Statics and Dynamics PDF

Statics Force: Vector, that can be moved on the line of action, and add them by the parallelogram rule. Couple: Two parallel forces that are equal in magnitude, opposite in sense and do not share a line of action. The moment (statics) / torque (dyn) of one force: (moment of a couple is ind...

Statics Force: Vector, that can be moved on the line of action, and add them by the parallelogram rule. Couple: Two parallel forces that are equal in magnitude, opposite in sense and do not share a line of action. The moment (statics) / torque (dyn) of one force: (moment of a couple is independent of the reference point) τ = k ⋅F (lever arm * force) generally: τ = r×F (by cross product) Wrench: For any point O, every force system can be reduced equivalent to a single force passing through O and a single couple. In unique case of simplification the force and couple are parallel. We refers it as wrench. Condition of Statics: ∑F = 0 i ∑τ i =0 Center of mass: also the point around which the resultant torque due to uniform gravity forces vanishes. rc = ∑m r i i ∑m i Quantities of Kinematics: translation and rotation displacement: s , t [m], [s] angular position: ϕ [rad] ds dϕ vt2 velocity: v= = s& [m/s] angular velocity: ω = = ϕ& [1/s] vtan = r ⋅ ω acp = dt dt r dv dω acceleration: a = = v& = &s& [m/s2] angular acceler.: α = = ω& = ϕ&& [1/s2] at = r ⋅ α dt dt Screw axis: The most general rigid body displacement can be produced by a translation and a rotation. In unique case of this simplification the axes of translation and rotation is the same (Chasles, 1830). Newton’s laws of Motion (1687): 1. Every object will remain at rest or in uniform motion in a straight line unless it is made to change its state by the action of an external force. 2. The resultant force acting on a body results in an acceleration which is in the same direction as the resultant force and is directly proportional to the magnitude of this force and inversely proportional to the mass of the object. 3. For every force or action there is an equal but opposite force or reaction. Quantities of Dynamics: translation and rotation (of rigid body or systems of particles) force (from statics) F = m ⋅ g [N=kg·m/s2] torque / moment τ = r × F [Nm] (τ = k ⋅ F lever arm * force) mass (súlyos és tehetetlen) [kg] moment of inertia I = m ⋅ r 2 2 [kg·m ] Steiner: I R = I 0 + M ⋅ R 2 2’nd law F = m⋅a è τ = I ⋅α linear momentum p = m ⋅ v [kg·m/s] angular momentum L = I ⋅ ω [kg·m2/s] L = r× p dp dL law of Momentum F= law of Ang. Mom. τ= dt dt Law of Momentum: The applied resultant force acting on an object is equal to the rate of change of the object's momentum and this force is in the direction of the change in momentum. (The product F ⋅ ∆t = ∆p is called impulse.) Principle of Conservation of Linear Momentum: The total linear momentum of an isolated system is constant. àIn an isolated system the total momentum before a collision is equal to the total momentum after the collision. dp F= dt ∑F i =0 è p = const Motion of the center-of-mass: The center of mass of a system of particles of mass M moves like an equivalent particle of mass M would move under the influence of the net external force on the system. ∑F ext = ∑ mi ⋅ &r&c Law of Angular Momentum: The net external torque τ acting on a system of particles is equal to the time rate of change of the system’s total angular momentum L. Principle of Conservation of Angular Momentum: If the net external torque acting on a system is zero, the angular momentum of the system remains constant, no matter what changes take place within the system. ∑τ dL τ= i =0 è L = const dt Gyroscope: symmetric -: free spinning pivots free -: nutation heavy -: gyroscopic torque, precession I max , I 2 = I 3 (orthogonal) ω∦L , L=const. τ =ωp ×L ω ×τ → ω p Firtha – Physics1 summary -1- Examples: free falling 1D 2D harmonic (spring, pendulum) 1 2π z = − g ⋅ t 2 + v z 0t + s z 0 x = vx 0t + sx 0 x = A ⋅ sin(ωt ) ω= ang. velocity 2 T vz = − g ⋅ t + vz 0 v x = vz 0 v = ω ⋅ A ⋅ cos(ωt ) az = − g ax = 0 a = −ω 2 A ⋅ sin(ωt ) diff.equ: &x& = −ω 2 x Work: Simple machines (lever,pulley,inclined plane,srew) output force can be bigger, but work is independent: dW = F ⋅ d s dW = τ ⋅ d ϕ [J=Nm] 1 1 1 2 cases of contant force: Wlift = mg ⋅ h Wacc = ma ⋅ s = ma ⋅ at 2 = mv 2 Wturn = Iω 2 2 2 1 2 1 Energy: capacity to do work E pot = mgh Ekin = mv E rot = Iω 2 2 2 The Principle of Conservation of Mechanical Energy: The total mechanical energy of an object remains constant as the object moves, provided that the net work done by external nonconservative forces is zero. 1 1 mgh1 + mv12 +... = mgh2 + mv22 +... 2 2 General: Energy is never created nor destroyed, but merely transformed from one form to another. (without friction) Types: mechanical, thermal, chemical (fossil), electrical, radiant, atomic, … solar, water, wind, biomass, geothermal Hydrostatics: pressure dF p= [Pa=N/m2] dA Pascal’s principle: Pressure applied on one point of liquid transmits equally in all direction. Models: weightless/heavy, incompressible/compressible, steady/unsteady flow, frictionless/lossy flow, … Hydrostatic pressure: in heavy, incompressible, homogeneous and static liquid p = p0 + ρgh Archimedes’ principle: Any object, totally or partially immersed in a fluid, is lifted up by a force equal to the weight of the fluid displaced by the object. Stability of floating body: Metacentre is the point at which the line of action of the force of buoyancy will meet the normal axis of the body when the body is given small angular displacement. Floating state is stabile, when the metacentre is above weight pont. Surface tension: the force acting over the surface per unit length of surface perpendicular to the force: dF = γ ⋅ ds [N/m] the energy required to increase the surface area by one square metre: dE A = γ ⋅ dA [J/m2] Laplace pressure (Young-Laplace): sphere: 1 1 2γ pc = γ ⋅ ( + ) pc ,s = r1 r2 r Angle of contact (Young-equation): near solid wall: γ 21 + cosθ ⋅ γ 13 = γ 23 2 degrees of freedom: γ 13 = cosθ3 ⋅ γ 12 + cos ϕ1 ⋅ γ 23 sin θ 3 ⋅ γ 12 = sin θ1 ⋅ γ 23 Dependency from temperature (Eötvös equation): γ ⋅ vm2 / 3 = k ⋅ (TC − T − 6) ; k: Eötvös constant, vm: molar volume of liquid, TC: critical temperature Measurements: Capillary height: rρg γ ⋅ 2rπ ⋅ cosϕ = r 2π ⋅ ρgh γ = h 2 cosϕ Stalagmometer: γ ⋅ dπ = mg Firtha – Physics1 summary -2-  Fluid dynamics: in tube: ~ flux: dV dV Volumetric flow rate ( qV = Q = V& ): IV := [m3/s] IV = A ⋅ v j V := =v dt dt ⋅ dA dm dm Mass flow rate ( qm = m & ): I m := [kg/s] Im = ρ ⋅ A⋅ v j m := = ρ ⋅v dt dt ⋅ dA Continuity equation (mass corservation): Any steady state process, the rate at which mass enters a system is equal to the rate at which mass leaves the system. For streamtube (one-dimensional flow, with one inlet and one outlet): I m1 = I m 2 è ρ1 ⋅ A1 ⋅ v1 = ρ 2 ⋅ A2 ⋅ v2 for incompressible fluid (liquid): ρ1 = ρ 2 è A1 ⋅ v1 = A2 ⋅ v2 Bernoulli’s principle (mechanical energy conservation): For steady flow on streamline: 1 1 p1 + ρ ⋅ g ⋅ h1 + ρ ⋅ v12 = p2 + ρ ⋅ g ⋅ h2 + ρ ⋅ v22 2 2 Level of liquid drained from the tank (if Δp=0, α contraction coefficient, A0 the area of hole): 2 dz − IV ( z ) − α ⋅ A0 2 ⋅ ∆p  α ⋅ A0  = = ⋅ + 2 gz è z (t ) =  h − ⋅ 2 g ⋅ t  dt Atan k At ρ  2 ⋅ At  Friction loss: Laminar flow: Newton’s law of viscosity (1687, Principia): viscous stresses are linearly proportional to the local strain rate. dv η Ft = η ⋅ A ⋅ ; η dinamic viscosity [Pa·s], ν= kinematic viscosity [m2/s] dx ρ Hagen-Poiseuille law: In case of stacionary, laminar flow of ideal (newtonial) fluid in long pipe the velocity profile (v(r) velocity as a function of the radius) is parabolic. ∆p π 1 ∆p 4 v(r ) = (R2 − r2 ) volumetric flow rate: IV = ⋅ ⋅ ⋅R 4 ⋅η ⋅ l 8 η l Stokes’s law: drag force on small sphere is proportional with velocity (in case of laminar flow!): F = 6π ⋅ η ⋅ r ⋅ v Reynolds number: flow in tube transitioned from laminar to turbulent, when the Reynolds number exceed the critical 2320 value. ρ ⋅d ⋅v 4A Re = 2320 hydraulic diameter: dh = η P dynamic similarity: v2 v inertial forces: I ≈ ρ ⋅ a ≈ ρ ⋅ frictional -: F ≈η ⋅ gravity -: G ≈ ρ⋅g l l2 v2 v2 ρ⋅ ρ⋅ Reynolds (flow space): Re = I ≈ l = ρ ⋅ d ⋅ v Froude (waves): Fr = I ≈ l = v F η⋅ v η G ρ ⋅g l⋅g l2 α ⋅l Biot/Nusselt (heat transfer): Nu = , Prandtl, Schmidt, Lewis, Sherwood, Stanton, Strouhal, Peclet, Knudsen, Euler, Chauchy, Grashof, λ... Turbulent flow: Drag equation: in case of turbulent flow the drag force is proportional with the square of velocity (with kinetic energy of vortex generated) F = CD ⋅ A ⋅ ρ ⋅ v 2 ; CD drag coefficient depends on shape and orientation Firtha – Physics1 summary -3- Lossy Bernoulli: in case of friction loss the mechanical energy at the right side is smaller. This loss can be expressed by a positive member. 1 1 loss pressure: p1 + ρ ⋅ g ⋅ h1 + ρ ⋅ v12 = p2 + p f + ρ ⋅ g ⋅ h2 + ρ ⋅ v22 2 2 1 1 head loss : p1 + ρ ⋅ g ⋅ h1 + ρ ⋅ v12 = p2 + ρ ⋅ g ⋅ ( h2 + h f ) + ρ ⋅ v22 2 2 1 1 1 loss coefficient: p1 + ρ ⋅ g ⋅ h1 + ρ ⋅ v12 = p2 + ρ ⋅ g ⋅ h2 + ρ ⋅ v22 + ξ ρ ⋅ v x2 2 2 2 l loss coefficient for tube: ξ =λ⋅ ; λ Darcy friction cooefficient d 64 0,3164 Darcy friction cooefficient: lam: λlam = turb (Blasius): λturb = 4 (2320 < Re < 105) Re Re Loss in pipe flow: 1 sudden expansion (Borda-Carnot equation): ∆pv = ρ (v12 − v22 ) 2 simple outlet: α = 0,66 → 1 (bell mouthed) ξ = 1 : v1 A valve, gate-, butterfly-: ξ = ( 2 − 1) 2 : v2 A1 A A 1 sudden contraction: α = 3 ≈ 0,6 + 0,4 ⋅ ( 2 ) 2 ξ = ( − 1) 2 : v2 A2 A1 α inlet, simple: α = 0,5 ξ =1 : v2 inlet, Borda-type: α = 0,6 → 1 (smooth) ξ = 0,44 : v2 1 diffuser (encreases p): ( p2 − p1 )ideális = ρ (v12 − v22 ) 2 ( p2 − p1 ) valódi 1 diffuser efficiency: η = ∆pv = (1 −η ) ⋅ ρ (v12 − v22 ) ( p2 − p1 ) ideális 2 Firtha – Physics1 summary -4-  Reology (deformation and flow of material): Local (characteristic for point, material) variables: dFn ∆x normal stress: σ= [Pa] normal strain: ε = [m/m] dA x dFt shear stress: τ= [Pa] shear strain: γ [rad] dA ideal bodies: 1.) Hooke body (ideal elastic): In case of small forces the deformation is reversible and proportional with stress. - normal stress: σ = E ⋅ε ; where E is the Young’ (or elastic) modulus [Pa] Bending is a special case of expansion and compression. For a beam caught on a side e.g.: 4 l3 s= ⋅ F E ab 3 - shear stress: τ = G ⋅γ ; where G is the shear modulus [Pa] Torsion is a special case of shearing. The torque of fiber is proportional with the angle of twist. D* is derived from G shear modulus. π ⋅ r4 τ = D∗ ⋅ ϕ D∗ = ⋅G 2⋅l ∆d ∆l - transverse change: = −µ ⋅ ; where µ is the Poisson-number (corkwood: 0, liquid 0,5) d0 l0 The volume change is derived from the expansion or compression: ∆V ∆l = (1 − 2 ⋅ µ ) V0 l0 ∆V - compressibility: = −κ ⋅ ∆p ; where κ is the compressibility V0 Converting the equation to the Hooke law like (force ~ effect) format: ∆V ∆p = K ⋅ ; where K is the compression (or Bulk) modulus V0 The so called Lamé constants are related in homogeneous and isotropic material: E 1− 2 ⋅ µ G= κ =3 2(1 + µ ) E 2.) Newton body (ideal liquid, viscous): Shear stress is proportional with speed gradient. Normal stress is proportional with time derivative of normal strain. Deformation is irreversible. - normal stress: σ = η ⋅ ε& ; η coefficient that is proportional with viscosity - shear stress: τ = η ⋅ v′ ; η is dynamic viscosity 3.) Saint Venant body (plastic): Below yeald stress (σh, τh) there’s no deformation, then the body breaks, slips. ∆p 2⋅l Shear stress in tube: τ (r ) = ⋅r è r0 = ⋅τ 0 2⋅l ∆p Yield criterions: Mohr (1882): τh(σ)=? Slope with α angle is in equilibrium state within the so called Mohr-circle: σ0 = 1 (σ 1 + σ 3 ) R= 1 (σ 1 − σ 3 ) (σ − σ 0 )2 + τ 2 = R 2 2 2 For normal stress: envelope outline of Mohr-circles is the Mohr-Coulomb line (a Mohr-körök burkológörbéje): τ h (σ ) = tgϕ ⋅ σ + τ 0 ; ϕ angle of internal friction τ0 cohesion Firtha – Physics1 summary -5-  Viscoelastic models: Maxwell-model: serial: σH =σN =σ ε = εH +εN σ& (t ) σ (t ) diffequ: ε& (t ) = + E η a.) when strain is constant: ε (t ) = ε 0 , ε& (t ) = 0 t E − diffequ: σ& (t ) = − ⋅ σ (t ) à σ (t ) = σ 0 ⋅ e τ ; σ 0 = E ⋅ε0 η η τ= (relaxation time) E b.) when stress is constant: σ (t ) = σ 0 , σ& (t ) = 0 σ0 σ σ0 diffequ: ε& (t ) = à ε (t ) = 0 ⋅ t + ε 0 ; ε0 = η η E Kelvin (Voigt)-model: parallel: ε H = ε N = ε σ = σH +σ N diffequ: σ (t ) = η ⋅ ε& (t ) + E ⋅ ε (t ) a.) strain is constant: ε (t ) = ε 0 , ε& (t ) = 0 diffequ: σ (t ) = E ⋅ ε 0 állandó b.) stress is constant: σ (t ) = σ 0 , σ& (t ) = 0 E E σ − t σ0 diffequ: ε& (t ) = − ⋅ ε (t ) + 0 à ε (t ) = ε sup ⋅ (1 − e η ) ; ε sup = η η E η τ= (retardation time) E Other viscoelastic models: gen. Maxwell gen. Kelvin Lethersich (xmas jelly) Jeffrey (jelly) Burger (pasta) Bingham model: nyomó/húzó: if σ < σ h σ = E ⋅ε else σ = η ⋅ ε& + σ h nyíró terhelésre: if τ < τ h τ = G ⋅γ else τ = η ⋅ v′ + τ h gen. Bingham (Herschel-Bulkley): τ (v′) = η ⋅ (v′) n + τ h : n is the consistency- / flow index n1: dilatancy (increasing viscosity): PVC paszta, szilikon changes by work: thixotropy (η decreasing by time): ketchup, jogurt hysteresis rheopexy (η increasing by time): corn starch τ −τ 0 dη pl B plastic viscosity: η pl := usually: ≈− ; where B is constant v′ dt t t after integration: η pl , 2 −η pl ,1 = − B ⋅ ln 2 t1 n +1 ∆p n n 1 2l gen. Bingham in tube: r0 = ⋅τ 0 v(r ) = ( ) ⋅ ⋅ (R − r) flow rate of cork: IV = r0 π ⋅ v( r0 ) n 2 ∆p 2lη n +1 ∆p ⋅ r 4 ⋅ IV 1 consistency variables: x = y= for newtonian fluid: y = ⋅x 2⋅l r 3 ⋅π η Firtha – Physics1 summary -6-  Optics: EM wave: c = λ ⋅v ; frequency, velocity à wavelength photon: E = h⋅v ; h is the Planck constant à mass E = m ⋅ c2 Geometrical optics Light is a ray, not a wave (wavelenth is not comparable with size of measurement instrument) and not a particle (frequency is not high, energy is not quantized) Laws: 1. Fermat’s Principle: Light travels through the (neighboring) path in which it can reach the destination in least time. 2. Reversibility of Light: The path taken by light from A to B or from B to A is identical. 3. Independency: Light rays passing through a point do not effect each other. (for incoherent light) Reflexion law: Angle of incident and reflected light is identical (BC. 1000) Similarity transformations: keeps lines, no distortion Real image: If the rays of light actually pass through the image. Orientation is inverted Virtual image: If image is seen at a point from which the rays appear to come to the observer. Orientation is upright. Gauss’ formula: magnification: 1 1 1 I i f i− f = + ; focal length, object -, image dist. M= = = = f o i O o o− f f Spherical mirrors: f = r / 2 , using small angle ( 0,008856) ? ( X ) : (7,787 ⋅ X 1 − 16 1/ 3 ) … 1 116 L* = 116 ⋅ Y2 − 16 a * = 500 ⋅ [ X 2 − Y2 ] b * = 200 ⋅ [Y2 − Z 2 ] 2 2 * LCh: C * = a * + b* h 0 = arctan(b ) a* Firtha – Physics1 summary -8-