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ENG1007 Mechanics Formula sheet: Dynamics and Mechanical Drives Non-rotational Dynamics: Statics (forces in equilibrium) of a body: sum of forces in any direction x: โ ๐น๐ฅ = 0 sum of moments about any point Q: โ ๐๐ = 0 Non-rotational Dynamics (forces not in equilibrium) of a body: sum of forces in any direction x: โ ๐น๐ฅ = ๐๐ฬ sum of moments about centre of mass G: โ ๐๐บ = 0 Body forces: an external force, e.g. gravitational force, for a rigid body, these behave as a single force acting on centre of mass Locating centre of mass: ๐๐ = ๐ด๐๐ ๐๐ ๐ โ ๐๐๐๐๐๐๐ก๐๐๐ก๐๐ ๐๐๐ ๐ ๐๐ โ ๐๐๐ ๐ก๐๐๐๐ข๐ก๐๐/๐๐๐๐๐๐๐ก๐๐ ๐๐๐ ๐ ๐ = ๐๐๐๐๐ก๐๐๐ ๐๐ ๐๐๐๐ก๐๐ ๐๐ ๐๐๐ ๐ ๐๐ = ๐๐๐๐ก๐๐ ๐๐ ๐๐๐ ๐ ๐๐ ๐๐๐๐๐ฃ๐๐๐ข๐๐ ๐๐๐๐๐๐๐ก๐๐ ๐๐๐ ๐ Friction: Body is not sliding (equilibrium) F = F (applied force); max F = ฮผR (ฮผ is coefficient of static friction) F F Body is sliding F = ฮผ R (ฮผ is coefficient of sliding friction) F s s Momentum and Impulse ๐ก2 โซ๐ก1 ๐น๐ฅ ๐๐ก = ๐2 โ ๐1 = ๐(๐ฃ2 โ ๐ฃ1 ) impulse of a force = change of momentum states Rotational Dynamics Rotational Dynamics of a body: โ ๐๐บ = ๐ผ๐บ ๐ฬ ๐ผ๐บ โ ๐๐๐๐๐๐ก ๐๐ ๐๐๐๐๐ก๐๐ ๐๐๐๐ข๐ก ๐บ ๐ฬ โ ๐๐๐๐ข๐๐๐ ๐๐๐๐๐๐๐๐๐ก๐๐๐ ๐๐ ๐กโ๐ ๐๐๐๐ฆ. 1 Moment of inertia ๐ผ = โซ ๐ 2 ๐๐ For a ring: ๐ผ = ๐๐ 2 1 For a disk: ๐ผ = 2 ๐๐ 2 1 For a slender bar ๐ผ๐บ = 12 ๐๐ 2 Parallel-Axis Theorem ๐ผ๐ = ๐ผ๐บ + ๐โ2 IG = moment of inertia of the body about the axis passing centre of mass m = mass of the body h = perpendicular distance between two parallel axes Mechanical drives: T2 N 2 ๏ท N = and 1 = โ 2 T1 N1 ๏ท2 N1 In practice there is a loss of power, e.g. due to friction ๐๐๐ข๐ก ๐ = ๐ ๐๐ข๐ก ๐๐๐ ๐๐๐ ๐๐๐ข๐ก , ๐๐๐ โ ๐๐ข๐ก๐๐ข๐ก ๐๐๐ ๐๐๐๐ข๐ก ๐ก๐๐๐๐ข๐, ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ ๐๐๐ข๐ก , ๐๐๐ โ ๐๐ข๐๐๐๐๐ ๐๐ ๐ก๐๐๐กโ ๐๐ ๐๐ข๐ก๐๐ข๐ก ๐๐๐ ๐๐๐๐ข๐ก ๐๐๐๐๐ , ๐๐๐ ๐๐๐๐ก๐๐ฃ๐๐๐ฆ ๐ โ ๐๐๐๐๐๐๐๐๐๐ฆ 2 D Quiz: Semester 2 Online Assessm X + – X ะฅ https://ncl.instructure.com/courses/25066/quizzes/38092/take aใ ร … ” Part B: Strength of Materials (taught by Graham Coates) Account Dashboard D Question 5 5 pts Courses At a point in a complex stress field, o xx -79.5 MPa, Oyy 50.9 MPa and Try – -25.4 MPa. Groups Determine the absolute maximum shear stress. Calendar Give your answer in MPa to 1 decimal place. E. Inbox History D ยฉ Help Question 6 10 pts A 45ยฐ strain gauge rosette placed on a stainless steel structure results in the following values being determined for the maximum and minimum principal strains: โฌ1 = 1,686.7 x 10-6 and โฌ2 = -774.2 x 10-6 Assuming plane stess, determine the factor of safety against the Tresca criterion. For stainless steel, use Young’s modulus E = 196G Pa , Poisson’s ratio v = 0.29 and yield stress Oy = 1120 MPa. Give your answer to 2 decimal places. K O ๅจ่ฟ้่พๅ ฅไฝ ่ฆๆ็ดข็ๅ ๅฎน o ืืื Ca x 14:09 2021/6/5 IMI ะ D Quiz: Semester 2 Online Assessm X + – X ะฅ https://ncl.instructure.com/courses/25066/quizzes/38092/take aใ ร … ” Account Dashboard Question 7 8 pts Courses A thin-walled cylindrical pressure vessel made from a grade of aluminium has an internal diameter d and a wall thickness t. Groups Determine the ratio of diameter to wall thickness of the vessel for a safe working pressure of 17.8 MPa and a factor of safety of 2.3 using the Tresca criterion. Calendar Take the uniaxial tensile strength of the grade of aluminium as 201 MPa. E. Give your answer to 1 decimal place. Inbox History ยฉ Help Question 8 12 pts A solid steel shaft of circular cross-section has diameter 54 mm, length 1,084 mm and shear modulus 79.3 GPa. If the allowable shear stress is 35 MPa and the allowable angle of twist is 1.7ยบ, determine the maximum permissible torque. Give your answer in Nm to 1 decimal place. K – O ๅจ่ฟ้่พๅ ฅไฝ ่ฆๆ็ดข็ๅ ๅฎน ืืื Ca x 14:10 2021/6/5 IM ะ D Quiz: Semester 2 Online Assessm X + – X ะฅ https://ncl.instructure.com/courses/25066/quizzes/38092/take aใ ร … ” Account Dashboard Part C: Dynamics (taught by Dehong Huo) Courses Groups Question 7 pts Calendar Which one of the following statements regarding mechanical drives is NOT correct? E. Inbox O Speed ratio (or gear ratio) is not affected by the efficiency. O Input power equals output power if there are no losses. History OTwo gears that are meshed must have same module. ยฉ Help O Helical gears are quieter than spur gears and have higher efficiency. O Bevel gears may be used to transmit motion between intersecting shafts, while spur gears transmit motion between parallel shafts. O Output torque is always higher than input torque in a gear train. O An idler gear does not affect the overall speed ratio. Question 10 13 pts A motor is driving an output shaft through a single-stage gears, as shown. K Input Gear, N, O ๅจ่ฟ้่พๅ ฅไฝ ่ฆๆ็ดข็ๅ ๅฎน ืืื Ca x 14:10 2021/6/5 IM ะ D Quiz: Semester 2 Online Assessm X + – X ะฅ https://ncl.instructure.com/courses/25066/quizzes/38092/take aใ ร … ” Account Question 10 13 pts Dashboard A motor is driving an output shaft through a single-stage gears, as shown. Courses Input Gear, N Groups Motor T; Calendar E. T. Inbox History Output Gear, N2 ยฉ Help Number of teeth of the input gear, N1 = 23; Number of teeth of the output gear, N2 = 47; Gear efficiency = 95%; If the output shaft is to deliver torque To = 13 Nm, calculate the torque required on the input shaft, Ti. Give your answer in Nm to 1 decimal places. Part D: Dynamics (taught b

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