular barrier, k = 18,000 N/m and c =20,000 N・s/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) 𝑥 0 2, 𝑥 0 5;b) 𝑥 0 2, 𝑥 0 5;c) 𝑥 0 2, 𝑥 0 0.And, draw x(t) for the three cases using a graph drawing software such as MS Excel.
Basic
Q7: Consider a disk that rotates on the floor without slippage. Its centeris connected to walls by way of spring‐damper systems withoutfriction of which spring and damping coefficients are k and c,respectively. The disk’s 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 being𝑥 0 0 and 𝑥 0 𝑣 .(2) Find t’ at which the displacement of the disk becomes
greatest.(3) When the system is underdamped (ζ < 1), solve the response of
the system with the initial conditions being 𝑥 0 0 and𝑥 0 𝑣 .
Intermediate
(4) Find the logarithmic decrement (対数減衰率) δ, which is thelogarithmic ratio of two successive amplitudes.
𝛿 ln 𝑎𝑎x(t)
t
a1 a2 a3
r
Q8: Obtain the response of the following equation of motion
for the following case:𝑥 0 3, 𝑥 0 5.And, draw x(t) for the three cases using a graph drawing software such as MS Excel.
Basic
Q9: Translate the following paragraph about spring‐mass‐dampersystems.
Consider a spring‐mass‐damper system described by𝑚𝑥 𝑐𝑥 𝑘𝑥 0where m, k, and c are the mass, spring coefficient, and dampingcoefficient, respectively. This equation is rewritten as𝑥 2𝜁𝑝𝑥 𝑝 𝑥 0where 𝑝 𝑘/𝑚 and 𝜁 𝑐/2 𝑚𝑘. 𝜁 is a dimensionless valuecalled the damping ratio and defined by the ratio of c and thecritical damping coefficient. The characteristic roots are𝑠 𝜁𝑝 𝑖𝑝 1 𝜁 .Its imaginary part is the frequency of the oscillation. This frequencyis called the damped natural angular frequency and is smaller thanundamped natural frequency.
Rk
α
x
θ 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 θ. When θ = 0, the spring is at itsnatural length. Assume that θ 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
θ
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
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.
θ
L/4
L/2
L
O
Intermediate
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.
テキスト P51‐2.9a
m
k
c
2k
Basic
Q15: 左端を支点として回転できる剛体棒の先端に質量の物体がついている。ばね定数 k ,粘性減衰係数 c .(1) 𝜃に関しての運動方程式を求めよ.(2) 臨界減衰係数を求めよ.
テキスト P51‐2.10a
m
l
l
l
Basic
Q16: Consider a pendulum supported by a spring‐damper system. A mass is attached at the end of a weightless rod. The rotation angle θ is zero at the equilibrium position. Let the gravity acceleration be g.(1) Answer the equation of motion about 𝜃.(2) Answer c such that the pendulum is critically damped (𝜃 is
small enough).
m
k
c
𝜃
𝑙𝑙
Basic
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 θ, and θ = 0 when the pendulum stands straight. The spring is at its natural length when θ = θ0. 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 θ.(3) Consider a case where θ0 = 0. The spring is at its natural length
when the pendulum stands straight.3‐1) Establish the equation of motion for a small angle (sinθ ~
θ).3‐2) Answer the characteristic root of the system.
Intermediate
3‐3) Depending on the k value, the behavior of the system can be categorized into three cases. Describe all of them.
3‐4) In the case where the pendulum vibrates around θ = 0, answer the natural frequency.
(4) Consider a case where the pendulum is statically equivalent at 𝜃 . We discuss the behavior at 𝜃 .4‐1) Answer the equation of motion about θ.4‐2) Answer the natural frequency of the system.
mθ
kl
Q18: If applicable, compute ζ, ωn, and ωd for the following characteristic roots of 1‐d.o.f damped systems.(1) 𝜆 2 6𝑖(2) 𝜆 10(3) 𝜆 , (or 1 )
Basic
Q19: A hammer strikes a metal plate at the initial speed of v0. Thehammer mass is 𝑚 and the plate mass is 𝑚 . The metal plate issupported by the stiffness 𝑘 and damping 𝑐. Answer the expressionof the plate about x, of which equilibrium position is 𝑥 0. Do thisfor two values of the coefficient of restitution (1) e = 0 and (2) e = 1.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 that𝑐 2 𝑚 𝑘 and 𝑐 2 𝑚 𝑚 𝑘.
Intermediate
Q20: 下記(1)—(3)は,減衰のある1自由度振動系の運動方程式である.x is the displacement of the system. 固有方程式の根を基に,それぞれが過減衰系,不足減衰系,臨界減衰系のいずれであるかを判断せよ.
(1) 3𝑥 2𝑥 𝑥 0(2) 2𝑥 5𝑥 2𝑥 0(3) 𝑥 4𝑥 4𝑥 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 𝑥 𝑡 0 𝑥 and 𝑥 𝑡 0 𝑣 . Answer the correct response of the system from X, Y, and Z when (1) 𝑐 2 𝑚𝑘(2) 𝑐 2 𝑚𝑘(3) 𝑐 2 𝑚𝑘.
Q2 𝑚𝑙 𝜃 2𝑘𝑙 𝑚𝑔𝑙 𝜃 0a) 2𝑘𝑙 𝑚𝑔𝑙 0b)c) The pendulum falls down or does not remain around 𝜃 0.The general solution is 𝜃 𝐴 exp 𝜆 𝑡 𝐴 exp 𝜆 𝑡.If 𝜆 or 𝜆 includes a positive real value, 𝜃 diverges over time.
Q3 𝜌 401.2 0.05 2003𝐼 𝜌 𝑟 𝑑𝑥𝑑𝑦. 19.23 kgm.
𝐼 𝜃 𝑐 𝜃 𝑘𝜃 0ζ 1 𝑐2 𝑀𝐾Equation of Motion
𝑐 2 𝐼𝑘 32.3 Nms/rad
Q4: Translate the following paragraph into Japanese.
減衰振動系の自由応答は,下記の式で表現される:
この特性方程式の根は,
ルートの中の符号によって,3つの場合がある.i 𝑐 4𝑚𝑘 (これは不足減衰であり,c は m と k に比べて小さい)ii 𝑐 4𝑚𝑘 (これは過減衰であり,c は m と k に比べて大きい)iii 𝑐 4𝑚𝑘 (これは臨界減衰であり,c は不足減衰と過減衰の調度間である).
Q11: Derive the equation of motion for θ. When θ = 0, the spring is atits natural length. Assume that θ is small. Mass of the lever isnegligible. Gravitational acceleration is g. Answer the dampednatural angular frequency.
m
l1
l2
l3
θ
c
k
𝜔 𝑝 1 𝜁𝑝 𝑚𝑔𝑙 𝑘𝑙𝑚𝑙𝜁 𝑐𝑙2 𝑚𝑙 𝑚𝑔𝑙 𝑘𝑙
𝑚𝑙 𝜃 𝑐𝑙 𝜃 𝑚𝑔𝑙 𝑘𝑙 𝜃 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.