Using Newton’s second Law, show how a standard 2nd order ordinary differential equation may be formulated to describe the dynamic performance of a spring-mass-dashpot system.
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- Using Newton’s second Law, show how a standard 2nd order ordinary differential equation may be formulated to describe the dynamic performance of a spring-mass-dashpot system.
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- 2. A pendulum is a mechanical system in which a mass m is attached to a massless incxtensible string of length which is in turn connected to a frictionless pivot, as shown at right. The swing angle (t) of the pendulum satisfies the second-order nonlinear differential equation d'e 22² + sin 0-0. 2sine The transformation v -- allows us to transform this equation into the separable first-order differential equa- tion (1) U + sin 0-0. sino pivot 1 v(0)' mass 77 (a) If the mass m is initally at rest at an angle of — T/6 degrees, solve the differ- ential equation (1) to find v = v(8). (b) Given that (2) de integrate both sides of this separable differential equation to find the general solution tt(0) in the form of an integral (with respect to 0). (c) Hence, write down a definite integral for the period 7' of the pendulum (that is, the time for the mass m to complete one full swing and return to its starting position. (Note: v <0 on this section because is decreasing.) (Hint: The period is…A mass less spring having a spring content k=185 N/m, is inside a horizontal barrel. It's left end is rigidly fixed at position z= 0.0 cm, and it's right end is at X=L, where L=19.7 cm. The spring is then squeezed so that the right end of the spring is at the X=s, where S=13 cm. A ball of mass 50g is put in front of the spring and the spring is released. Calculate the speed of the ball in m/s. as it comes out of the barrel. Ignore friction Inside the barrel.You are asked to estimate the maximum human body temperature if the metabolic heat produced in your body could escape only by tissue conduction and later on the surface by convection. Simplify the human body as a cylinder of L=1.8 m in height and ro= 0.15 m in radius. Further, simplify the heat transfer process inside the human body as a 1-D situation when the temperature only depends on the radial coordinater from the centerline. The governing dT +q""=0 dr equation is written as 1 d k- r dr r = 0, dT dr =0 dT r=ro -k -=h(T-T) dr (k-0.5 W/m°C), ro is the radius of the cylinder (0.15 m), h is the convection coefficient at the skin surface (15 W/m² °C), Tair is the air temperature (30°C). q" is the average volumetric heat generation rate in the body (W/m³) and is defined as heat generated per unit volume per second. The 1-D (radial) temperature distribution can be derived as: T(r) = q"¹'r² qr qr. + 4k 2h + 4k +T , where k is thermal conductivity of tissue air (A) q" can be calculated…
- Consider the mass-spring system shown in the figure below. Assume the natural frequency of the system is wn = 10 rad/s. The system is subject to a time-dependent force Fo cos (wt) N, where Fo = 0.3m N and w = 2wn rad/s (m: mass in kg). The system is subjected to initial displacement and velocity, i.e. xo = 10 mm and vo = 30 mm/s. The response of the system at t = 24 s is: Displacement x(t) F = F,cos(wt)I m 21.63 mm 5.77 mm 7.21 mm 3,61 mm 43.26 mm ОООООA system in Figure has m = 1 kg, T= 10N and an equation of motion x = (1 - cos 8rt)mm. Determine: (a) spring constant k (b) рeriod (c) frequency (d) displacement, velocity, acceleration, and spring force at t=0.15s. 3 I T= f(1) ww- kIn mass spring system experiment, , spring with 15 turns, outer diameter is 3cm and inner diameter is 2.3mm if the natural frequency is 18 rad/sec ; the mass of system is Rigidity = 80 GPa Elasticity =200 GPa g=9.81m/sec2 O 38.3 kg O 34.5 kg O 2.1 kg O 1.7 kg O 5.7 kg
- Consider a mass spring system with an external force ?(?) = ?0sin?0?. Let's assume ? is the mass of the object attached to the spring, ? is the spring constant, and ? is the damping coefficient. Let's also assume that ?(0) and ?′(0) are the location and speed of the object attached to the spring at time ? = 0. a) Find the general solution of the differential equation governing the motion of the spring, in terms of ?,?,?,?(0),?"(0),?0, and ?o. Your solution shall cover all possible cases including 1. the case with no damping and ?! is not equal to natural frequency of the system, ? =sqrt?/?. 2. the case with no damping and ?! is equal to natural frequency of the system, ? =sqrt?/?. 3. the three cases with damping (overdamped, critically damped, and underdamped).This is regarding Forced Vibrations and Ordinary Differential Equations.How long should it take to boil an egg? Model the egg as a sphere with radius of 2.3 cm that has properties similar to water with a density of = 1000 kg/m3 and thermal conductivity of k = 0.606 Watts/(mC) and specific heat of c = 4182 J/(kg C). Suppose that an egg is fully cooked when the temperature at the center reaches 70 C. Initially the egg is taken out of the fridge at 4 C and placed in the boiling water at 100 C. Since the egg shell is very thin assume that it quickly reaches a temperature of 100 C. The protein in the egg effectively immobilizes the water so the heat conduction is purely conduction (no convection). Plot the temperature of the egg over time and use the data tooltip in MATLAB to make your conclusion on the time it takes to cook the egg in minutes.
- Derive the governing differential equation for the system shown in Figure 2 subject to gravity. m1 m2 X1 X2 Œ Figure 2: System for problem 24. Consider a mass of 500 g placed at the end of a spring with stiffness constant 100 N/m and hanging at rest. The mass is displaced downward by 1.0 cm and released from rest. When the motion sets in, a time-varying force F(t) = cos³ 2t is applied to the spring-mass system. Solve the nonhomogeneous 2nd order differential equation representing the motion. That is, solve for x(t). [Note: Take the downward direction as the negative x-axis.]A metallic ball of radius ro = 5 mm, is initially in equilibrium at 400 C in a furnace. It is suddenly dropped to water at 20 C (hw =last 5 digits of your student number W/m K, e.g your student number is 201912345 then hw = 12345). The properties of the sphere are density= 3000 kg/m", k = 20 W/m_ K, c = 1000 J/kg K, and a= 6.66E-6 m/s. Calculate the time required for the center of the sphere to cool to 50 C. %3D 2